Table of Contents
See also: 1. The chronological list.
2. All scripts of the companion videos.

Mathematical Proofs

1. Proof by inspection for finitely applicable statements.
2. Only a negative deserves a proof.
3. Proving by induction the truth of infinitely many things.
4. Stochastic proofs leave only a vanishing uncertainty.
5. Heuristic arguments state the likelihood of a conjecture.
6. Computer-aided proofs: 4-color theorem (1976).
7. Lack of a good approach doesn't invalidate a statement.
8. The most interesting theorems are general tools for proving other theorems.
9. Treacherous patterns. What holds for a long time may not hold forever.
10. Fermat's last theorem: One example of a botched proof.

    Measurements and Units

11. All metric prefixes: Current SI prefixes, obsolete prefixes, bogus prefixes...
12. Prefixes for units of information (multiples of the bit). Brontobyte hoax.
13. Density one. Relative and absolute density precisely defined.
14. Acids yielding a mole of H⁺ per liter are normal (1N) solutions.
15. Calories: Thermochemical calorie, gram-calorie, IST calorie (and Btu).
16. Horsepowers: hp, metric horsepower and boiler horsepower.
17. The standard acceleration of gravity has been 9.80665 m/s² since 1901.
18. Reciprocal time. Frequency (beat) and rate of phase change vs. activity.
    Time:
19. Tiny durations; zeptosecond (zs, 10^-21s) & yoctosecond (ys, 10^-24s).
20. A jiffy is either a light-cm or 10 ms (tempons and chronons are shorter).
21. A shake (10 ns) is much shorter than two shakes of a lamb's tail.
22. The length of a second. Solar time, ephemeris time, atomic time.
23. The length of a day. Solar day, atomic day, sidereal or Galilean day.
24. Scientific year = 31557600 SI seconds (» Julian year of 365.25 solar days).
    Length:
25. The International inch (1959) is 999998/1000000 of a US Survey inch.
26. The typographer's point is exactly 0.013837" = 0.3514598 mm.
27. How far is a league? Land league, nautical league.
28. Radius of the Earth and circumference at the Equator.
    • Distance between two points at the surface of the Earth.
    • The figure of the Earth. Geodetic and geocentric latitudes.
29. Extreme units of length. The very large and the very small.
    Surface Area:
30. Acres, furlongs, chains and square inches...
    Volume, Capacity:
31. Capitalization of units. You only have a choice for the liter (or litre ).
32. Drops or minims: Winchester, Imperial or metric. Teaspoons and ounces.
33. Fluid ounces: US ounces (fl oz) are about 4% larger than British ounces.
34. Gallons galore: Winchester (US) vs. Imperial gallon (UK), dry gallon, etc.
35. US bushel and Winchester units of capacity (dry = bushel, fluid = gallon).
36. Kegs and barrels: A keg of beer is half a barrel, but not just any "barrel".
    Mass, "Weight":
37. Tiny units of mass. A hydrogen atom is about 1.66 yg.
38. Solar mass: The unit of mass in the astronomical system of units.
39. Technical units of mass. The slug and the hyl.
40. Customary units of mass which survive in the electronic age.
41. British weights: Avoirdupois pound & ounce (lb & oz). Troy ounce (ozt).
42. The poids de marc system: 18827.15 French grains to the kilogram.
43. A talent was the mass of a cubic foot of water.
44. Tons: Short ton, long ton (displacement ton), metric ton (tonne), assay ton...
    • Other tons: Energy (kiloton, toe, tce), cooling power, thrust, speed.

    The Art of Rounding Numbers:

45. Scientific notation: Nonzero numbers given as multiples of powers of ten.
46. So many "significant" digits imply a result of limited precision.
47. Meaning of inequalities when rounded bounds are involved.
48. Standard deviation specifies the uncertainty in the precision of a result.
49. Engineering notation reduces a number to a multiple of a power of 1000.
50. The quadratic formula is numerically inadequate in common cases.
51. Devising robust formulas which feature a stable floating-point precision.
52. Underflow: 0.0 stands for a small number not known to be exactly zero.

    Scales and Ratings: Measuring without Units

53. The Beaufort scale is now defined strictly in terms of wind speed.
54. The Saffir-Simpson Hurricane Scale (SSHS) became a wind scale in 2010.
55. The Fujita scale for tornadoes.
56. Decibels: A general-purpose logarithmic scale for relative power ratios.
57. Sound intensity level (SIL) is well-defined; SPL is an approximation.
58. Pitch. An octave is 1200 cents.
59. Apparent and absolute magnitudes of stars.
60. Acidity. The pH scale was invented by Søren Sørensen in 1909.
61. The organoleptic Scoville scale is used to rate the pungency of hot spices.
62. Oven temperatures. Cooks may use gas marks or traditional descriptions.
63. The Richter scale for earthquakes and other sudden energy releases.
64. Volcanic Explosivity Index (VEI) Chris Newhall and Stephen Self (1982).
65. Pencil hardness. Clay makes harder graphite leads (lighter swatching).

    Scaling and Scale Invariance

66. The scale of animals according to Galileo Galilei.
67. Jumping fleas... compared to jumping athletes...
68. Kleiber's law: Metabolic rates of animals as a function of their sizes.
69. Drag coefficient of a sphere as a function of the Reynolds number R.
70. Benford's Law. Leading digits of quantities expressed in random units.

    Model Trains: Scales, gauges & other mathematical aspects.

71. Lexicon of railroad jargon: North-American, British and French usage.
72. A short history of the scales of model trains...
73. Gauges of worldwide lines. 60% are standard gauge (56.5'' = 1.4351 m).
74. Gauges of model tracks. HO and OO models can share the same tracks.
75. Types of model trains. Naming usual combinations of a scale and a gauge.
76. Large model trains are often used outdoors. Some are ridable.
77. Gauges are almost in geometric progression: Z,N,TT, HO, S,O,1,3...
78. The height of a miniature rail, in thousands of an inch, is called its code.
79. Composition of modern miniature rails: Nickel-silver contains no silver.
80. Travelage is the number of crossties per unit of distance.
81. Luminous power of sources at scale 1/s should be 1/s² that of prototypes.
82. Scenery scale. 81 is the geometric mean between 87 (HO) and 76 (OO).
83. Mirrors and shadows. Overlooked aspects of scenery in confined spaces.
84. Couplers.
85. Loading gauge.
86. Radius of curved track available from different manufacturers.
87. Least separation between curved parallel tracks, for specific rolling stock.
88. Turnouts. The frog number is the cotangent of the frog angle.
89. Sectional tracks. The need for compensator and/or correctors.
90. DC control. Traditional control of model trains, by Direct current.
91. DCC. Digital command control of several locomotives on the same track.
92. Sound. Some DCC locomotives feature on-board sound effects.
93. Blocks. A layout can be divided into blocks powered separately.
94. Detection & transponding. Locomotives located by the power they use.
95. OpenLCB (NMRAnet). Local control bus = Control area network (CAN).
96. Famous trains and locomotives and their miniature counterparts.
97. Dream lines. Legendary railroad services, past and present.
98. A quick survey of a few grand layouts. Great modeling achievements.

    Numerical Constants: Mathematical & Physical Constants

    Physical Constants:
99. For the utmost in precision, physical constants are derived in a certain order.
100. Primary conversion factors between customary systems of units.
     6+1 Basic Dimensionful Physical Constants (Proleptic SI)
101. Speed of Light in a Vacuum (Einstein's Constant): c = 299792458 m/s.
102. Magnetic Permeability of the Vacuum: An exact value defining the ampere.
103. Planck's constant: The ratio of a photon's energy to its frequency.
104. Boltzmann's constant: Relating temperature to energy.
105. Avogadro's number: The number of things per mole of stuff.
106. Mechanical equivalent of light (683 lm/W @ 540 THz) defines the lumen.
107. Newton's constant of gravitation and a futuristic definition of the second.
     Dimensionless Physical Constants:
108. Galileo's constant. What Galileo measured is now known to be 8 / p²
109. Sommerfeld's fine-structure constant a = 1 / 137.036 (or so).
110. The large number W. A dimensionless number pondered by Dirac.
     Fundamental Mathematical Constants:
111. 0: Zero is the most fundamental and most misunderstood of all numbers.
112. 1 and -1: The unit numbers.
113. p ("Pi"): The ratio of the circumference of a circle to its diameter.
114. ࡞: The diagonal of a square of unit side. Pythagoras' Constant.
115. ࡟: Diameter of a cube of unit side. Constant of Theodorus.
116. f: The diagonal of a regular pentagon of unit side. The Golden Number.
117. Euler's e: Base of the exponential function which equals its own derivative.
118. 63.2% (1-1/e) of a sudden level shift is achieved after one time constant.
119. ln(2): The alternating sum of the reciprocals of the integers.
120. An engineering favorite: The decimal logarithm of 2.
121. Euler-Mascheroni Constant g : Limit of [1 + 1/2 + 1/3 +...+ 1/n] - ln(n).
122. Catalan's Constant G : The alternating sum of the reciprocal odd squares.
123. Apéry's Constant z(3) : The sum of the reciprocals of the perfect cubes.
124. Imaginary i: If "+1" is a step forward, "+ i" is a step sideways to the left.
     Exotic Mathematical Constants:
125. Delian constant: 2^1/3 is the solution to the duplication of the cube.
126. Gauss's constant: Reciprocal of the arithmetic-geometric mean of 1 and Ö2.
127. Rayleigh factor for the diffraction limit of angular resolution.
128. Mertens constant: The limit of [1/2 + 1/3 + 1/5 +...+ 1/p] - ln(ln p)
129. Artins's constant is the proportion of long primes in decimal or binary.
130. Ramanujan-Soldner constant (m): Positive root of the logarithmic integral.
131. Landau-Ramanujan constant. Asymptotic density of sums of two squares.
132. The Omega constant: W(1) is the solution of the equation x exp(x) = 1.
133. Feigenbaum constant (d) and the related reduction parameter (a).
134. Fransén-Robinson constant: Inverse-Gamma integral.
135. Landau's constant.
136. Bloch's constant.
     Some Third-Tier Mathematical Constants:
137. Gelfond's Constant raised to the power of i is -1.
138. Brun's Constant: A standard uncertainty (s) means a 99% level of ±3s
139. Prévost's Constant: The sum of the reciprocals of the Fibonacci numbers.
140. Grossman's Constant: One recurrence converges for only one initial point.
141. Ramanujan's Number: exp(p Ö163) is almost an integer.
142. Viswanath's Constant: Mean growth in random additions and subtractions.
143. Copeland-Erdös Number: Almost all numbers are normal, like this one.

     Counting, Combinatorics, Probability

144. Always change your first guess if you're always told another choice is bad.
145. The Three Prisoner Problem predated Monty Hall and Marilyn by decades.
146. Seating N children at a round table in (N-1)! different ways.
147. How many Bachet squares? A 1624 puzzle using the 16 court cards.
148. Choice Numbers: C(n,p) is the number of ways to choose p items among n.
149. Multichoice Numbers: Putting n objects into distinct boxes of fixed sizes.
150. C(n+p-1,p) distinct ways to put p identical balls into n numbered bins.
151. C(n+2,3) three-scoop sundaes. Several ways to count them (n flavors).
152. C(n+p-1,p) choices of p items among n different types, allowing duplicates.
153. How many new intersections of straight lines defined by n random points.
154. Catalan numbers are the results of many different enumerations.
155. Face cards. The probability of getting a pair of face cards is less than 5%.
156. Homework Central: Aces in 4 piles, bad ICs, airline overbooking.
157. Binomial distribution. Defective units in a sample of 200.
158. Siblings with the same birthday. What are the odds in a family of 5?
159. Covariance: A generic example helps illustrate the concept.
160. Variance of a binomial distribution, derived from general principles.
161. Standard deviation. Two standard formulas to estimate it.
162. Markov's inequality is used to prove the Bienaymé-Chebyshev inequality.
163. Bienaymé-Chebyshev inequality: Valid for any probability distribution.
164. Inclusion-Exclusion: One way to find the probability of a union of 3 events.
165. The "odds in favor" of poker hands: A popular way to express probabilities.
166. Probabilities of a straight flush in 7-card stud (generalized to "q-card stud").
167. Probabilities of a straight flush among 26 cards (or any other number).
168. The exact probabilities in 5-card, 6-card, 7-card, 8-card and 9-card stud.
169. Rearrangements of CONSTANTINOPLE so no two vowels are adjacent...
170. Four-letter words from POSSESSES: Counting with generating functions.
171. How many positive integers below 1000000 have their digits add up to 19?
172. Polynacci Numbers: Flipping a coin n times without p tails in a row.
173. Winning in finitely many flips or losing endlessly...
174. 252 decreasing sequences of 5 digits (2002 nonincreasing ones).
175. How many ways are there to make change for a dollar? Closed formulas.
176. Partitioniong an amount into the parts minted in a certain currency.
177. The number of rectangles in an N by N chessboard-type grid.
178. The number of squares in an N by N grid: 0, 1, 5, 14, 30, 55, 91, 140...
179. Screaming Circles: How many tries until there's no eye contact?
180. Average distance between two random points on a segment, a disk, a cube...
181. Average distance between two points on the surface of a sphere.

     Randomness & Randomizing

182. Normal sequences feature any string with equal probability.
183. Fair coin generated from the output of a Geiger counter.
184. Random integer below n generated from a fair coin.
185. Burning cards may defeat cheating but doesn't affect random outcomes.
186. Shuffling cards to give every permutation the same probability.
187. Legally cheating at online poker by taking advantage of poor programming.
188. Probability distributions: Measurability.

     Bayesian Statistics

189. Bayes' Theorem. A formula which applies to classical probabilities.
190. The Bayesian universe: Probabilities quantify beliefs.
191. Aggregation paradox: Confusion factors and meaningless correlations.
192. Raw product ratings. How do they compare to each other?
193. Quantum theory is not Bayesian.
194. The human brain is not a Bayesian engine but there's still hope...

     Diamonds Hearts Spades Clubs Playing Cards

195. Short history of playing cards: From China to Europe, to the New World.
196. Sizes of playing cards: French, Bridge, Poker, French tarot, Patience, etc.
197. How playing cards are made: Either 2 layers of paper or 100% plastic.
198. Suits: Spades, hearts, diamonds & clubs (swords, cups, coins & wands).
199. The four court cards: Ace, king, queen, jack (king, queen, cavalier, page).
200. The Mameluke 52-card standard deck with 3 figure cards per suit.
201. 78-card tarot deck: 21 trumps, 1 fool, 4 suits of 14 (incl. 4 court cards).
202. The Major Arcana: Trumps and fool of the tarot deck, in occult parlance.
203. Names of the court cards in the French tradition. Hundred Years' War.
204. 48-card Aluette deck: Latin suits, mimicks and names of special cards.
205. Jokers from Euchre (1857) found their way into Poker in the 1870's.
206. The 40-card Spanish baraja deck lacks 8, 9 & 10.
207. The 32-card piquet deck lacks 2-6. French Belote and German Skat.
208. Skat: The most popular German card game (32-card deck).
209. 24-card deck for Euchre (single deck) and Pinochle (double deck).
210. Happy Families: 44-card British deck of 11 families of 4 (1851).
211. Jeu des 7 familles: 42-card French deck of 7 families of 6 (1876).
212. 1000 Bornes: 106 cards for a boardless car-racing game (1954).
213. Set® cards: Combinatorics of a modern 81-card ternary deck (1974).
214. New-deck order for the whole 81-card SET deck (or the 27 solids only).
215. Zener cards were invented in the early 1930s for (deprecated) ESP tests.

     The Card Games Played in Casinos (Banked Games)

216. Gaming chips: Color coding, shapes & sizes, designs. Jetons & plaques.
217. Casino edge: Gambling beyond the cost of entertainement is foolish.
218. Faro was the most popular banking game from 1825 to 1915.
219. Baccarat (Punto-Banco).
220. EZ Baccarat(™). The original Dragon-7 and newer Panda-8 side bets.
221. Carribean Stud.
222. Three-Card Poker.

     The Game of Blackjack (Twenty-One)

223. Glossary: A few specialized term used in blackjack.
224. Casino rules for Blackjack.
225. Basic strategy against an infinite shoe.
226. Pair of aces (soft 12). What to do if you're not allowed to split it?
227. Blackjack enumerations using polynomials.
228. History of Blackjack counting.

     Poker 101 : Rules, Odds & Glossary

229. 5-card draw: The simplest form of poker is the basis for video poker.
230. The 2598960 poker hands come in 9 or 10 types, rarest ones first.
231. Kickers may help break ties between hands bearing the same name.
232. Perfect Poker: "Deuces or better" have 1 to 1 odds with the full-wheel rule.
233. Poker chips: Color, size and weight.
234. Poker chip sets: Practical repartitions into various denominations.
235. Structure sheet of the most exclusive invitation-only tournament.
236. Handling chips: Counting them, stacking them, betting with them.
237. Poker chip tricks.
238. 7-card stud was the most popular variant of poker before NLHE and PLO.
239. 7-card combos: Odds of best 5-card hands extracted from 7 random cards.
240. Betting rules: Antes & blinds, checking, opening, calling and raising.
241. Glossary: The jargon of poker.

     Poker 102 : No-Limit Hold 'em & Pot-Limit Omaha

242. Texas hold 'em: Two hole cards (hand) and five community cards (board).
243. Preflop probabilities (win or tie a showdown) for all 169 starting hands.
244. If you have kings in an m-handed game, how often do you run into aces?
245. Trips or quads from a lone board pair.
246. With 3 clubs on the board: If a player has a flush, does someone else?
247. Nontransitivity of matches: Pairwise showdowns can be inconsistent.
248. How much to bet depends on the goal (make money or avoid elimination).
249. Omaha Hold 'em: Use 2 hole cards (out of 4) and 3 board cards (out of 5).

     Stochastic Processes & Stochastic Models

250. Poisson Processes: Random arrivals happening at a constant rate (in Bq).
251. Simulating a poisson process with a uniform random number generator.
252. Markov Processes: When only the present influences the future...
253. The Erlang B Formula assumes callers don't try again after a busy signal.
254. Markov-Modulated Poisson Processes may look like Poisson processes.

     "Utility" and Decision Analysis

255. The Utility Function: A dollar earned is usually worth less than a dollar lost.
256. St. Petersburg's Paradox: What would you pay to play the Petersburg game?
257. Two-envelope problem. Don't misuse random variables !

     Fair Division

258. Sperner's lemma. A discrete version of Brouwer's fixed-point theorem.
259. Sharing a necklace with N types of beads, in just N cuts.

     Social Choice Theory

260. One man, one vote! That's conclusive only when there are just two options.
261. The case for a rigorous approach, when the majority rule doesn't suffice.
262. Condorcet's Paradox: A group of rational voters need not behave rationally.
263. Tallying systems based on additive preference points are misguided at best.
264. Plurality voting is just another point-system, possibly the worst of them.
265. Runoff systems: Ad hoc improvements on plurality voting.
266. The Borda method, the foremost point system, is finally being phased out.
267. Cardinal voting could usefully supplement ordinary ordinal voting.
268. Conform Condorcet methods always produce the right choice, if it's clear.
269. Llull's method turns any preliminary order into a conform voting system.
270. Tallies: Summaries of voting data that allow the outcome to be computed.
271. Manipulations: Taking advantage of the weaknesses of a voting system.
272. A composite system aggregating preference lists and discretionary points.
273. Stability of a slate and stabilization of slate primaries..
274. Incumbents play a key rôle in any political election.
275. Aggregating lists of preferences from voters, when that's all we have...
276. Voter transition matrix from a primary to a general election.
277. Copeland scores provide an appealing conform Condorcet tally.
278. The second-order Copeland tallying method has some theoretical virtues.
279. The Schulze method uses a transitive relation based on voter wishes.
280. Apportioning whole numbers in lieu of fractional values, for a given total.
281. The Big Picture: What an elected government should and should not do.

     Elementary Euclidean Geometry

282. Center of an arc determined with straightedge and compass.
283. Surface areas: Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
284. Simson line of a circumcircle point contains its projections along the sides.
285. Isogonal conjugates Concurrent bissector reflections of concurrent lines.
286. Euler's line goes through the orthocenter, the centroid and the circumcenter.
287. Euler's circle is tangent to the incircle and the excircles (Feuerbach, 1822).
288. Power of a point with respect to a circle (Jakob Steiner, 1826).
289. Ptolemy's theorem relates the diagonals and the sides of a cyclic tetragon.
290. Barycentric coordinates & trilinears exemplify homogeneous coordinates.
291. Elliptic arc: Length of the arc of an ellipse between two points.
292. Perimeter of an ellipse. Exact formulas and simple ones.
     • Circumference of an ellipse: Unabridged discussion.
293. Surface of an ellipse.
294. Volume of an ellipsoid (either a spheroid or a scalene ellipsoid).
295. Surface area of a spheroid (oblate or prolate ellipsoid of revolution).
     • Surface area of a general ellipsoid.
296. Quadratic equations in the plane describe ellipses, parabolas, or hyperbolas.
297. Centroid of a circular segment. Find it with Guldin's (Pappus) theorem.
298. Parabolic arc of given extremities with a prescribed apex between them.
299. Focal point of a parabola. y = x ² / 4f (where f is the focal distance).
300. Parabolic telescope: The path from infinity to focus is constant.
301. Make a cube go through a hole in a smaller cube. Prince Rupert's paradox.
302. Octagon: The relation between side and diameter.
303. Constructible regular polygons and constructible angles (Gauss).
304. Areas of regular polygons of unit side: General formula & special cases.
305. For a regular polygon of given perimeter, the more sides the larger the area.
306. Curves of constant width: Reuleaux Triangle and generalizations.
307. Irregular curves of constant width. With or without any circular arcs.
308. Solids of constant width. The three-dimensional case.
309. Constant width in higher dimensions.
310. Fourth dimension. Difficult to visualize, but easy to consider.
311. Volume of a hypersphere and hyper-surface area, in any dimensionality.
312. Hexahedra. The cube is not the only polyhedron with 6 faces.
     • Unabridged discussion.
313. Descartes-Euler Formula: F-E+V=2 but restrictions apply.
314. Symmetries of the plane. Another approach to Euclidean axioms.

     Triangles

315. Napoleon's theorem: Three equilateral triangles around a scalene triangle...
316. Morley's theorem: The 3 intersections of adjacent trisectors are equidistant.
317. 6-point Conway circle. Every vertex prolongated by the opposite distance.

     Projective Geometry

318. Polarity: A geometric duality due to Apollonius of Perga (c. 200 BC).
319. The invention of perspective by Renaissance artists.
320. Projective spaces: Projective line, projective plane, etc.
321. Homogeneous coordinates. Feuerbach, Möbius, Plücker (1827 & 1828).
322. Projective duality: Points are incident to lines. Lines are incident to points.
323. Pappus' theorem.
324. Pascal's theorem was proven by Blaise Pascal when he was 16.
325. Brianchon's theorem: The dual of Pascal's theorem.
326. Desargues' theorem (c. 1639).
327. The two cyclic points (I and J) of Jean-Victor Poncelet (1820).
328. Fano plane PG(2,2). Projective geometry in dimension 2 and order 2.

     Topology

329. Metric spaces: The motivation behind more general topological spaces.
330. Abstract topological spaces are defined by calling some subsets open.
331. Limits in abstract topological spaces.
332. Nets (generalized sequences). Moore-Smith sequences (1922).
333. Basis of a topology: A set is open iff it's a union of sets from the base.
334. Closed sets are sets (of a topological space) whose complements are open.
335. Subspace F of E: Its open sets are the intersections with F of open sets of E.
336. Separation axioms: Flavors of topological spaces, according to Trennung.
337. Compactness of a topological space: Any open cover has a finite subcover.
338. Paracompact space: Any open cover has a locally-finite open refinement.
339. Locally compact sets contain a compact neighborhood of every point.
340. Extreme-value theorem: The continuous image of a compact is compact.
341. Borel sets. Tribes form the topological foundation for measure theory.
342. General properties of sequences characterize topological properties.
343. Continuous functions let the inverse image of any open set be open.
344. Homeomorphic spaces. An homeomorphism is a bicontinuous function.
345. Hilbert curve. A continuous function from ]0,1[ to the square ]0,1[×]0,1[.
346. Restrictions remain continuous. Continuous extensions may be impossible.
347. The product topology makes projections continuous on a cartesian product.
     • Tychonoff's Theorem: Any product of compact spaces is compact.
348. Connected sets can't be split by open sets. The empty set is connected.
349. Intermediate-value theorem.
350. Path-connected sets are a special case of connected sets.
351. Arc-connected spaces are path-connected. The converse need not be true.
352. Homotopy: A progressive transformation of a function into another.
353. The fundamental group: The homotopy classes of all loops through a point.
354. Homology and Cohomology. Poincaré duality.
355. Descartes-Euler Formula: F-E+V = 2, but restrictions apply.
356. Euler Characteristic: c (chi) extended beyond its traditional definition...
357. Winding number of a continuous planar curve about an outside point.
     • The dog on a leash lemma. (Rouché's theorem, 1862.)
     • A topological proof of the fundamental theorem of algebra.
358. Fixed-point theorems by Brouwer, Shauder and Tychonoff.
359. Turning number of a planar curve with a well-defined oriented tangent.
360. Real projective plane and Boy's surface.
361. Classification Theorem for connected closed surfaces.
362. Hadwiger's additive continuous functions of d-dimensional rigid bodies.
363. Eversion of the sphere. An homotopy can turn a sphere inside out.
364. Classification of surfaces: "Zero Irrelevancy Proof" (ZIP) by J.H. Conway.
365. Braid groups: Strands, braids and pure braids.
366. Zariski topology (algebraic geometry).  Only algebraic subsets are closed.
367. Fiber bundles and fibrations. Locally homeomorphic to a direct product.

     Completeness:

368. Complete metric space: A space in which all Cauchy sequences converge.
369. Continuity and uniform continuity.
370. Uniformity is sufficient to define completeness (or lack thereof).
371. Flawed alternatives to completeness.
372. Banach spaces are complete normed vector spaces.
373. Fréchet spaces are generalized Banach spaces.

     Fractal Geometry:

374. Fractional exponents were first conceived by Nicole d'Oresme (c. 1360).
375. The von Koch curve (and snowflake): Dimension of self-similar objects.
376. The Julia set and the Fatou set of an analytic function are complementary.
377. The Mandelbrot set was so named by Adrien Douady & John H. Hubbard.
378. Hausdorff dimension is revealed by a covering with balls of radius < e.

     Angles and Solid Angles:

379. Planar angles (from one direction to another) are signed quantities.
380. Bearing: Unless otherwise specified, this is the angle west of north.
381. Solid angles are to spherical patches what planar angles are to circular arcs.
382. Circular measures: Angles and solid angles aren't quite dimensionless...
383. Solid angle formed by a trihedron : Van Oosterom & Strackee (1983).
384. Solid angle subtended by a rhombus. Apex of a right rhombic pyramid.
385. Formulas for solid angles subtended by patches with simple shapes.
386. Right ascension and declination. Precession of celestial coordinates (a,d).

     Curvature and Torsion:

387. Curvature of a planar curve: Variation of inclination with distance dj/ds.
388. Curvature and torsion of a three-dimensional curve.
389. Distinct curvatures and geodesic torsion of a curve drawn on a surface.
390. The two fundamental quadratic forms at a point of a parametrized surface.
391. Lines of curvature: Their normal curvature is extremal at every point.
392. Geodesic lines. Least length is achieved with zero geodesic curvature.
393. Meusnier's theorem: Tangent lines have the same normal curvature.
394. Mean curvature at a point: Half-sum of the two principal curvatures.
395. Gaussian curvature of a surface. The Gauss-Bonnet theorem.
396. Parallel-transport of a vector around a loop. Holonomic angle of a loop.
397. Total curvature of a curve. The Fary-Milnor theorem for knotted curves.
398. Linearly independent components of the Riemann curvature tensor.

     Planar Curves:

399. Cartesian equation of a straight line passing through two given points.
400. Confocal Conics: Ellipses and hyperbolae sharing the same pair of foci.
401. Spiral of Archimedes: Paper on a roll, or groove on a vinyl record.
402. Hyperbolic spiral: The inverse of the Archimedean spiral.
403. Catenary: The shape of a thin chain under its own weight.
404. Tractrix: Meridian of Beltrami's pseudosphere. Involute of the catenary.
405. Witch of Agnesi. How the versiera (Agnesi's cubic) got a weird name.
406. Folium of Descartes.
407. Lemniscate of Bernoulli: A quartic curve shaped like the infinity symbol.
408. Cassini oval: The product of the distances to the two foci is constant.
409. Limaçons of Pascal: The cardioid (unit epicycloid) is a special case.
410. Cartesian oval: The weighted average distance to two poles is constant.
411. The envelope of a family of curves is everywhere tangent to one of them.
412. The evolute of a curve is the locus of its centers of curvature.
413. Involute of a curve: Trajectory of a point of a line rolling on that curve.
414. Parallel curves share their normals, along which their distance is constant.
415. The nephroid (or two-cusped epicycloid ) is a catacaustic of a circle.
416. Freeth's nephroid: A special strophoid of a circle.
417. Bézier curves are algebraic splines. The cubic type is the most popular.
418. Piecewise circular curves: The traditional way to specify curved forms.
419. Intrinsic equation [curvature as a function of arc length] may have spikes.
420. The quadratrix (trisectrix) of Hippias squares the circle and trisects angles.
421. The parabola is constructible with straightedge and compass.
422. Mohr-Mascheroni constructions use the compass alone (no straightedge).
423. Osculating curves. Beautiful extensions of the Tait-Kneser theorem.

     Surfaces in Three Dimensions:

424. Cartesian equation of a plane, knowing its closest point from the origin.
425. Helicoid. Surface of minimal area first studied by Meusnier in 1776.
426. Ruled surfaces: Cone, conic hyperboloid, helicoid, etc.
427. Developable surfaces.
428. Surfaces of revolution: Parallel and meridians are lines of curvature.
429. Guldin's theorems: Surface area and volume of a solid of revolution.
430. Euler's catenoid: Surface of revolution of least surface area.
431. Surfaces of constant mean curvature (CMC). The shapes of soap film.
432. Beltrami's pseudosphere: Surface of revolution with constant curvature.
433. Quadric surfaces = quadratic equations (degree n-1 in n dimensions).
434. Perfect monkey saddle: A minimal surface with ternary axial symmetry.
435. Convex combinations of two surfaces.
436. Dual pair of surfaces.
437. Intrinsic equation of a surface.

     Gears:

438. Glossary of terms related to gears.
439. Gear ratio: Ratio of the input rotation to the output rotation (signed).
440. Planar curves rolling without slipping while rotating about two fixed points.
441. Congruent ellipses roll against each other while revolving around their foci.
442. Elliptic gears: A family of gears which include ellipses and sine curves.
443. Watchmaker gears: Ogival surfaces for pinions & radial planes for wheels.
444. La Hire's theorem : An hypocycloid of ratio 2 is a straight line.
445. Cycloidal gear: Epicycloidal addendum curve & hypocycloidal dedendum.
446. The law of conjugate action was formulated by Leonhard Euler (c. 1754).
447. Involute tooth profiles provide a uniform rotational speed ratio.
448. Harmonic Drive: A flexspline with 2 fewer teeth than the circular spline.
449. Circular arc helical gears: E. Wildhaber (1923) & M.L. Novikov (1956).
450. Double circular arc helical gears were standardized by the Chinese in 1981.

     Horology:

451. Sidereal time: 46879 sidereal days = 46751 mean solar days.
452. Daniels coaxial escapement: A major horological innovation.

     Polyhedra (3D), Polychora (4D), Polytopes (nD)

453. Hexahedra. The cube is not the only polyhedron with 6 faces.
454. Fat tetragonal antiwedge: Chiral hexahedron of unit volume and least area.
455. Duality: To a face of a polyhedron corresponds a vertex of its dual.
456. Enumeration of polyhedra: Convex polyhedra with n faces and k edges.
     • Counting Polyhedra (1): Table, comments, references...
     • Counting Polyhedra (2): Larger table!
457. The 5 Platonic solids: Cartesian coordinates of the vertices.
458. Symmetries may equate all commensurate components of a polyhedron.
459. Equimetric polyhedra feature constant measures for all elements of a kind.
460. There are 75 or 76 nonprismatic uniform polyhedra (18 of them convex).
461. The 13 Archimedean solids and their duals (Catalan solids).
462. Every isogonal family is typified by a uniform polyhedron.
463. Polyhedra in certain families are named after one prominent polygon.
464. Some special polyhedra may have a traditional (mnemonic) name.
465. Deltahedra have equilateral triangular faces. Only 8 deltahedra are convex.
466. Johnson Polyhedra and the associated nomenclature.
467. Polytopes are the n-dimensional counterparts of 3-D polyhedra.
468. A simplex of touching unit spheres may allow a center sphere to bulge out.
469. Regular Antiprism: Height and volume of a regular n-gonal antiprism.
470. The Szilassi polyhedron features 7 pairwise adjacent hexagonal faces.
471. Wooden buckyball: Cutting 32 blocks to make a truncated icosahedron.
472. Zonogons, zonohedra, zonotopes. Zones and zonoids.
473. Plesiohedra are space-filling: Cuboctahedron, truncated octahedron, etc.
474. Pyritohedron: One dodecahedral pentagonal isohedron is space-filling.

     Dice

475. 16 possible standard dice (opposite faces add up to 7). Two handedness.
476. 30 labelings of a die: For 16 of them, opposite faces never add up to 7.
477. 3-sided spindle: A 9-hedron with 6 unstable faces.
478. Nontransitive dice: Every die is dominated by another die from the set.
479. Sicherman dice yield any total with the same probability as a regular pair.
480. Percentile dice and other ways several dice can have equiprobable sums.
481. Polyhedral dice were popularized by rôle playing games.
482. Commonly available dice sizes: Small, medium, large, jumbo and giant.
483. Convex isohedra are fair dice, by reason of symmetry between faces.
484. Round dice: Outer isohedral marks & steel ball in an isogonal cavity.
485. Scalene isogonal polyhedra. Their duals are scalene isohedra.
486. Juryeonggu: Korean die with 8 hexagonal faces and 6 square ones.
487. Fairness of a non-isohedral die may depend on the way it's tossed.
488. Are there any intrinsically fair dice which aren't isohedral?
489. Necessary conditions an absolutely fair die must satisfy.
490. Quasistatic probability is proportional to the solid angle a face subtends.
491. Thermal tossing puts a face of minimal height at the bottom.
492. Balanced mesohedral dice are fair for both quasistatic and thermal tossing.
493. Mesodecahedron: Mesohedron with 10 faces.
494. Mesopentahedron: The mesohedral proportions of a rhombic pyramid.
495. Mesoheptahedron: Mesohedron with 7 faces.
496. Statistical bias of unfair dice.

     Isohedra. The symmetry of fair dice.

497. Classification of all convex isohedra. Intrinsic fair dice.
498. Disphenoids are tetrahedra where opposing edges have equal length.
499. The hexakis icosahedra (120 faces) include the disdyakis triacontahedra.
500. Two chiral fair dice: No mirror symmetry, 24 or 60 pentagonal faces.
501. A pseudo-isohedral die. Its faces are congruent, but is it fair?

     Graph Theory

502. The bridges of Königsberg: Eulerian graphs and the birth of graph theory.
503. Undirected graphs are digraphs with symmetrical adjacency matrix.
504. Adjacency matrix of a directed graph (digraph) or of a bipartite graph.
505. The 3-utilities problem: Providing 3 cottages with water, gas & electricity.
506. Silent Circles: An enumeration based on adjacency matrices (Alekseyev).
507. Silent Prisms: Modifying the screaming game for short-sighted people.
508. Tallying markings of one edge per node where no edge is marked twice.
509. Line graph: Nodes of L(G) are edges of G (connected iff adjacent in G).
510. Transitivity: Vertex-transitive and/or edge-transitive graphs.
511. Desargues graph and distance transitivity.
512. Tensor product of two graphs (directed or not).
513. Hedetniemi's conjecture (1966). Disproved by Yaroslav Shitov, in 2019.
514. The brain as a graph. Blue-Brain project (2005-) of Henry Markam.
515. Heterosexuality: Male promiscuity is the same as female promiscuity!

     Algebra

516. Factorial zero is 1, so is an empty product. An empty sum is 0.
517. Anything to the power of 0 is equal to 1, including 0 to the power of 0.
518. Zero is divisible by anything but it divides only itself!
519. Idiot's Guide to Complex Numbers.
520. Using the Golden Ratio (f) to express the 5 [complex] fifth roots of unity.
521. "Multivalued" functions are functions defined over a Riemann surface.
522. Square roots are inherently ambiguous for negative or complex numbers.
523. The difference of two numbers, given their sum and their product.
524. All symmetric polynomials of 3 variables are determined by the first three.
525. Geometric progression of 6 terms. Sum is 14, sum of squares is 133.
526. Quartic equation involved in the classic "Ladders in an Alley" problem.

     Special Polynomials

527. Chebyshev polynomials express cos nq as a function of cos q
528. Chebyshev polynomials of the second kind obey the same recurrence...
529. Legendre polynomials and zonal harmonics.
530. Laguerre polynomials. Hypergeometric confluent function.
531. Hermite polynomials. Eigenstates of the quantum harmonic oscillator.
532. Bessel polynomials.
533. Bernoulli polynomials.
534. Faulhaber polynomials.
535. Euler polynomials.
536. Mittag-Leffler polynomials.
537. Cyclotomic polynomials are irreducible over the rationals.
538. Lucas coefficients form polynomials dividing cyclotomic polynomials.
539. The Art of Polynomial Factorizations. Manufacturing remarkable identities.

     Matrices and Determinants

540. Permutation matrices include the identity matrix and the exchange matrix.
541. Gil's matrix. The favorite square matrix of professor W. Gilbert Strang.
542. Operations on matrices are conveniently defined using Dirac's notation.
543. Triangular matrices. Proper isomorphism between upper and lower ones.
544. The determinant is proportional to any completely antisymmetrical form.
545. Minors. First minors are obtained by deleting one row and one column.
546. Adjugate of a matrix: Transpose of its cofactor matrix A adj(A) = det(A) I
547. Eigenvectors and eigenvalues of an operator or a matrix.
548. The numerical range of a complex matrix. Locus of its Rayleigh ratios.
549. The characteristic polynomial of an operator doesn't depend on the basis.
550. The minimal polynomial has the same zeroes but no multiple roots.
551. Cayley-Hamilton theorem: A matrix vanishes its characteristic polynomial.
552. Normal matrices and diagonalizable matrices.
553. Totally positive generalized Vandermonde matrices (fractional powers).
554. Cholesky decomposition L L* of an Hermitian matrix.
555. Toeplitz matrix: Constant diagonals.
556. Circulant matrix: Cyclic permutations of the first row.
557. Linear recurrences of order k-1 which define sequences of period k.
558. Wendt's Determinant: The circulant of the binomial coefficients.
559. Hankel matrix: Constant skew-diagonals.
560. Catberg matrix: Hankel matrix of the reciprocal of Catalan numbers.
561. Hadamard matrix: Unit elements and orthogonal columns.
562. Sylvester matrix of two polynomials has their resultant for determinant.
563. The discriminant of a polynomial is the resultant of itself and its derivative.
564. Pfaffian: Polynomial whose square equals an antisymmetric determinant.
565. The numerical range of a complex matrix. Locus of its Rayleigh ratios.
566. Matrices with coefficients in a noncommutative ring.
567. Matrix exponential. The exponential of a matrix is an invertible matrix.
568. Matrix gamma function. The reciprocal gamma is an entire function.
569. Matrix Mittag-Leffler functions. Applications to fractional calculus.
570. Values of analytic functions for nilpotent matrices reduce to finite sums.

     Trigonometry

571. Trigonometric functions: Memorize a simple picture for 3 basic definitions.
572. Solving triangles with the law of sines, law of cosines & law of tangents.
573. Spherical trigonometry: Triangles drawn on the surface of a sphere.
574. Sum of tangents of two half angles, in terms of sums of sines and cosines.
575. The absolute value of the sine of a complex number.
576. All positive rationals (& square roots) as trigonometric functions of zero!

     Elementary Functions and Special Functions

577. Function types: Polynomial, rational, algebraic, transcendental, special.
578. Exact solutions to transcendental equations.
579. The sine function: How to compute it numerically.
580. Chebyshev economization saves billions of steps in common computations.
581. The Gamma function: Its definitions, properties and special values.
582. Lambert's W function is used to solve practical transcendental equations.
583. Dilogarithm, trilogarithm and polylogarithms. Jonquière's function.
584. Deformed exponential function. Entire function with a complex parameter.

     Gamma Function

585. The Gamma function: Its definitions, properties and special values.
586. Euler's integral of the second kind can define G(z) when Re(z) > 0.
587. Euler's reflection formula. G(z) G(1-z) = p / sin pz .
588. Stirling's approximation: An asymptotic expansion for factorials.
589. Stirling's expansion is a divergent asymptotic series.
590. Hölder's theorem: G doesn't satisfy any algebraic differential equation.
591. Kümmer's series and the integral representation of Log G (x).
592. Knar's formula.
593. Euler's beta function (Euler, 1730). Euler integral of the first kind. B(x,y).
594. Digamma (Gauss' psi-function): Logarithmic derivative of G.

     Hypergeometric Functions

595. Pochhammer's symbol: Upper factorial of k increasing factors: x(x+1)...
596. Gauss's hypergeometric function: 2+1 parameters (and one variable).
597. Kummer's transformations relate different hypergeometric expressions.
598. Sum of the reciprocal of Catalan numbers, in closed hypergeometric form.
599. Appell series. Two-variable generalizations.
600. WZ pairs. Zeilberger's algorithm. Gosper's algorithm.

     Theta Functions & Elliptic Functions

601. Adrien-Marie Legendre (1643-1727) Prototypical theta function.
602. Legendre's elliptic integrals.
603. Carlsons's canonical symmetric elliptic integrals.
604. Jacobi theta functions:
605. q-analogs.
606. Ramanujan theta function.
607. False theta functions, (L.J. Rogers, 1894, 1917).
608. Mock theta functions were introduced by S. Ramanujan, in 1920.
609. Partial theta functions.

     Perimeter of an Ellipse

610. Circumference of an ellipse: Exact series and approximate formulas.
611. Ramanujan I and Lindner formulas: The journey begins...
612. Ramanujan II: An awesome approximation from a mathematical genius.
613. Hudson's Formula and other Padé approximations.
614. Peano's Formula: The sum of two approximations with cancelling errors.
615. The YNOT formula (Maertens, 2000. Tasdelen, 1959).
616. Euler's formula is the first step in an exact expansion.
617. Naive formula: p ( a + b ) features a -21.5% error for elongated ellipses.
618. Cantrell's Formula: A modern attempt with an overall accuracy of 83 ppm.
619. From Kepler to Muir. Lower bounds and other approximations.
620. Relative error cancellations in symmetrical approximative formulas.
621. Complementary convergences of two series. A nice foolproof algorithm.
622. Elliptic integrals & elliptic functions. Traditional symbols vs. computerese.
623. Padé approximants are used in a whole family of approximations...
624. Improving Ramanujan II over the whole range of eccentricities.
625. The Arctangent Function as a component of several approximate formulas.
626. Abed's formula uses a parametric exponent to fine-tune the approximation.
627. Zafary's formula. Improved looks for a brainchild of Shahram Zafary.
628. Rivera's formula gives the perimeter of an ellipse with 104 ppm accuracy.
629. Better accuracy from Cantrell, building on his own previous formula
630. Rediscovering a well-known exact expansion due to Euler (1773).
631. Exact expressions for the circumference of an ellipse: A summary.

     Surface Area of an Ellipsoid

632. Surface Area of a Scalene Ellipsoid: The general formula isn't elementary.
633. Thomsen's Formula: A simple symmetrical approximation.
634. Approximate formulas for the surface area of a scalene ellipsoid.
635. Nautical mile: "Average" minute of latitude on an oblate spheroid.
636. Great ellipses have the same center as the ellipsoid they are drawn on.
637. Area enclosed by a curve drawn on the surface of an oblate spheroid.
638. Pseudo-straight boundaries of areas varying quadratically with longitude.

     Calculus

639. Derivative: The slope of a function and/or something more abstract.
640. The logarithmic derivative of a product is the sum of those of its factors.
641. Integration: The Fundamental Theorem of Calculus.
642. Integration by parts: Reducing an integral to another one.
643. Wallis' integrals. Integration by parts yields a recurrence relation.
644. Length of a parabolic arc.
645. Top height of a curved bridge with a 5280 ft span and a 5281 ft length.
646. Sagging: A cable which spans 28 m and sags 30 cm is 28.00857 m long.
647. The length of the arch of a cycloid is 4 times the diameter of the wheel.
648. Integrating the cube root of the tangent function.
649. Changing inclination for a particle moving along a parabola.
650. Algebraic area of a figure 8 may be the sum or the difference of its lobes.
651. Area surrounded by an oriented planar loop which may intersect itself.
652. Linear differential equations of higher order and/or in several variables.
653. Theory of Distributions: Convolution products and their usage.
654. Laplace Transforms: The Operational Calculus of Oliver Heaviside.
655. Integrability of a function and of its absolute value.
656. Analytic functions of a linear operator. Defining f (D) when D is d/dx.
657. Generalizing the Cauchy-Schlömilch substitution (for definite integrals).
658. Feynman's trick: Put a parameter in the integrand and differentiate along it.
659. Malmsten integrals: Tough family involving Log Log x (Malmsten, 1842).

     Differential Equations

660. Ordinary differential equations. Several examples.
661. A singular change of variable may not be valid over a maximal domain.
662. Vertical fall against fluid resistance (valid for viscous and quadratic drag).
663. Jet propulsion: Expelling stuff at speed u makes (u-v) m remain constant.
664. Riccati equation: When y' is a quadratic function of y...
665. Sturm-Liouville problem has nontrivial solutions only for some parameters.
666. Frobenius method to solve a second-order differential equation.
667. Fuchsian conditions for the applicability of the method of Frobenius.

     Differential Forms & Vector Calculus

668. Differential forms and partial derivatives.
669. Generalizing the fundamental theorem of calculus.
670. Vectorial surface dotted into an observing direction gives apparent area.
671. Practical identities of vector calculus.

     Optimization: Operations Research, Calculus of Variations

672. Stationary points (saddlepoints) are where all partial derivatives vanish.
673. Classical optimizations: Heron's problem, Dido's problem, etc.
674. Single-variable optimization: Derivative vanishes at any interior extremum.
675. The angle maximization problem of Regiomontanus. Historical case study.
676. Extrema of a function of two variables obey a second-order inequality.
677. Saddlepoints of a multivariate function. One equation for each variable.
678. Lagrange multipliers: Constrained optimization of an objective function.
679. Minimizing the lateral surface area of a cone of given base and volume.
680. Euler-Lagrange equations hold along the path of a stationary integral.
681. Noether's theorem: One conserved quantity for each Lagrangian symmetry.
682. Geodesics in a 2-dimensional surface are curves of least length.
683. The brachistochrone curve is a cycloid (in a uniform gravitational field).
684. Dido's problem. A geometric solution to the oldest variational problem.
685. Isoperimetric Inequality: The largest area enclosed by a loop of unit length.
686. Plateau extended the calculus of variations from paths to membranes.
687. Embedded minimal surfaces: Plane, catenoid, helicoid, Costa's surface, etc.
688. Connecting blue dots to red dots in the plane, without any crossings...
689. Torricelli points and the shortest way to connect three guven dots.
690. The Honeycomb Theorem: A conjecture of old, proved by Thomas Hales.
691. Symmetrical monkey saddles of zero mean curvature.
692. Counterexamples to Kelvin's conjecture. Unit spatial tiles of least area.
693. J.J. Thomson's problem: N repelling charges on a spherical surface.
694. Monge-Kantorovich optimal transport problem.

     Analysis, Convergence, Series, Complex Analysis

695. Cauchy sequences help define real numbers rigorously.
696. Borderline convergence. Prototypes of series which are barely convergent.
697. A simple proof of convergence. Part of the mathematical folklore.
698. Permuting the terms of a series may change its sum arbitrarily.
699. Two decreasing divergent series may have a convergent minimum!
700. Uniform convergence of continuous functions makes the limit continuous. Augustin Cauchy (1789-1857) Joseph Fourier (1768-1830)
701. Defining integrals: Cauchy, Riemann, Darboux...
702. Lebesgue integrals. Horizontal slices, not vertical ones !
703. Cauchy principal value of an integral.
704. Fourier series. A simple example.
705. Infinite sums evaluated with Fourier series.
706. A double sum is often the product of two sums (possibly Fourier series).
707. At a jump, a Fourier series is the half-sum of its left and right limits.
708. Gibbs phenomenon; 9% overshoot of partial Fourier series near a jump.
709. Method of Frobenius about a regular singularity of a differential equation.
710. Laurent series of a function about one of its poles.
711. Cauchy's Residue Theorem is helpful to compute difficult definite integrals.
712. Tame complex functions: Holomorphic and meromorphic functions.
713. Cauchy-Riemann equations. Also known as d'Alembert-Euler conditions.
714. Wirtinger derivatives. Calculus methods for several complex variables.
715. Cauchy integral formula. Fundamental theorem of complex analysis.
716. Argument principle. The logarithmic integral counts zeros ans poles.
717. Rouché's theorem: Nearby holomorphic functions have nearby zeros.
718. Riemann mapping theorem and biholomorphic functions.
719. Univalent functions (analytic injections) have nonvanishing derivatives.
720. Schlicht functions are normalized analytic injections from the unit disk.
721. Starlike schlicht functions & Bieberbach's conjecture restricted to these.
722. Maximum-modulus principle: No local maxima for holomorphic functions.

     Limits

723. Epsilon-Delta. Elementary concept of a limit in a metric space.
724. A function sandwiched between two functions of limit L has limit L.
725. L'Hospital's rule on the limit of the ratio of two vanishing quantities.

     Power Series & Analytic Continuations

726. Taylor's expansion of a differentiable function as a power series.
727. The remainder of a Taylor expansion can be expressed in several ways.
728. Lagrange used Taylor's expansion to redefine Calculus algebraically.
729. Radius of convergence. The convergence disk of a complex power series.
730. Stolz sector: Slice of the disk of convergence with its apex on the boundary.
731. Composition of two (formal) power series.
732. Lagrange inversion formula and Lagrange-Bürmann inversion formula.
733. The exponential series. Proving that exp (x) exp (y) = exp (x+y)
734. Analytic continuation: Power series that converge on overlapping disks.
735. Decimated power series are equal to finite sums involving roots of unity.
736. Splitting a power series into its k classes of indices modulo k.
737. Calculus of finite differences. Analog of calculus for discrete sequences.

     Asymptotic Analysis

738. Fundamentals of asymptotics. The simple definition of a powerful tool.
739. The big O notation is named after Paul Bachman and Edmund Landau.
740. Solving aymptotic equations. Method of dominant balance.
741. Asymptotic expansions about a limit point may or may not converge.
742. Moments, Stieltjes functions and Stieltjes series.
743. Extracting information from an asymptotic series.
744. Stirling's approximation and Stirling's series.
745. Hyper-asymptotics: Aymptotics beyond all orders.
746. Stokes phenomenon: Wedge where an asymptotic approximation is valid.
747. WKB method: Asymptotic order-reduction of a linear differential equation.

     Summation of Divergent Series

748. Summing geometric series: Equating things that match over some domain.
749. Definitions and notations. Distinguishing a formal series from its sum.
750. Desirable properties of summation methods yield rules for handling them.
751. Stability of geometric series is often just assumed.
752. All series with vanishing terms are stable, under any linear summation.
753. Cauchy product of two series. It's a stable series if at least one factor is.
754. Dirichlet convolution: Is the sum of a product the product of the sums?
755. Infinite products: Exponentials of infinite series.
756. Double summation. Series whose terms are themselves sums of series.
757. Original Bernoulli numbers. Generated by z / (1-exp(-z)). So, B1 = ½.
758. Formal series are kets. Linear summation methods are bras.
759. Summation by convergence is compatible with regular summations.
760. Functional analysis: Using the Hahn-Banach extension theorem.
761. Euler summation (1746).
762. Cesàro summations (1890).
763. Borel summation (1899).
764. Nørlund summations All are linear, regular and consistent (1919).
765. Abel summation .
766. Lindelöf summation (1903).
767. Mittag-Leffler summation method (1908).
768. Valiron summability (1917).
769. Generalized summation methods, using arbitrary convergence factors.
770. When does a series have a stable sum? Stability theorems.
771. Weierstrass summation: Summation by analytic continuation (1842).
772. Resummation with Meijer G-functions.
773. Zeta-function regularization (1916). Dubbed heat-kernel regularization.
774. Wonders of unstable summations. Linear summations of unstable series.
775. Decimated power series can be worked out for divergent series too.
776. Stretching a series doesn't change its sum, except when it does...
777. Invariants of a series and the series of the sums of its decimations.
778. Divergent Fourier series. An example of a stretched divergent series.
779. Parametrized differential equations and their solutions as asymptotic series.
780. The Mercator series is the integral of the geometric series (1668).
781. Decimating the harmonic series, by taming a multivalued continuation.
782. Sum of the harmonic series: ln 2p = 1.83787706640934548356065947...
783. Darboux's summation formula generalizes the Euler-Maclaurin formula.
784. Ramanujan's irregular summation (1913).
785. Summations of p-adic integers for a special radix or for all of them.
786. Moments, Stieltjes functions and Stieltjes series.
787. Shanks' transformation greatly accelerates an alternating convergence.
788. Richardson extrapolation.
789. Index-free acceleration of sequences with harmonic convergence.
790. Parametrized acceleration based on the expected type of convergence.
791. Padé approximant. Simplest rational function of given Taylor expansion.
792. Compensated derivatives. In any ring or power-associative algebra.

     Infinite Products and Pole Expansions.

793. Wallis product (1655) due to John Wallis (1616-1703).
794. Euler's product formula for sine, using the normalized sinc function.
795. Factorization of the reciprocal Gamma function by Karl Weierstrass.
796. Weierstrass factorization theorem (1876).
797. Euler's partial-fraction expansion of the cotangent function.
798. Mittag-Leffler's theorem (1876).
799. Well-known pole expansions of a few meromorphic functions.

     Harmonic Analysis & Fourier Expansions

800. Euler's formulas extract a harmonic component from a trigonometric sum.
801. Fourier expansion of a function: The sum of all its harmonic components.
802. Dirichlet's conditions are sufficient for a Fourier expansion to converge.
803. Divergent Fourier expansion of the tangent function.
804. Cantor's proofs of unicity and the introduction of U-sets.
805. Simples rectangular waves. From periodic pulses to square waves.
806. Sawtooth functions can be obtained by integration of rectangular waves.
807. Clipped sinusoids. The harmonic distortion introduced by hard clipping.

     Fourier Transform & Tempered Distributions

808. The direct product of two functions is a function of two variables.
809. Convolution as an inner operation among numerical functions. Joseph Fourier (1768-1830)
810. Duality: The product of a bra by a ket is a (complex) scalar.
811. A distribution associates a scalar to every test function.
812. Schwartz functions are smooth rapidly decreasing test functions.
813. Tempered distributions are continuous functionals over Schwartz functions.
814. The Fourier Transform associates a tempered distribution to another.
815. Competing definitions of the Fourier transform. For the record.
816. Parseval's theorem (1799). The Fourier transform is unitary.
817. Noteworthy distributions and their Fourier transforms: Antoine Parseval (1755-1836)
     • Dirac's d and the uniform distribution ( f (x) = 1).
     • The signum function sgn(x) and its transform: i / ps
     • The Heaviside step function H(x) = ½ (1+sgn(x)) and its transform.
     • The square function P(x) = H(x+½)-H(x-½) and sinc ( ps )
     • The triangle function L(x) and sinc² ( ps )
818. The normalized Gaussian distribution is its own Fourier transform.
819. Central Limit Theorem (CLT).
820. Far image of a picture on translucent film is its Fourier transform.
821. Sampling formula: The unit comb (Shah function) is its own Fourier transform.
822. Crystallography using X-ray diffraction (Max von Laue, Nobel 1914).
823. Spectrum: Support of the Fourier transform.
824. Quasicrystals: Distributions with a discrete support and a discrete spectrum.
825. The Radon transform (used in lateral tomography) is easily inverted.

     Discrete Fourier Transforms & Fast Fourier Transform

826. Discrete Fourier Transform, defined as a unitary involution.
827. Discrete Cosine Transform. JPEG lossy compression.

     Set Theory and Logic

828. The logic of Aristotle. Syllogisms about categorical propositions.
829. The Barber's Dilemma is not a paradox, if analyzed properly.
830. What is infinity? There's more to it than a pretty symbol (¥).
831. Peano's axioms provide a rigorous definition of the set of natural integers.
832. Ordered numbers: From integers to rational, real and surreal numbers.
833. There are more real than rational numbers. Cantor's diagonal argument.
834. Cantor's ternary set. A vanishing set of reals equipollent to the whole line.
835. The language of set theory: Symbols and idioms useful to anybody.
836. The axioms of set theory: Basic ZF axioms (without any choice principle).
837. The Axiom of choice (AC) is the most powerful choice principle (ZFC).
838. The Axiom of restriction isn't needed in ZFC set theory (AC implies it).
839. Equivalents and lesser alternatives to the Axiom of choice.
840. The existence of nonmeasurable sets is guaranteed by the Axiom of choice.
841. Should we drop the Axiom of choice ?
842. Binary cartesian products. Multiple cartesian products. Infinite ones.
843. Binary relations between two sets are subsets of their cartesian product.
844. Functions and applications are special types of binary relations.
845. A set is smaller than its powerset: A simple proof applies to all sets.
846. Transfinite cardinals describe the various sizes of infinite sets.
847. The continuum hypothesis: Is the continuum the smallest uncountable set?
848. Transfinite ordinals: Counting to infinity... and beyond.
849. Hartogs number of a set (well-ordered or not). A well-defined ordinal.
850. Surreal numbers include reals, transfinite ordinals, infinitesimals & more.
851. Multidimensional (hypercomplex) numbers: To octonions and beyond.
852. Naming hypercomplex numbers beyond sedenions.
853. Power associativity: What multiplication needs to allow exponentiation.
854. A set belongs to a class in NBG (a conservative extension of ZFC).
855. Tarski-Grothendieck theory (TG) is a nonconservative extension of ZFC.

     Category Theory

856. Categories: History, motivation, definition. Objects & morphisms.
857. Initial objects and final objects. A zero object is both initial and final.
858. Finite categories.
859. Set : Category of sets and functions.
860. Rel : Category of sets and relations.
861. Example of categories: Large or small, abstract or concrete.
862. Constructing new categories from pre-existing ones.
863. Functors are homomorphisms between categories.
864. Cat : Category of small categories and functors.
865. Natural transformations are morphisms in a category of functors (1942).
866. Duality. The opposite of a category is obtained by reversing all arrows.
867. Isomorphisms in a category are reversible morphisms (arrows) in it.
868. Groupoid: Small category in which all morphisms are reversible.
869. The product of two objects, if defined, is an equivalence class of objects.
870. Exponential of two objects. Counterpart of function spaces for sets.
871. Cartesian-closed categories (CCC). Well-defined products & exponentials.
872. Yoneda lemma and Yoneda embedding.
873. Adjoint functors.
874. Monads.
875. Abelian categories.
876. Regular categories.
877. Exact categories.
878. Exact sequences of morphisms. The image of one is the kernel of the next.
879. Homologies and cohomologies. How sequences fail to be exact.
880. Should categories replace sets at the foundation of all mathematics?
881. Categorial quantum mechanics. Describing physics with category theory.

     Topos (Category Theory)

882. Presheaves and sheaves.
883. Bohr topos.
884. Elementary topos (Lawvere).
885. Grothendieck topos.
886. Kochen-Specker theorem. Categorial expression of quantum theory.

     Alain Connes' Noncommutative Geometry

887. Von Neumann's algebras.
888. Connes-Consani plane connection.

     Integer Arithmetic, Number Theory

889. The number 1 is not prime. Good definitions allow simple theorems.
890. Composite numbers aren't prime, but the converse need not be true...
891. Two prime numbers whose sum is equal to their product.
892. Gaussian integers: Factoring into primes on a two-dimensional grid.
893. The least common multiple, obtained without factoring into primes.
894. Standard factorizations: n⁴ + 4 is never prime for n > 1 because...
895. High-order Aurifeuillian factorizations using polynomial identities.
896. Euclid's algorithm gives the GCD and the related Bézout coefficients.
897. Bézout's Lemma: The GCD of p and q is of the form u p + v q.
898. Greatest Common Divisor (GCD) defined for all commensurable numbers.
899. Linear equation in integers can be solved using Bézout's lemma.
900. Pythagorean Triples: Right triangles whose sides are coprime integers.
901. The number of divisors of an integer.
902. Perfect squares are the only integers with an odd number of divisors.
903. The product of all divisors is often a perfect square.
904. Even perfect numbers are well-known. Are there any odd ones out there?
905. Mersenne primes are the odd parts of the even perfect numbers.
906. Multiperfect & hemiperfect numbers. Whole or half-integral abundancies.
907. Fast exponentiation by repeated squaring.
908. Partition function. How many collections of positive integers add up to 15?
909. A Lucas sequence whose oscillations never carry it back to -1.
910. A bit sequence with intriguing statistics. Counting squares between cubes.
911. Binet's formulas: N-th term of a sequence obeying a linear recurrence.
912. The square of a Fibonacci number is almost the product of its neighbors.
913. D'Ocagne's identity relates conjugates products of Fibonacci numbers.
914. Catalans's identity generalizes Cassini's Identity.
915. Faulhaber's formula gives the sum of the p-th powers of the first n integers.
916. Multiplicative functions: If a and b are coprime, then f (ab) = f (a) f (b).
917. Moebius function: Getting N values with O(N Log(Log N)) additions.
918. Dirichlet convolution is especially interesting for multiplicative functions.
919. Fractional Dirichlet powers of arithmetic functions with positive lead.
920. Dirichlet powers of the Möbius function and/or its inverse u = 1,1,1,1,1...
921. Convolutive subgroup generated by u = 1,1,1,1,1... and N = 1,2,3,4,5...
922. Dirichlet powers of multiplicative functions are given by a superb formula.
923. Totally multiplicative functions are the simplest multiplicative functions.
924. Dirichlet characters are important totally multiplicative functions.
925. Euler products for L-functions. Generalized Riemann hypothesis.
926. Logarithms of L-functions. Sums of series indexed by the prime powers.
927. Expressing L-functions in term of Hurwitz zeta functions and vice-versa.
928. Additive functions: If a and b are coprime, then f (ab) = f (a) + f (b).

     Cryptography: Ciphers and Codebreaking

929. Simple shift ciphers: Caesar's cipher, Augustus cipher. ROT13.
930. Substitution cipher.
931. Tranposition ciphers. Tri-code.
932. Disk ciphers.
933. The Vigenère cipher can be broken with the Babbage-Kasisky method.
934. Bazeries cylinders consist of many stacked coding disks (Jefferson, 1795).
935. Rotor machines were invented in 1915 and still used in the 1980s.
936. The German Enigma. Broken by Marian Rejewski and Alan Turing.
937. Backdoor in elliptic curve cryptography. NSA surveillance controversy.
938. The Voynich manuscript (c. 1420): The book that nobody could read.

     Modularity: Elliptic curves and modular forms

939. Fermat's Last Theorem (FLT) is a consequence of the modularity theorem.
940. Elliptic functions are doubly-periodic functions of a complex variable.
941. Elliptic curves: Nonsingular cubics in the projective plane.
942. Modular forms.
943. Modularity theorem: Every elliptic curve is modular.
944. Ramanujan's Tau Function and the Sato-Tate conjecture (proved in 2011).

     Positional Numeration & Number Systems

945. Modular arithmetic can tell the last digits of extremely large numbers.
946. Leading digits of insanely large numbers can't be found by logarithms.
947. Powers of ten expressed as products of two factors without zero digits.
948. Divisibility by 7, 13, and 91 (or by B²-B+1 in base B).
949. Lucky 7's. Any integer divides a number composed of only 7's and 0's.
950. The decimal representation of rational numbers is ultimately periodic.
951. Midy's theorem: Properties of periods in radix-B numeration.
952. Numbers with two decimal expansions. E.g., 1 and 0.99999999999999...
953. Binary and/or hexadecimal numeration for floating-point numbers as well.
954. Extract a square root the old-fashioned way.
955. Ternary system: Is base 3 really the best radix for positional numeration?
956. Dozenal counting: (Roman) ounce, as, dozen, gross, great-gross.
957. Sexagesimal numeration is still with us. Cuneiform writing isn't.

     Prime Numbers

958. Divisors can be defined in any additive semigroup.
959. Relatively prime integers: The GCD of two coprime integers is 1.
960. A prime number is a positive integer with 2 distinct divisors (1 and itself).
961. Lemma: Every integer larger than 1 has at least one prime factor.
962. Fundamental theorem of arithmetic. The factorization into primes is unique.
963. Euclid's proof: There are infinitely many primes.
964. Dirichlet's theorem: There are infinitely many primes of the form kN+a.
965. Green-Tao theorem: Arbitrarily long arithmetic progressions of primes.
966. Von Mangoldt's function is Log p for a power of a prime p, 0 otherwise.
967. Prime Number Theorem: The probability that N is prime is roughly 1/ln(N).
968. Riemann's power-prime counting function (J). Riemann's explicit formula.
969. The average number of factors of a large number N is Log N.
970. The average number of distinct prime factors of N is Log Log N.
971. The largest known prime: Historical records, old and new.
972. The Lucas-Lehmer Test checks the primality of a Mersenne number fast.
973. Proth primes are almost as fast to check as Mersenne primes.
974. Sierpinski numbers are moduli for which no Proth number is prime.
975. Pratt primality certificates give quick automated proofs of primality.
976. The AKS test determines the primality of any number in polynomial time.
977. Formulas giving only primes may not help with new primes.
978. Ulam's Lucky Numbers and other sequences generated by sieves.

     Modular Arithmetic

979. Chinese remainder theorem: Remainders define an integer, within limits.
980. Modular arithmetic: The formal algebra of congruences, due to Gauss.
981. Fermat's little theorem: For a prime p not dividing a, a^p-1 is 1 modulo p.
982. Euler's totient function: f(n) counts the integers coprime to n, from 1 to n.
983. Fermat-Euler theorem: If a is coprime to n, a to the f(n) is 1 modulo n.
984. Carmichael's reduced totient function (l) : A special divisor of the totient.
985. 91 is a pseudoprime to half of the bases coprime to itself.
986. Carmichael Numbers: An absolute pseudoprime n divides aⁿ-a for any a.
987. Chernik's Carmichael numbers: 3 prime factors (6k+1)(12k+1)(18k+1).
988. Other products that yield a Carmichael number iff every factor is prime.
989. Large Carmichael numbers may be obtained in various ways.
990. Conjecture: Any odd integer coprime to its totient has Carmichael multiples.

     Group Theory and Symmetries

991. Monoids feature an associative operation and a neutral element.
992. The inverse of an element comes in 2 flavors that coincide when both exist.
993. Free monoid: All the finite strings (words) in a given alphabet.
994. Raising something to the power of an integer.
995. Groups are monoids in which every element is invertible.
996. A subgroup is a group contained in another group.
997. Ideals of a semigroup are subsemigroups.
998. Generators of a group are not contained in any proper subgroup.
999. Presentation of a group: A set of generators followed by relators.
1000. Lagrange's theorem: The order of a subgroup divides the order of the group.
1001. Cauchy's theorem: If a prime p divides |G|, some element of G has order p.
1002. Sylow's theorems. On the possible orders of subgroups of a finite group.
1003. Normal subgroups and their quotients in a group.
1004. Wielandt's symbol asserts that a set is a normal subgroup of a group.
1005. Homomorphism: The image of a product is the product of the images.
1006. The symmetric group on E consists of all the bijections of E onto itself.
1007. Inner automorphisms: Inn(G) is isomorphic to G modulo its center.
1008. Outer automorphism group: The automorphisms of G modulo Inn(G).
1009. Complete groups (unrelated to completeness in metric or uniform spaces).
1010. The conjugacy class formula uses conjugacy to tally elements of a group.
1011. Simple groups are groups without nontrivial normal subgroups.
1012. The derived subgroup of a group is generated by its commutators.
1013. Direct product of two groups (called a direct sum for additive groups).
1014. Finite abelian groups are either cyclic or direct sums of cyclic groups.
1015. Holomorph of G : The canonical semi-direct product of G and Aut(G).
1016. Groups of small orders. Basic families: Cyclic groups, dihedral groups, etc.
1017. Q8: The quaternion group. Quaternion numbers (Hamilton, 1843).
1018. The gamma group of order 32 is generated by Dirac's 4 gamma matrices.
1019. D4: An incomplete group of order 8, isomorphic to its automorphisms.
1020. Enumeration of small groups. How many groups of order n?
1021. Classification of finite simple groups, by Gorenstein and many others.
1022. Sporadic groups: Tits Group, 20 relatives of Fischer's Monster, 6 pariahs.
1023. Torsion of a group: The set of all elements whose orders are finite.
1024. Linear representations are homomorphisms into a group of matrices.
1025. Classical groups: Their elements depend on parameters from a field.
1026. Projective qualifier may denote a group modulo its own center.
1027. The Möbius group consists of homographic transformations of CÈ{¥}.
1028. The modular group G is the projective special linear group PSL(2,Z).
1029. Group structure of an elliptic curve.
1030. Degenerate elliptic curve consisting of a circle and a straight line.
1031. Amenable groups are locally compact topological groups allowing a mean.
1032. The Chameleon Groups (F, T and V) of Richard J. Thompson (1965).
1033. Abelian sandpile groups. The neutral element is not immediately obvious.
1034. Lorentz transformations may change spatial orientation or time direction.
1035. Symmetries of the laws of nature: A short primer.
1036. The renormalization group is a subgroup of the cosmic Galois group.

      Ring Theory: Commutative & Noncommutative Rings

1037. A Brief History of Rings.
1038. Rings are sets endowed with addition, subtraction and multiplication.
1039. Divisors of zero include all nilpotents. Zero-divisors (if any) are nonzero.
1040. Units are invertible elements. That's also meaningful in non-unital rings.
1041. Ideals within a ring are multiplicatively absorbent additive subgroups.
1042. Quotient ring, modulo an ideal: The residue classes modulo that ideal.
1043. Ring homomorphisms and isomorphisms. All kernels are ideals.
1044. Nonzero characteristic: Least n for which all sums of n like terms vanish.
1045. The SUN ring has 8 elements. Smallest unital noncommutative ring.
1046. Idempotent elements allow the Peirce decomposition of a ring.
1047. Finite rings. Smallest examples, enumeration and classification.
1048. Simple rings. A simple ring has only two ideals; {0} and itself.
1049. Coprime ideals and generalized Chinese remainder theorem.
1050. Bezout rings: Rings where the sum of two principal ideals is principal.
1051. Primary ideals are to ideals what prime-powers are to integers.
1052. Cauchy multiplication is well-defined for formal power series over a ring.
1053. Ring of univariate polynomials A[X] with coefficients in a given ring A.
1054. Noetherian rings don't have any infinite ascending chain of ideals (ACC).
1055. Artinian ring: Noetherian with every element either invertible or nilpotent.
1056. Lasker-Noether theorem. Generalized fundamental theorem of arithmetic.
1057. Greatest common divisor (GCD) always exists for a pair in a GCD domain.
1058. Factorial domain (UFD). GCD domain with ACC for principal ideals.
1059. Galois rings. Residues of modular polynomials, modulo one of them.
1060. Hilbert rings, where every prime ideal is an intersection of primitive ideals.
1061. Abelian rings are rings where every idempotent element is central.
1062. Commutative local rings and their modules: Local algebra (1938).
1063. Topological rings Both operators are continuous.
1064. Involutive rings are endowed with an involutive anti-automorphism.
1065. Etale homomorphisms of rings.
1066. Morita equivalence (on rings).

      Fields, Galois Fields and Skew Fields

1067. Vocabulary: We consider skew fields to be noncommutative. Some don't.
1068. Fields are commutative rings where all nonzero elements are invertible.
1069. Quotient field of a ring without divisors of zero.
1070. Wedderburn's Theorem: Finite division rings are necessarily commutative.
1071. Every finite integral domain is a field.
1072. Galois fields are the finite fields. Their orders are powers of primes.
1073. Polynomials in a Galois field. Irreducible and primitive polynomials.
1074. The trivial field is a singleton. It's the only field where 0 is invertible.
1075. Splitting field of P in F[x] : Smallest extension of F where P fully factors.
1076. Perfect fields include fields of characteristic zero and all finite fields.
1077. The field of Laurent series.
1078. Conway's Nim-Field is algebraically complete. It contains infinite ordinals.
1079. Ternary multiplication compatible with ternary addition (without "carry").

      Vector Spaces (over a field) and Algebras

1080. Vectors were originally just differences between points in ordinary space...
1081. Abstract vector spaces: Vectors can be added, subtracted and scaled.
1082. Dimension of a vector space: The number of its independent generators.
1083. Subspaces. Intersection. Sum. Direct sums of supplementary subspaces.
1084. Linear maps between vector spaces respect addition and scaling.
1085. Quotient of two vector spaces. Hyperplanes have codimension 1.
1086. Fundamental theorem of linear algebra and rank theorem.
1087. Normed vector spaces. The fundamental properties of a norm.
1088. Inner-product spaces over the field of complex numbers.
1089. Dual space: The set of all [continuous] linear functions with scalar values.
1090. Lebesgue spaces. Sequence spaces exemplify more general types.
1091. Tensors: Multilinear functions of vectors and covectors with scalar values.
1092. Graded linear spaces are direct sums of homogeneous spaces.
1093. Algebra: A vector space with a scalable and distributive internal product.
1094. Lie algebra: Anticommutative algebra obeying Jacobi's identity.
1095. Jordan algebra: Commutative and alternative algebras (Jordan, 1933).
1096. Albert algebra: Exceptional 27-dimensional Jordan-algebra (Albert, 1934).
1097. Clifford algebra: Unital associative algebra endowed with a quadratic form.
1098. Dual numbers with a second component squaring to zero (Clifford, 1873).
1099. Spacetime algebra: Cl(1,3) is the Clifford algebra with Minkowski metric.
1100. Things that are not vectorial because they're not defined intrinsically.
1101. David Hestenes proposed geometric calculus as a denotational unification.

      Modules (over a ring)

1102. Modules are vectorial structures over a ring of scalars (instead of a field).
1103. Free modules have a basis similar to that of vector spaces.
1104. Injective modules. The rationals form an injective module over the integers.
1105. Projective modules. Due to Eilenberg & Cartan (1956).
1106. Flat modules. Devised by Jean-Pierre Serre in 1956.

      Tensors and Tensor Calculus

1107. Definition of a tensor.
1108. Metric spaces: Linking contravariant and covariant representations.
1109. Gradient: Tensor of rank 1 in covariant form.

      Convex Geometry

1110. Convex sets in a real vector space.
1111. Real-valued convex functions of a vectorial variable.
1112. Uniformly convex spaces.
1113. Aspect ratio: The length divided by the height (i.e., the smallest width).
1114. A norm is characterized by a closed convex body, symmetric about 0.
1115. Convex hull: Conv (S) is the smallest convex set containing the set S.
1116. Closed halfspaces generate all closed convex sets by intersection.
1117. Polar of a closed convex set. Dot-product duality among convex bodies.
1118. Separating hyperplane (in the loose sense) between disjoint convex sets.
1119. A compact convex can be strictly separated from a disjoint closed convex.
1120. Two disjoint open convexes are separated by a nonintersecting hyperplane.

      Functional Analysis

1121. Functionals assign scalar values to some functions over an infinite set I.
1122. Topological vector spaces. Banach spaces are normed and complete.
1123. Eduard Helluy (1912): The space C[a,b] (continuous functions over [a,b]).
1124. Sublinear functionals are merely subadditive and positively homogeneous.
1125. Hahn-Banach extension theorem. Extending a dominated linear functional.
1126. Hahn-Banach separation theorem. A different view of the same result.
1127. Generalization of Hahn-Banach to complex or quaternionic linear spaces.
1128. The two Baire category theorems (1899).
1129. Uniform boundedness principle. The Banach-Steinhaus theorem.
1130. Open mapping theorem. The Banach-Schauder theorem.
1131. Weak convergence and weak* convergence.
1132. Banach-Alaoglu theorem (1940).
1133. Krein-Milman theorem: Retrieving a compact convex set from its extremes.
1134. Nuclear spaces cover practical cases besides Banach spaces.
1135. Nuclear operators.
1136. Compact operators.
1137. Schauder basis.

      Ring of p-adic Integers, Field of p-adic Numbers

1138. The ring of p-adic integers. Objects with infinitely many radix-p digits.
1139. Polyadic integers: Greek naming of p-adic integers.
1140. What if p isn't prime? Dealing with zero-divisors.
1141. Decadic integers: The strange realm of 10-adic integers (composite radix).
1142. Decadic puzzle: A tribute to the columnist J.A.H. Hunter (1902-1986).
1143. The field of p-adic numbers. Quotient field of the ring of p-adic integers.
1144. Dividing two p-adic numbers looks like long division, only backwards...
1145. Overbar notation, for p-adic and rational numbers alike.
1146. The p-adic metric can be used to define p-adic numbers analytically.
1147. The reciprocal of a p-adic number computed by successive approximations.
1148. Ratios of rational integers have two representations: g-adic and radix-g.
1149. Solving algebraic equations in p-adic integers.
1150. Q-linear maps between R and Q_p are discontinuous at every point.
1151. Hasse's local-global principle. Established for the quadratic case in 1920.
1152. Rotating digit patterns (in base g) may double the corresponding values.
1153. Rotating digits one place to the left divides some integers by k.

      Nim Arithmetic

1154. Ordinary fractions added with Nim rules.
1155. Multiplicatiion of ordinary fractions added with Nim rules.
1156. Nim square-root.

      Pseudoprimes

1157. Pseudoprimes to base a. Poulet numbers are pseudoprimes to base 2.
1158. Weak pseudoprimes to base a : Composite integers n dividing (aⁿ-a).
1159. Counting the bases to which a given composite number is a pseudoprime.
1160. Strong pseudoprimes to base a are less common than Euler pseudoprimes.
1161. The witnesses of a composite number: At least 75% of nontrivial bases.
1162. Rabin-Miller Test: An efficient and trustworthy stochastic primality test.
1163. The product of 3 primes is a pseudoprime when all pairwise products are.
1164. Super-pseudoprimes: All their composite divisors are pseudoprimes.
1165. Maximal super-pseudoprimes have no super-pseudoprime multiples.
1166. Wieferich primes are scarce but there ought to be infinitely many of them.

      Factoring into Primes

1167. Jevons Number. Factoring 8616460799 is now an easy task.
1168. Challenges help tell special-purpose and general-purpose methods apart.
1169. Special cases of a priori factorizations are helpful to number theorists.
1170. Trial division may be used to weed out the small prime factors of a number.
1171. Ruling out factors can speed up trial divison in special cases.
1172. Pocklington's lemma: Conditions imposing the form of unknown factors.
1173. Recursively-defined sequences (over a finite set) are ultimately periodic.
1174. Pollard's rho factoring method is based on ultimately periodic sequences.
1175. Pollard's p-1 Method finds prime factors p for which p-1 is smooth.
1176. Williams' p+1 Method is based on the properties of Lucas sequences.
1177. Lenstra's Elliptic Curve Method generalizes Pollard's p-1 approach.
1178. Dixon's method: Combine small square residues into a solution of x ² º y ²
1179. Shors's algorithm would work in polynomial time on a quantum computer.

      Quadratic Reciprocity

1180. Motivation: On the prime factors of some quadratic forms...
1181. Quadratic residues: Half of the nonzero residues modulo an odd prime p.
1182. Euler's criterion: A quadratic residue raised to the power of (p-1)/2 is 1.
1183. The Legendre symbol (a|p) extends to values of p besides odd primes.
1184. The law of quadratic reciprocity states a simple but surprising fact.
1185. Gauss' Lemma expresses a Legendre symbol as a product of many signs.
1186. Eisenstein's Lemma: A variation of Gauss's lemma allows a simpler proof.
1187. One of many proofs of the law of quadratic reciprocity.
1188. Artin's Reciprocity.

      Continued Fractions (and related topics)

1189. What is a continued fraction? Example: The expansion of p.
1190. The convergents of a number are its best rational approximations.
1191. Large partial quotients allow very precise approximations.
1192. Regular patterns in the continued fractions of some irrational numbers.
1193. In almost all cases, partial quotients are ≥ k with probability lg(1+1/k).
1194. Elementary operations on continued fractions.
1195. The Baire space: Continued fraction expansions of irrationals in [0,1].
1196. Expanding functions as continued fractions.
1197. Expanding functions as continued exponentials.
1198. Engel expansions of positive numbers are nondecreasing integer sequences.
1199. Pierce expansions of numbers from 0 to 1. Strictly increasing sequences.
1200. Continued fractions in the complex realm. Algorithm of Asmus L. Schmidt.

      Recreational Mathematics

1201. Counterfeit Coin: In 3 weighings, find an odd object among 12, 13 or 14.
      • Unabridged discussion.
1202. Counterfeit Penny Problem: Find an odd object in the fewest weighings.
      • Unabridged discussion.
1203. Seven-Eleven: Four prices with a sum and product both equal to 7.11.
1204. Equating a right angle and an obtuse angle, with a clever false proof.
1205. Choosing a raise: Trust common sense, beware of fallacious accounting.
1206. 3 men pay 30ドル for a 25ドル hotel room, the bellhop keeps 2ドル... Is 1ドル missing?
1207. Chameleons: A situation is unreachable because of an invariant quantity.
1208. Sam Loyd's 14-15 puzzle also involves an invariant quantity (and 2 orbits).
1209. Einstein's riddle: 5 distinct colors, nationalities, drinks, smokes and pets.
1210. Numbering n pages of a book takes this many digits (formula).
1211. The Ferry Boat Problem (by Sam Loyd): To be or not to be ingenious ?
1212. Hat overboard ! What's the speed of the river?
1213. All digits once and only once: 48 possible sums (or 22 products).
1214. 2-people bridge crossed by 4 people (U2). Four paces, one flashlight!
1215. Managing supplies to travel 6 days while carrying enough for only 4 days.
1216. Go south, east, north and you're back... not necessarily to the North Pole!
1217. Icosapolis: Put 1 to 20 in a 5 by 4 grid so neighbors differ by at least 4.
1218. Unusual mathematical boast for people born in 1806, 1892, or 1980.
1219. Puzzles for extra credit: From Chinese remainders to bookworms.
1220. Simple geometrical dissection: A proof of the Pythagorean theorem.
1221. Early bird saves time by walking to meet incoming chauffeur.
1222. Sharing a meal: A man has 2 loaves, the other has 3, a stranger has 5 coins.
1223. Fork in the road: Find the way to Heaven by asking only one question.

      ---

1224. Proverbial Numbers: Words commonly associated with some numbers.
1225. Riddles: The Riddle of the Sphinx and other classics, old and new.

      Trick Questions and Lateral Thinking

1226. Crossing the Panama Canal east to west to reach the Pacific Ocean.

      The Mathematical Games of Martin Gardner

1227. Martin Gardner (1914-2010) described himself as "strictly a journalist".
1228. Flexagons: Hexaflexagons were popularized by Martin Gardner in 1956.
1229. Polyominoes: The 12 pentominoes and other tiles invented by Sol Golomb.
1230. Soma: 7 nonconvex solids consisting of 3 or 4 cubes make a larger cube.
1231. Tessellations by convex pentagons. The contributions of Marjorie Rice.
1232. Kites and Darts. The aperiodic tilings of Roger Penrose.
1233. Ambigrams: Calligraphic spellings which change when rotated or flipped.
1234. The Game of Life. John Conway's endearing cellular automaton (1970).
1235. Rubik's Cube: Ernõ Rubik (1974) Singmaster (1979) Gardner (1981).
1236. It's impossible to tie a knot without letting go of the ends of the string.
1237. On the limited knowledge of Man. An Indian legend...

      Mathematical "Magic" Tricks

1238. 1089: Subtract a 3-digit number and its reverse, then...
1239. Multiples of Nine: A secret symbol is revealed.
1240. Casting Out Nines: A missing digit is revealed.
1241. Triple threat mind reading.
1242. Mass media mentalism by David Copperfield (1992).
1243. Grey Elephants in Denmark: Classroom mental magic.
1244. Fitch Cheney's 5-card trick: 4 cards tell the fifth one.
1245. Generalizing the 5-card trick and Devil's Poker...
      border
      border
      border
      Clubs Hearts Spades Diamonds
1246. Kruskal's Count.
1247. Paths to God.
1248. Stacked Deck.
1249. Enigma Card Trick.
1250. Magic Age Cards.
1251. Ternary Cards.
1252. Magical 21 (or 27).
1253. The Final 3 are the chosen cards.
1254. Boolean Magic.
1255. Perfect Faro Shuffles.
1256. Relocate the top card to any given position using at most 7 faro shuffles.
1257. Equal Numbers of Heads !
1258. Gilbreath principle: Predictability survives a riffle-shuffle (1958, 1966).
1259. Divination by counting. A self-working trick by Paul A. Lelikis (c. 1970).
1260. Last card: Tell the last card after being shown all the others.

      Illusions and Deceit

1261. Deceit and lying.
1262. Misdirection.
1263. Find the Lady.
1264. Cups and balls. One of the most ancient tricks.
1265. Chop cup. Invented by "Chop-Chop" Wheatley in 1954.
1266. Invisible Thread Reel (ITR) by James George (1992).
1267. Force and Reveal: A whole class of magic tricks.
1268. Bill Simon's Prophecy Move (1952).

      Mathematical Games (Strategies)

1269. Dots and Boxes: The "Boxer's Puzzle" position of Sam Loyd.
1270. The Game of Nim: Remove items from one of several rows. Don't play last.
1271. Sprague-Grundy numbers are defined for all positions in impartial games.
1272. Moore's Nim: Remove something from at most (b-1) rows. Play last.
1273. Normal Kayles: Knocking down a pin or two adjacent pins may split a row.
1274. Grundy's Game: Split a row into two unequal rows, if at all possible.
1275. Wythoff's Game: Take either from one heap or equally from both heaps.

      The Noble Game of Chess

1276. Origins of Chess: Chaturanga (India) & Shatranj (Persia). 7th century.
1277. Chess boards. Tables, boards and mats.
1278. Chess men. Styles and sizes of chess pieces. Weighted or magnetized.
1279. Wood and other materials for chessmen and chessboards.
1280. Time controls. Use of modern chess clocks for tournament-like play.
1281. Chess bag to hold full-sized chessmen, a clock and a rolled-up mat.
1282. Chess notation. Squares: a1 to h8. Pieces: K, Q, R, B and N.
1283. Openings: Structured list of common opening lines (names & links).
1284. Combinatorics of chess.
1285. Single-byte encoding of most half-moves in chess, with rare extra byte.
1286. Systems of play: Desired configurations of several pieces.
1287. Chess maxims: Guiding principles in the form of Proverbs.
1288. Textbook endgames: Foolproof recipes for some well-known endgames.
1289. Nalimov Tables: Perfect computerized analysis of endgame situations.
1290. Evaluation function: Estimating a quiescent position statically.
1291. Minimax search tree: The basic paradigm for analyzing two-player games.
1292. Alpha-beta pruning. In a minimax search, some alternatives can be ignored.
1293. Hash tables. How to avoid analyzing the same position more than once.
1294. Shortest chess games. Checkmates occurring during the opening moves.
1295. Mate in N moves.. The quintessential type of chess puzzles.
1296. Traxler's Counterattack. Analyzing a wild strategy against the fried-liver.
1297. Miniature games, including the Immortal Game of 23 moves.
1298. Classic traps: Fishing pole, etc.
1299. Elo ratings: Idealized probabilities and actual frequencies.
1300. Odds Chess: The old-fashioned handicapping system.
1301. Over-the-board chess titles from FIDE or national federations.
1302. Leading chess centers and famous chess venues, throughtout History.
1303. World champions. Historical dominance and formal championships.
1304. Glossary of common chess terms, classified by topic.

      The Ancient Game of Go

1305. A brief history of Go.
1306. The board (standard goban). 361 intersections of lines in a 19 by 19 grid.
1307. 361 stones. 181 black pieces and 180 white ones (usually lenticular).
1308. Go bowls (gosu). Japanese wooden bowls. Baskets from China.
      Rules of the Game of Go:
1309. Komi: Points received by White as compensation for starting second.
1310. Score = (surrounded territory) + (captured stones) + komi.
1311. Liberty: A chain of stones is captured when all its liberties vanish.
1312. Rules of ko: Japanese and Chinese rules differ in the rare case of triple ko.
1313. Playing Go on unusual boards. Big or small, rectangular or not... 5 by 5 goban
1314. Combinatorics of Go. Enumerating legal Go configurations.
1315. Nalimov tables. How Go has been solved for small grids.
      Playing Go. Concepts, Tactics and Strategy:
1316. Opening moves (fuseki). What to play first (and why).
1317. Mistakes to avoid. What not to do and why not to do it.
1318. Shapes: Some localized patterns which skilled players are all familiar with.
1319. Ladders and loose ladders. Repeatedly prevent increase in opponent's air.
1320. Nets. Loose limits which make captures inescapable. Elementary reading.
1321. Snapback. A sacrifice may allow a greater capture in return.
1322. Eyes and eye-shapes are the keys to life-and-death problems.
1323. Seki. Shared life. Configurations that would backfire if you touch them.
1324. Tesuji. Finding key moves.
1325. Joseki. The Go term for a standard sequence of moves worth memorizing.
1326. Framework (moyo) to consolidate and/or defend against invasions.
1327. Invasions. Challenging the opponent's claim to a territory.
1328. Endgame. Squeezing the most of the situation when the end is near.
      Go Players and the World of Go:
1329. Kyu (k) & dan (d). Pro dan or ping (p). Pecking order in Go.
1330. Competitions: Amateur and professional tournaments.
1331. Championships and champions: Past and present stars of the Go world.
1332. Machines that play Go. In March 2016, AlphaGo beat Lee Sedol, 9p.
1333. Go jargon: A short glossary of Go words indispensable in English.

      The Royal Game of Ur

1334. 4500-year history of the 20-square race game and its rosettes.
1335. Normal route and short lap: The two variants of the basic rules.
1336. Complex variants: Pieces are flipped for the second part of their routes.
1337. All variants ever played and then some. Classification and naming.
1338. Number of positions with n pieces per player.
1339. Traditional tetrahedral dice. Using 3 or 4 dice changes everything.
1340. Nalimov table for the single-lap Royal Game of Ur (7 pieces per player).
1341. Northwest Corner devised new rules for the Mesopotamian equipment.

      Ramsey Theory

1342. The pigeonhole principle: What's entailed by fewer holes than pigeons.
1343. Among 70 distinct integers between 1 and 200, two must differ by 4, 5 or 9.
1344. n+1 of the first 2n integers always include two which are coprime.
1345. Largest sets of small numbers with at most k pairwise coprime integers.
1346. Ramsey's Theorem: Monochromatic complete subgraphs of a large graph.
1347. Infinite alignment among infinitely many lattice points in the plane? Nope.
1348. Infinite alignment in a lattice sequence with bounded gaps? Almost...
1349. Large alignments in a lattice sequence with bounded gaps. Yeah!
1350. Van der Waerden's theorem: Long monochromatic arithmetic progressions.
1351. Happy-Ending problem: Unavoidable convex n-gons among m points.
1352. Unavoidable monochromatic Pythagorean triples with 2 colors in 1-7825.

      Computer Science: Computability and Undecidability.

1353. Finite-state automata. The simplest type of computing machines.
1354. Deterministic and nondeterministic machines.
1355. Pushdown automata (PDA) recognize context-free languages (CFL).
1356. Two-way deterministic pushdown automata simulated in linear time.
1357. Turing-machines run a finite program on an infinite read-write tape.
1358. Universal Turing machine executes a program stored on its data tape.
1359. Almost all decision problems are not computable. The sad truth.
1360. The halting problem. Fundamental limitation of computers.
1361. The Ackermann function is total computable but not primitive recursive.
1362. The busy-beaver function (Radó's Sigma function) isn't computable.
1363. Church-Turing Thesis: If it can be done, a Turing machine can do it.
1364. Turmites. Langton's Ant and other two-dimensional Turing machines.
1365. Cellular automata (1D or 2D) can be Turing-complete.

      Subreal Numbers: The field of computable numbers.

1366. Subreal numbers are computed by convergent two-tape Turing machines.
1367. Subreal numbers are countable because Turing machines are.
1368. The set of subreal numbers is not complete.
1369. Equality of subreal numbers isn't computable by any general procedure.

      Irrational Numbers.

1370. Rational numbers are quotients of two integers.
1371. Some irrational numbers: Surds. Logarithmic ratios of coprime integers.
1372. Constructible numbers can be constructed with straightedge and compass.
1373. Algebraic numbers are roots of polynomials with integer coefficients.
1374. Subreal numbers: Only countably many real numbers can be computed.
1375. Transcendental numbers. Almost all of them aren't even computable.
1376. Irrationality measure.
1377. Louville's (transcendental) numbers have infinite irrationality measure.
1378. Pickover's Flint Hills series. Governed by the irrationality measure of p.

      Algorithms

1379. Stable marriages: Two people shouldn't prefer each other to their spouses.
1380. Ford-Fulkerson algorithm. Best network flow from source to sink (1956).
1381. Heap priority queues allow sorting in O( n Log n ) worst time (1964).
1382. Alphabetic sorting is often not performed by pairwise comparisons.
1383. Radix sorting: Sorting n integers in linear time with radix-n numeration.
1384. Best path in a network: Dijkstra's algorithm and Bellman-Ford algorithm.
1385. " A* " best-first search uses a heuristical underestimate of the cost to a goal.
1386. Alpha-Beta Prunning Algorithm: Finding a minimax value in optimal time.
1387. String-matching in sublinear time. The Knuth-Morris-Pratt algorithm.
1388. Union-find: Linked partitions allow component merging and fast retrievals.
1389. Dynamic programming. Store partial results to avoid duplication of efforts.
1390. Minimum spanning tree. Least costly set of edges retaining connectivity.
1391. Linear programming in polynomial time. Foregoing the simplex algorithm.

      Miscellaneous

1392. Ford circles: Kissing circles touching the real line at rational points.
1393. Farey series: The rationals from 0 to 1, with a bounded denominator.
1394. The Stern-Brocot tree features every positive rational once and only once.
1395. Eisenstein-Stern sequence (Calkin-Wilf tree) enumerates the rationals too!
1396. Any positive rational is a unique ratio of two consecutive Stern numbers.
1397. Pick's formula gives the area of a lattice polygon by counting lattice points.

      History, Nomenclature, Vocabulary, etc.

      History :
1398. Earliest mathematics on record. Before Thales was Euphorbus...
1399. Indian numeration became a positional system with the introduction of zero.
1400. Roman numerals are awkward for larger numbers. [ Unabridged version ]
1401. The invention of logarithms: Napier, Bürgi, Briggs, St-Vincent, Euler.
1402. The earliest mechanical calculators. W. Shickard (1623) & Pascal (1642).
1403. The Fahrenheit Scale: 100°F was meant to be the normal body temperature.
1404. The revolutionary innovations which brought about new civilizations.
      Nomenclature & Etymology:
1405. The origin of the word algebra, and also that of algorithm.
1406. The name of the avoirdupois system is from a pristine form of French.
1407. Long Division: Cultural differences in long division layouts.
1408. Is a parallelogram a trapezoid? In a mathematical context, yes it is...
1409. Naming polygons. Greek only please; use hendecagon not "undecagon".
1410. Chemical nomenclature: Sequential names are systematic or traditional.
1411. Fractional prefixes: hemi (1/2) sesqui (3/2) hemipenta (5/2) hemisesqui (3/4).
      • Matches, phosphorus, and phosphorus sesquisulphide.
1412. Zillion. Naming large numbers.
1413. Zillionplex. Naming huge numbers.

      Style and Usage

1414. Abbreviations: Abbreviations of scholarly Latin expressions.
1415. "Resp." is a mathematical symbol whose syntax isn't that of respectively.
1416. Typography of long numbers.
1417. Intervals denoted with square brackets (outward for an excluded extremity).
1418. Dates in the simplest ISO 8601 form (with customary time stamps or not).
1419. The names of operands in common numerical operations.
1420. Spoken numbers.
1421. Pronouncing mathematical expressions, like native English speakers do.
1422. PEMDAS: A mnemonic for a rule that should not be taught.
1423. Physical units: Their products and their ratios.

      Setting the Record Straight

1424. The heliocentric system was known two millenia before Copernicus.
1425. The assistants of Galileo and the mythical experiment at the Tower of Pisa.
1426. Switching calendars: Newton was not born the year Galileo died.
1427. The Lorenz Gauge is due to Ludwig Lorenz (1829-1891) not H.A. Lorentz.
1428. Special Relativity was first formulated by Henri Poincaré (1854-1912).
1429. Radioactivity was discovered in 1857, by Abel Niépce de Saint-Victor.
1430. The Fletcher-Millikan "oil-drop" experiment isn't entirely due to Millikan.
1431. Collected errata about customary physical units of measurement.
1432. Portrait of Legendre: The mathematician was confused with a politician.
1433. Incorrect depictions of Ambroise Paré don't match one authentic portrait.
1434. The iconography used for Apollonius of Perga was meant for another man.
1435. The missing portrait of Robert Hooke. Did Newton really destroy it?
1436. Lisa Jardine's enduring blunder hinders the depictions of two scientists.
1437. Tribulations of a great portrait of Jan Baptist Van Helmont (1577-1644).
1438. Dubious quotations: Who really said that?

      Ancient Knowledge

1439. Exact sexagesimal ratios in Pythagorean triples. Set in clay (c. 1800 BC).
1440. Extant mathematical papyri betray the Egyptian taste for recreational mathematics.
1441. Classical geometry describes an homogeneous space indifferent to scale.
1442. Anthyphairesis is more elementary than factorization into primes.
1443. Obliquity of the ecliptic in the time of Eratosthenes (276-194 BC).
1444. Vertical wells at Syene are completely sunlit only once a year, aren't they?
1445. Eratosthenes sizes up the Earth: 700 stadia per degree of latitude.
1446. Knowing the Earth is round. Astronomical and terrestrial obsrvations.
1447. The distance to the Moon was computed by Aristarchus and Hipparchus.
1448. Latitude and longitude: The spherical grid of meridians and parallels.
1449. Itinerary units: The land league and the nautical league.
1450. Amber, compass and lightning: Glimpses of electricity and magnetism.
1451. The Antikythera Mechanism (c. 87 BC) is the oldest known orrery.
1452. Music theory: Design of musical instruments and study of harmony.
1453. The cult of Pythagoras.

      The Scientific Method

1454. On the nature of physical laws: The example of gravitation.
1455. Controlled Experiment: A concept attributed to Sir Francis Bacon (1590).
1456. History of the Scientific Method.
1457. Distinguishing between Science and Pseudoscience.
1458. Faster-than-light neutrinos? How the media butchers the scientific method.

      The Arrow of Time

1459. What is time? Why don't we remember the future?
1460. The beginning of time. Was there anything before that?
1461. Time machines: Unavoidable microscopically, impossible macroscopically.
1462. Determinism precludes the arrow of time.
1463. GPS time is now universally available, fairly inexpensively.
1464. Cosmic time is the time kept by a free-falling clock at rest in its local CMB.

      Physics

1465. Introduction: Geometry, statics, kinematics, dynamics and beyond...
1466. The notion of force. Statics, mechanical advantage and virtual work.
1467. Speed. Allowing the division of unlike quantities (distance and time).
1468. Mean-speed theorem. The distance traveled at constant acceleration.
1469. The timing experiments of Galileo: From the pendulum to falling bodies.
1470. The true period of a pendulum is proportional to 1 / agm ( 1 , cos A/2 ).
1471. The parabola of a cannonball, compared to Aristotle's triangular path.
1472. Conservation of momentum is key to Newton's three laws of motion.
1473. The work done to a point-mass equals the change in its kinetic energy.
1474. Relativistic work done and the corresponding change in relativistic energy.
1475. Relativistic thermodynamics: A point-mass endowed with internal heat.
1476. Spacecraft speeds up upon reentry into the upper atmosphere.
1477. Lewis Carroll's monkey climbs a rope over a pulley, with a counterweight.
1478. Two-ball drop can make one ball bounce up to 9 times the dropping height.
1479. Normal acceleration = Square of speed divided by the radius of curvature.
1480. Roller-coasters must rise more than half a radius above any loop-the-loop.
1481. Conical pendulum: A hanging bob whose trajectory is an horizontal circle.
1482. Conical pendulum constrained by a hemisphere: The string tension.
1483. Ball in a Bowl: Pure rolling increases the period of oscillation by 18.3%.
1484. Hooke's Law: Simple harmonic motion of a mass suspended to a spring.
1485. Speed of an electron estimated with the Bohr model of the atom.
1486. Hardest Stuff: Diamond is no longer the hardest known material.
1487. Hardness is an elusive nonelastic property, distinct from stiffness.
1488. Hot summers, hot equator! The distance to the Sun is not the explanation.
1489. Kelvin's Thunderstorm: Using falling water drops to generate high voltages.
1490. The Coriolis effect: A dropped object falls to the east of the plumb line.
1491. Terminal velocity of an object falling in the air.
1492. Angular momentum and torque. Spin and orbital angular momentum.
1493. Ad hoc conserved quantities unrelated to energy or momenta.

      Motion of Rigid Bodies (Classical Mechanics)

1494. Rotation vector of a moving rigid body (and/or "frame of reference").
1495. Angular momentum equals moment of inertia times angular velocity.
1496. Kinetic energy of a solid: Sum of its translational and rotational energies.
1497. Moments about a point or a plane are convenient mathematical fictions.
1498. Perpendicular Axis Theorem: Axis of rotation perpendicular to a lamina.
1499. The Parallel Axis Theorem: Moment of inertia about an off-center axis.
1500. Moment of inertia of a thick plate, derived from the parallel axis theorem.
1501. Moment of inertia of a right cone or conical frustum.
1502. Momenta of homogeneous bodies. List of common examples.
1503. Rigid pendulum moving under its own weight about a fixed horizontal axis.
1504. Reversible pendulum. The same period around two distinct axes.
1505. Moment of inertia of a spherical distribution or an homogeneous ellipsoid.
1506. Moment of inertia of the Earth is equal to 0.330695 M a ².
1507. Second dynamic form factor (J2) of a mass distribution.
1508. Axial precession: Reaction of a gyroscope to a torque across its axis.
1509. Elbow in free space: Muscle acting on two solids around a common axis. Isaac Newton 1643-1727

      Newtonian Gravity

1510. All physical theories have a limited range of validity.
1511. Gravity vs. Electrostatics: Straight comparisons.
1512. Binet's formulas: Deriving Kepler's laws for two orbiting bodies.
1513. The celerity hodograph of a body in elliptical orbit is a perfect circle !
1514. Airy weighs the Earth by timing a pendulum deep in a mine.
1515. Rigid equilateral triangle formed by three gravitating bodies.
1516. The five Lagrange points of two gravitating bodies in circular orbit.
1517. Geosynchronous Orbit: Semimajor radius of 36000 km around the Earth.
1518. Hohmann transfer orbit: From one circular orbit to another in two kicks.
1519. The gravitational self-energy of a ball (mass M, radius R) is -1.2 GM²/R
1520. Orbital Mechanics. Description of orbital motion using orbital elements.
1521. Tides on Earth: Dominant rôle of the Moon. Lesser rôle of the Sun.
1522. Attraction between rigid bodies, not necessarily spherically symmetric.
1523. Asteroid 99942 Apophis: Near-Earth objects and gravitational keyholes.
1524. Mass distributions of galaxies. Evidence for the existence of dark matter.

      Friction & Dissipative Mechanics

1525. Coefficients of friction: The static coefficent exceeds the kinetic one.
1526. Example involving a nontrivial choice between static and kinetic regimes.
1527. Minimum inclination of a ladder leaning against a frictionless wall.
1528. Spinning cylinder on an horizontal plane: The skidding before pure roll.
1529. Walter Lewin's Effect: Drawing a dotted line on a blackboard.
1530. Coefficient of restitution (e) Ratio of initial to final closing speed.
1531. Leonardo da Vinci's Friction Arch : Straight planks, no nails...

      The Physics of Billiards (Classical Mechanics)

1532. Billiards and pool tables: Nominal & quoted size, play area and clearance.
1533. Slate slab: The playing surface is cloth-covered rock.
1534. Billiard balls: Phenolic resin binding a dense powder has replaced ivory.
1535. Cue sticks: Butts and shafts. Basic construction. Anti-squirt technology.
1536. The contents of a cue case reflect the player's basic choices.
1537. Cue tips. Leather and phenolic tips.
1538. Two types of billiard chalk to reduce hand friction or increase tip friction.
1539. Silicon spray can be used by trickshot artists to reduce ball-cloth friction.
1540. Normal trajectory of a billiard ball: A parabola followed by a straight line.
1541. Making the cue ball stop after hitting the object ball.
1542. The stun path ("tangent line").
1543. The impossible 90° cut-shot made possible with extreme english.
1544. Squirt between cue and cue ball with extreme English (vertical spin axis).
1545. Jump shots. Legal and illegal ways to send the cue ball up in the air.

      Geometrical Optics

1546. Concave mirrors create enlarged virtual images of objects in front of them.
1547. Thin-lens equation. How the positions of an object and its image are tied.
1548. Focusing distance: The distance between an object and its image.
1549. Hyperfocal distance. Nearest in-focus objects when lens is set to infinity.
1550. Matrix methods: Transformations of a ray's inclination and radial distance.
1551. A crystal ball (index n and radius R) has focal length f = R / (2n-2).
1552. Lens-maker's formula: Focal lens as a function of signed curvatures.
1553. Thin-lenses are rectilinear: The image of a straight line is straight.
1554. Galileo's refractor (1609). Based on a design by Hans Lippershey (1608).
1555. Reflecting telescopes: The simplest design is due to Isaac Newton (1668).
1556. The compound microscope: Combining an object lens and an eyepiece.
1557. Distortion: When the image of a straight line ain't straight.
1558. The image of a tilted plane is a plane, which it intersects on the lens plane.
1559. Gullstrand's formula: Power of two lenses separated by a distance.
1560. Light falls off as cos⁴ q (where q = angular distance to the image center).
1561. Retrofocus allows a wide-angle image to be produced by a remote lens.
1562. Numerical aperture doesn't depend on the refraction index.
1563. Angular resolution varies as the wavelength and inversely as the diameter.
1564. Opposition effect increases albedo by eliminating micro-shadows.
1565. Honeycomb grids attenuate drastically the peripheral parts of a light beam.
1566. Schlieren imaging shows how shockwaves disturb the refractive index (n).

      Light and Color Vision

1567. A brief history of light: From Empedocles to modern times.
1568. Luminous units of measurement: Light as the human eye sees it.
1569. Monochromacy. In dim light, humans see in black-and-white.
1570. Dichromacy. 8% of men and 0.64% of women are color-blind.
1571. Trichromacy. Normal photopic vison (bright-light). 3 primary colors.
1572. Tetrachromacy. Mysteries of the fourth primary color.
1573. Aphakia. The eye of Claude Monet was sensitive to UV light.
1574. Light polarization is easy to demonstrate with sheet polarizers (sunglasses).

      Technical Aspects of Basic Photography

1575. Technical jargon: Different manufacturers promote different terms.
1576. Pinhole camera: The simplest camera doesn't even have a lens.
1577. Simple lens. Focal length. Aperture. f-stops.
1578. Microphotography. Using a lens backward in front of another.
1579. Depth of field. Circle of confusion. Hyperfocal distance.
1580. Bokeh: The aspect of out-of-focus areas.
1581. Defocus control (DC). Enhancing the bokeh of either near or far regions.
1582. Smooth transition focus. Optical apodization provides perfect bokeh.
1583. Lens sharpness. Center resolution, corner resolution. Vignetting.
1584. Stabilizers: Image stabilization (IS) = vibration reduction (VR).
1585. Color-corrected lenses. Color dispersion of various types of glass.
1586. Zoom lens. Adjustable focal length (can be internal or not).
1587. Focusing. Manual focusing. Focus detection.
1588. Professional cameras. Evolution toward ruggedness and full sensor size.
1589. Autofocus: Focusing motors attached to the lens or the camera.
1590. Focus breathing. Effective focal length may depend on focusing distance.
1591. Macro darkening. Effective aperture is lower in close-up photography.
1592. ISO sensitivity is directly derived from the old ASA & DIN standards.
1593. Light-sensitive films: Chemistry, sensitivity and grain size.
1594. Electronic photocells. Capturing an image one pixel at a time.
1595. Signal to noise ratio is limited by the number of photons per pixel.
1596. Bayer filters give color-vision to large arrays of photodiodes.
1597. Digital image sensors. Sizes, spatial resolution and color depth.
1598. Crop factor: 43.2666153 mm divided by the diagonal of the sensor.
1599. Handheld shots: Shutter speed should exceed equivalent focal length.
1600. Spatial filters. Lowpass filters or lack thereof. ADC resolution loss.
1601. Formats. Digital encoding of still pictures: raw, jpeg, etc.
1602. Exposure. Exposure index (and corrections for long exposure).
1603. Mounts: The various ways of attaching a body to interchangeable lenses.
1604. Screw-on filters have a 0.75 mm pitch and a few standard diameters.
1605. Neutral density filters. Changing exposure at constant speed and aperture.
1606. Photographing the Sun. ISO 100, 1/4000 s, f/32, with an ND1000 filter.
1607. Cut filters: Blocking parts of the IR, visible and/or UV spectra.
1608. Color temperature. Light sources and white balance.
1609. Shadows on a sunny day. How dark and how blue are they really?
1610. Color conversion: Converting one color temperature to another.
1611. Flash photography. Technology. Guide number (GN) & duration control.

      Waves

1612. Huygens' Principle. A convenient fiction to describe wave propagation.
1613. Diffraction occurs when when a wave emanates from a bounded source.
1614. Young's double-slit experiment demonstrates the wavelike nature of light.
1615. Celerity is the speed with which phase propagates.
1616. Standing waves feature stationary nodes and antinodes.
1617. Snell's Law gives the angle of refraction (Thomas Harriot, July 1601).
1618. Total internal reflection (TIR) at incidences exceeding the critical angle.
1619. Birefringence. Discovery of polarization (Erasmus Bartholinus, 1669).
1620. Fresnel equations: Reflected or refracted intensities of polarized light.
1621. Brewster's angle is the incidence which yields a 100% polarized reflection.
1622. Stokes parameters: A standard description of the state of polarization.
1623. Transverse wave on a rope: (celerity) ² = (tension) / (linear mass density).
1624. Chladni patterns: The lines formed by nodes in an oscillating plate.
1625. Wave inertia: The idea behind the Hemispherical Resonator Gyroscope.

      Colors & Dispersion

1626. Dispersion relation: Pulsatance vs. wave number; frequency vs. wavelength.
1627. Empirical approximations often give wavelength as a function of frequency.
1628. Group velocity is the traveling speed of a beat phenomenon.
1629. Rayleigh scattering makes the sky blue and sunsets red.
1630. Index of refraction of water for light of different colors.
1631. A spherical drop reflects light back (red up to 42.34° & violet up to 40.58°).
1632. The length of a rainbow: Mathematical digression.

      Infrared Transmissions (of remote control codes & data)

1633. Wavelength: 940 nm (319 THz) is the most common specification for IR.
1634. Modulation: 38 kHz (38.4 kHz). Also: 30, 33, 36, 36.7, 40, 56, 455 kHz.
1635. Remote shutter release for cameras. The simplest type of infrared control.
1636. On/off patterns to encode data bits and the start/stop of data frames.
1637. Unexplained datasheet mysteries. Why are Rohm's specs slightly off?
1638. Discrete IR control codes provide critical functions for automated control.
1639. RECS-80: An obsolete system, proposed by Philips in 1988.
1640. RC-5 and RC-6. Philips and the well-documented European protocol.
1641. NEC Protocol. The Japanese format.
1642. SIRC Protocol by Sony.
1643. RCA Protocol: 64 ms to send a 4-bit address and 8-bit data at 56 kHz.
1644. HP 82240B: The standard printer for HP scientific calculators, since 1989.
1645. Philips Pronto universal remotes. Pronto codes for learned commands.
1646. Databases of IR codes: Defending consumers & creating flexibility.
1647. Serial protocol: 2400 Bd, 1 start bit, 8 data bits, 1 stop bit, odd parity.
1648. Resurrrecting 455 kHz modulation to transmit at high-speed (19200 Bd).

      Lasers : From masers to laser beams

1649. Stimulated emission is crucial to blackbody equilibrium (Einstein, 1916).
1650. Bose-Einstein Statistics is what explains stimulated emission of bosons.
1651. Optical pumping is the key to creating a population inversion.
1652. Population inversion : When energetic states are abnormally abundant.
1653. LASER Cavity. Light Amplification by Stimulated Emission of Radiation.
1654. Gaussian beam. The shape of an ideal laser beam.
1655. Tunable lasers.
1656. Negative tenperature of a laser.

      Analytical Mechanics & Classical Field Theory

1657. Fermat's principle (least time) for light (c.1655) predates Newton.
1658. Maupertuis principle of least action (1744).
1659. Virtual Work: A substitute for Newton's laws that cancels constraint forces.
1660. Phase Space: Phase describes completely the state of a classical system.
1661. Either velocities or momenta are added to configuration to specify a phase.
1662. The Lagrangian is a function of positions and velocities.
1663. The Hamiltonian depends on positions and momenta.
1664. Poisson brackets: An abstract synthetic view of analytical mechanics.
1665. Liouville's theorem: The Hamiltonian phase volume doesn't change.
1666. Noether's theorem: Conservation laws express the symmetries of physics.
1667. Relativistic point-mass: Lagrangian, Hamiltonian and free momentum.
1668. Charge in a magnetic field: The canonical momentum isn't the linear one.
1669. Fokker lagrangian: Lagrangien formulation of general relativity (1924).
1670. Field theory: Lagrangian function of a continuum of values and velocities.

      Electromagnetism (Maxwell's Equations)

1671. Clarifications: Vector calculus (Heaviside) & microscopic view (Lorentz).
1672. The vexing problem of units is a thing of the past if you stick to SI units.
1673. The Lorentz force on a test particle defines the local electromagnetic fields.
1674. Electrostatics (1785): The study of the electric field due to static charges.
1675. Electric capacity is an electrostatic concept (adequate at low frequencies).
1676. Electrostatic multipoles: The multipole expansion of an electrostatic field.
1677. Birth of electromagnetism (1820): Electric currents create magnetic fields.
1678. Biot-Savart Law: The static magnetic induction due to steady currents.
1679. Magnetic scalar potential: A multivalued static scalar field.
1680. Magnetic monopoles do not exist : A law stating a fact not yet disproved.
1681. Ampère's law (1825): The law of static electromagnetism.
1682. Faraday's law (1831): Electric circulation induced by magnetic flux change.
1683. Self-induction received by a circuit from the magnetic field it produces.
1684. Ampère-Maxwell law: Dynamic generalization (1861) of Ampère's law.
1685. Putting it all together: Historical paths to Maxwell's electromagnetism.
1686. Maxwell's equations unify electricity and magnetism dynamically (1864).
1687. Continuity equation: Maxwell's equations imply conservation of charge.
1688. Waves: Predicted by Faraday, Maxwell & FitzGerald. Observed by Hertz.
1689. Electromagnetic energy density and the flux of the Poynting vector.
1690. Planar electromagnetic waves: The simplest type of electromagnetic waves.
1691. Maxwell-Bartoli radiation pressure. First detected by P. Lebedev in 1899.
1692. Electromagnetic potentials are postulated to obey the Lorenz gauge.
1693. Solutions to Maxwell's equations, as retarded or advanced potentials.
1694. Electrodynamic fields corresponding to retarded potentials.
1695. Electrodynamic fields corresponding to advanced potentials.
1696. The gauge of retarded potentials: is it really the Lorenz gauge?
1697. Power radiated by an accelerated charge: The Larmor formula (1897).
1698. Lorentz-Dirac equation for the motion of a point charge is of third order.

      Capacitors and Electrostatic Distributions

1699. Capacitance is measured in farads (F).
1700. Capacity of a sphere. Single-electrode capacitor.
1701. Parallel-plates: Two large planar electrodes separated by a small surface d.
1702. Cylindrical wires or shielding. Many possible configurations.
1703. Leyden jars. Earliests high-voltage capacitors.
1704. Ultracapacitors: One farad or more.

      Electromagnetic Dipoles

1705. Molecular electric dipole moments. First studied by Peter Debye in 1912.
1706. Force exerted on a dipole by a nonuniform field.
1707. Torque on a dipole is proportional to its cross-product into the field.
1708. Electric and magnetic dipoles: Dipolar solutions of Maxwell's equations.
1709. Static distributions of magnetic dipoles can be emulated by steady currents.
1710. Static distributions of electric dipoles are equivalent to charge distributions.
1711. Field at center of a uniformly magnetized or polarized sphere of any size.
1712. Sign reversal in magnetic and electric fields from matching dipoles.
1713. Relativistic dipoles: A moving magnet develops an electric moment.

      Magnetism; Electromagnetic Properties of Matter

1714. Magnetization and polarization describe densities of bound dipoles.
1715. Distinct magnetization and polarization gauges may yield the same field.
1716. Maxwell's equations in matter: Electric displacement & magnetic strength.
1717. Electric susceptibility is the propensity to be polarized by an electric field.
1718. Electric permittivity and magnetic permeability. Related to susceptibilities.
1719. Paramagnetism: Weak positive susceptibility.
1720. Diamagnetism: Lorentz force turns orbital moments against an external B.
1721. Magnetic levitation: How to skirt the theorem of Samuel Earnshaw (1842).
1722. Pyrolytic carbon: The most diamagnetic substance, at room temperature.
1723. Bohr-van Leeuwen Theorem: Diamagnetism and paramagnetism cancel ?!
1724. Thermodynamics of dielectric matter: dU = E.dD + ...
1725. Ferromagnetism: Permanent magnetization without an external field.
1726. Antiferromagnetism: When adjacent dipoles tend to oppose each other...
1727. Ferrimagnetism: With two kinds of dipoles, partial cancellation may occur.
1728. Magneto-optical effect discovered by Faraday on September 13, 1845.
1729. Ohm's Law: Current density is proportional to electric field: j = s E.
1730. Bloch equations: Longitudinal and transverse magnetization relaxation.
1731. Cotton effect: Interaction of light with chiral molecules.

      Permanent Magnets

1732. The relativistic origin of magnetism.
1733. Magnetometry. How do you measure a permanent magnet?
1734. Make a magnet without using electric current or another magnet.
1735. Halbach arrays: Field is reinforced on one side and vanishes on the other.

      Motors and Generators

1736. Homopolar motor: The first electric motor, by Michael Faraday (1831).
1737. Faraday's disk can generate huge currents at a low voltage.
1738. Magic wheels: Two repelling ring magnets mounted on the same axle.
1739. Beakman's motor. Current switches on and off as the coil spins.
1740. Tesla turbine. Stack of spinning disks with outer intake and inner outflow.

      Tesla's World

1741. Tesla coil. A low-current high-voltage generator.
1742. Schumann's cavity is resonating at 8 Hz, below the ionosphere.

      Electronics 101

1743. Mechanical switches. Working out proper snubber networks.
1744. Mechanical relays. Flyback diodes in parallel with DC-controlled coils.
1745. Diodes: PN, PIN, Silicium, Germanium, Schottky, Zener, varicap, etc.
1746. Temperature coefficients (tempco) and temperature compensation (TC).
1747. Operational amplifiers A quick survey of those indispensable analog ICs.
1748. Bipolar Junction Transistors (BJT) : NPN and PNP polarity types.
1749. Thyristor. GE's silicon-controlled rectifier (SCR) has 4 silicon layers.
1750. Triacs conduct in either direction once triggered, until current drops to zero.
1751. A diac is turned on by a large voltage. It turns off when current is too low.
1752. Field-programmable gate arrays (FPGA). Digital flexibility at high speed.
1753. Hall effect. The classical effect discovered by Edwin Hall in 1879.

      The Vacuum

1754. Aristotle's plenism. Downfall of the Horror Vacui doctrine (17^th century).
1755. Sprengel's pump (1865) made Crookes tubes and lightbulbs possible.
1756. Vacuum tubes. Heated filaments, grids and electrons moving in a vacuum.
1757. Dirac's equation predicted positrons as holes in a bizarre vacuum.
1758. The Quantum Vacuum. The vacuum isn't empty. Structure of the vacuum.

      Special Relativity

1759. Observers in motion: An elementary derivation of the Lorentz Transform.
1760. Combining parallel velocities never results in a speed exceeding c.
1761. Combining velocities when they're not collinear.
1762. The headlight effect: An isotropic source will radiate forward if it moves.
1763. Closing speed: The distance between objects may change faster than c.
1764. Fizeau's empirical relation between refractive index (n) and Fresnel drag.
1765. Harress-Sagnac effect. Measuring angular motion with fiber optic cable.
1766. The rapidity concept simplifies the relativistic addition of speeds.
1767. Relative velocity of two photons: Undefined if they have the same direction
1768. Minkowski spacetime. Lorentz transform applies to 4-vector coordinates.
1769. The Lorentz transform expressed vectorially for a boost of speed V.
1770. Wave vector: The 4-dimensional gradient of the phase describes a wave.
1771. Doppler shift: The relativistic effect is not purely radial.
1772. Relativistic momentum and Einstein's relation between mass and energy.
1773. Kinetic energy: At low speed, the relativistic energy varies like ½ mv ².
1774. Photons and other massless particles: Finite energy at speed c.
1775. The de Broglie celerity (u) is inversely proportional to a particle's speed.
1776. Compton diffusion: The result of collisions between photons and electrons.
1777. The Klein-Nishina formula: gives the cross-section in Compton scattering.
1778. Compton effect is suppressed for visible light and bound electrons.
1779. Elastic shock: Energy transfer is v.dp. (None is seen from the barycenter.)
1780. Photon-photon scattering is like an elastic collision of two photons.
1781. Cherenkov effect: When an electron exceeds the celerity of light...
1782. Constant acceleration over an entire lifetime will take you pretty far.
1783. Langevin's twins paradox. Confirmed by the Häfele-Keating experiment.
1784. Terrell effect. Apparent rotation of a fast-moving body.

      Photonics

1785. Photons are quanta of light. They're both wavelike and corpuscular.
1786. The photoelectric effect was explained by Albert Einstein in 1905.
1787. Signal-to-noise ratio of light sensors: The ultimate physical limit.
1788. Soft photons carry (almost) no energy but still have unit spin.

      Nuclear Physics

1789. Henri Becquerel and the [second] discovery of natural radioactivity (1896).
1790. Pierre & Marie Curie: The discovery of new radioactive elements (1898).
1791. Rutherford lead-block experiment. The three types of ionizing radiations.
1792. Radioactive-decay law was formulated by Rutherford and Soddy, in 1902.
1793. Geiger-Marsden experiment: There's a tiny dense nucleus inside the atom!
1794. Alpha-decay: Polonium (Po-210, Z=84) decays into Lead (Pb-206, Z=82).
1795. Mass Defect: In a nuclear reaction, the Q-value balances the mass change.
1796. The standard decay modes: a, b-, 2b-, b+, e (electron capture) or IT.
1797. The 4 radioactive series: Thorium, Neptunium, Uranium and Actinium.
1798. Other decay modes: Proton or neutron emission, fission and spallation.
1799. The Geiger counter measures the activity flux of ionizing radiation.
1800. Scintillation allows quantitive measurements of a gamma spectrum.
1801. Cross-section: A target looks as if its size depends on the projectile's speed.
1802. Artificial radioactivity: Neutron bombardment creates unstable nuclides.
1803. Chain reactions: When neutron-induced decays produce more neutrons...
1804. Critical mass: The smallest mass that will allow runaway chain reactions.
1805. Thermonuclear bombs. Nuclear fusion ignited by fission devices.
1806. Carbon-dating: Radiocarbon ratio starts decaying when an organism dies.
1807. Fusion of deuterons: Helium is formed with liberation of energy.
1808. The Proton-Proton chain fusion powers all stars less than 1.5 solar masses.
1809. Catalytic nuclear reactions. The CNO cycle yields 7% of the Sun's power.
1810. Triple-Alpha process explains the abundance of Carbon and Oxygen,
1811. Tokamak reactors: Deuterium-Tritium fusion (DT) is the easiest to ignite.
1812. Farnsworth-Hirsch fusor: Controlled fusion on a desktop. Neutron source.
1813. Polywell reactor: The design advocated by the late Robert Bussard.
1814. Amateur nuclear physics: Demystifying nuclear energy and radioactivity.
1815. The Radioactive Boyscout and other misguided experimenters.
1816. Natural fission reactors. Predicted in 1956. Discovered in Gabon in 1972.
1817. Safe reactors: Fast automatic shutdown using UZrH fuel rods.
1818. Muon-facilitated nuclear fusion occurs at low temperatures (Frank, 1947).

      Ionizing Radiation

1819. Cathode-ray tubes (CRT, 1875). Due to William Crookes (1832-1919).
1820. X-rays were discovered in 1895 by Wilhelm Röntgen (1845-1923).
1821. Bragg peak (1903). An ionizing ray loses a lot of energy near its demise.
1822. X-rays crystallography (XRC) was founded by Max von Laue, in 1912.
1823. Synchrotron radiation is produced by bending a beam of charged particles.
1824. Brehmsstrahlung: The radiation emitted by decelerating charged particles.
1825. Dose deposited by radiation in human tissue. 1 Sv = 1 J/kg = 100 Rem.

      The Weak Nuclear Force

1826. Pauli deduced the existence of the neutrino from conservation laws.
1827. Fermi theory of beta decay (1934).
1828. Direct detection of neutrinos. Cowan-Reines neutrino experiment (1956).
1829. Parity violation was established by the Wu experiment (1956).
1830. Discovery of the muon-neutrino. Steinberger, Lederman, Schwartz (1962).
1831. Neutrino oscillations. Solar neutrinos revealed nonzero neutrino masses.

      The Strong Nuclear Force: Quarks, Gluons & Color SU(3)

1832. Confined color charges. Only color-neutral particles can be observed.
1833. Isospin (isotopic spin) is like spin in a disembodied Hilbert space.
1834. Bootstrap principle by Geoffrey Chew and Steve Frautschi (1961).
1835. The 8 Gell-Mann matrices. SU(3) is an eight-dimensional Lie group.
1836. Quantum Chromodynamics (QCD). The way color charges are traded.
1837. Deep inelastic scattering of electrons by protons or bound neutrons (1967).
1838. Renormalization of Yang-Mills (gauge) theories ('t Hooft, 1971).
1839. Asymptotic freedom of the strong force at small distances (1973).

      Particle Detectors

1840. Scintillation counter. Invented by William Crookes in 1903 (using ZnS).
1841. Geiger counter. Counting ionizing particles by the avalanches they cause.
1842. Cloud chamber. Mist condenses first along the tracks of ionizing particles.
1843. Nuclear emulsions: Latent images are produced along particle tracks.
1844. Bubble chamber. Nucleation occurs along trails left by ionizing particles.
1845. Wire chamber (1968). Multi-wire proportional chamber (MWPC).

      Particle Accelerators

1846. Electrostatic generators. Accelerating charged particle with electric fields.
1847. Linear accelerators (1924). Using waves rather than static electric fields.
1848. Cyclotron (1929). A constant magnetic field creates spiraling trajectories.
1849. Betatron (1934). Accelerating electrons with a changing magnetic field.
1850. Microtron (1944). Modified cyclotron needing only small electrodes.
1851. Synchrotrons (1944, 1945). Modern circular particle accelerators.
1852. Strong focusing of charged beams, using alternating gradients.
1853. Large Hadron Collider (LHC). The largest circular accelerator ever.
1854. Wakefield accelerators. Particles surfing the wake of a laser in a plasma.

      Supersymmetry between bosons and fermions

1855. History of Supersymmetry (SUSY).
1856. The Wess-Zumino-Witten model (WZW).

      String Theory

1857. Unification: Consistency is required. Actual high-energy unification is not.
1858. Kaluza-Klein Theory: Postulating an extra dimension for electromagnetism.
1859. 1960's hadron physics: Regge trajectories begat constant-tension strings.
1860. Gabriele Veneziano: The magic of Euler's beta and gamma functions.
1861. Leonard Susskind (1940-): The basic idea of a fundamental string.
1862. Joël Scherk (1946-1979) & John Schwarz: Rediscovering gravity.
1863. Michael Green & John Schwarz: Hoping for a Theory of Everything.
1864. String Quintet: Five different consistent string theories!
1865. M-Theory: Ed Witten's 11-dimensional brainchild, unveiled at String '95.
1866. The brane world scenarios of Lisa Randall and Burt Ovrut.

      Quantum Gravity: Toward a theory of everything (TOE).

1867. Wheeler-DeWitt equation (WdW). The basis for loop quantum gravity.
1868. Loop Quantum Gravity (LQG).
1869. Anti-de Sitter space (AdS).
1870. Conformal field theory (CFT).
1871. Maldacena duality: AdS/CFT correspondence (1997).
1872. AMPS Firewall: Almheiri, Marolf, Polchinski, and Sully (2012).

      Gauge Theories

1873. Fluid mechanics: Tracking actual particles.

      Physics of Gases and Fluids

1874. Atmospheric pressure varies vertically in proportion to the density of air.
1875. The Magdeburg hemispheres are held together by more than a ton of force.
1876. The ideal gas laws of Boyle, Mariotte, Charles, Gay-Lussac, and Avogadro.
1877. Joule's law: Internal energy of an ideal gas depends only on temperature.
1878. Deflating a tire: Releasing a pressurized gas into the atmosphere.
1879. The Van der Waals equation and other interesting equations of state.
1880. Virial equation of state. Virial expansion coefficients. Boyle's temperature.
1881. Viscosity is the ratio of a shear stress to the shear strain rate it induces.
1882. Permeability and permeance: Vapor barriers and porous materials.
1883. Resonant frequencies of air in a box.
1884. The Earth's atmosphere. Pressure at sea-level and total mass above.
1885. Composition of dry air at sea level (for 450 ppm carbon dioxide).
1886. Humidity. The moisture content of clear atmospheric air.
1887. The first hot-air balloon (Montgolfière) was demonstrated on June 4, 1783.
1888. Sulfur hexafluoride is a very heavy gas and a good electrical insulator.

      Virial

1889. Virial of force: A dynamic quantity defined by Rudolf Clausius in 1870.
1890. Virial. The classical virial of momentum is a conserved quantity.
1891. The relativistic virial is defined at constant time in the observer's frame.
1892. Quantum virial: The quantum counterpart of the classical virial.

      Transport Properties of Matter

1893. Viscosity: The transport of microscopic momentum.
1894. Brownian motion and Einstein's estimate of molecular sizes.
1895. Thermal Conductivity: The transport of microscopic energy.
1896. Diffusivity: The transport of chemical concentration.
1897. Boltzmann Transport Equation: Solve for a continuous random variable.
1898. Speed of Sound: Reversible transport of a pressure disturbance in a fluid.

      Sound and Acoustics

1899. The speed of sound in the atmosphere varies with altitude.
1900. Newton's formula is off by 15.48% under (bad) isothermal assumptions.
1901. Speed of sound in a fluid, computed under (good) isentropic assumptions.
1902. Perceived loudness differs from absolute (physical) loudness of sound.
1903. Acoustical limits: Sound waves have a limited frequency range.

      Mathematical Aspects of Music

1904. Western origins. Gregorian modes just denoted specific vocal octaves.
1905. Note durations follow a binary progression. Dotting prolongs by 50%.
1906. Tempo: The speed of music. What the metronome measures.
1907. Beats of the metronome. Bar (measures) and time signatures.
1908. Triplet: Group of 3 notes equally splitting twice their common duration.
1909. Dynamics. Pianissimo to fortissimo. Diminuendo or crescendo.
1910. The Frequency Domain: Sound reduces to a superposition of tones.
1911. Perfect pitch: Native music speakers can easily name any absolute pitch.
1912. Legacy pitch and marketing hype.
1913. Staves and clefs. Transcribing and reading musical notes.
1914. Common keyboards. From double-octave (25 keys) to full piano (88 keys).
1915. Musical intervals are ratios of frequencies.
1916. Quantifying the beauty of consonances: Euler's gradus function (1730).
1917. The circle of fifths: Best fit for the two simplest harmonies (2:1 and 3:2).
1918. Tritone: Half an octave. A dissonant interval (F to B or B to F).
1919. Scales. The key of C major (or A minor) only uses white piano keys.
1920. The 7 modes of the major scale (diatonic modes) including natural minor.
1921. Brightness and darkness. Canonical quantification of an elusive concept.
1922. Non-diatonic minor scales: Melodic minor and harmonic minor.
1923. Harmonic major scale. The mirror inverse of the harmonic minor scale.
1924. Other noteworthy heptatonic scales and Bebop scales (with passing tones).
1925. Hexatonic scales: Whole-tone. Augmented. Prometheus. Blues scale.
1926. Two diminished scales: Whole-half diminished & half-whole diminished.
1927. Chords are separate notes played together.
1928. Harmonization.
1929. Cadence: Resolution of tension. Musical punctuation and closure.
1930. Modulation and key changes.
1931. Composing 101: Some basic principles composers have been perusing.
1932. Musical Dice: Composing music without human intervention (1757).
1933. Ornaments and embellishments: Superficial changes respecting the melody.
1934. Octave displacement (Octave dispersion).
1935. Jazz: Lydian Chromatic Concept of Tonal Organization (George Russell).
1936. MIDI: Musical Instrument Digital Interface (1980).
1937. Microtonal instruments and exploration of polychromatic music.
1938. Guitars and other fretted musical instruments.
1939. Classical fretless 4-string instruments. The violin family.
1940. Recorders. Wooden flipple flutes produce pure tones with little harmonics.

      Heptatonic Musical Scales

1941. 4 heptatonic scales have signatures without double alterations in all 12 keys.
1942. The 7 diatonic modes include the major mode and natural minor.
1943. Brightness and darkness. Canonical quantification of an elusive concept.
1944. Non-diatonic minor scales: Melodic minor and harmonic minor.
1945. Harmonic major scale. The mirror inverse of the harmonic minor scale.
1946. Double-harmonic major scale. Each of its 7 modes has one cursed key.
1947. Hungarian major scale and Romanian major scale are non-cohemitonic.
1948. Neapolitan Major is palindromic. Neapolitan minor is Major b2 inverted.
1949. The Blues leading-tone scale is the inverse of the Persian scale.
1950. The Major flat-5 scale and its interesting modes.
1951. Half a dozen Bebop modes in three distinct scales.
1952. Enigmatic scales: Minor, ascending, descending and (octonic) mixed.
1953. The Egyptian Crater scale of Jeff Buset (2009).
1954. Petra's Crater scale. (2019). The most imperfect heptatonic scale.

      Filters and Feedback

1955. Complex pulsatance: s = s+iw (damping constant + imaginary pulsatance)
1956. Complex impedance: Resistance and reactance.
1957. Quality Factor (Q). Ratio of maximal stored energy to dissipated power.
1958. Nullators and norators: Strange dipoles for analog electronic design.
1959. Corner frequency of a simple first-order low-pass filter. -3 dB bandwidth.
1960. Second-order passive low-pass filter, with inductor and capacitor.
1961. Two cascaded RC low-pass filters can almost achieve critical damping.
1962. Sallen-Key filters: Active filters do not require inductors.
1963. Lowpass Butterworth filter of order n : The flattest low-frequency response.
1964. Linkwitz-Riley crossover filters used in modern active audio crossovers.
1965. Chebyshev filters: Ripples in either the passband or the stopband.
1966. Elliptic (Cauer) filters encompass all Butterworth and Chebyshev types.
1967. Legendre filters maximal roll-off rate for monotonous frequency response.
1968. Gegenbauer filters: From Butterworth to Chebyshev, via Legendre.
1969. Phase response of a filter.
1970. Bessel-Thomson filters: Phase linearity and group delay.
1971. Gaussian filters: Focusing on time-domain communication pulses.
1972. Linear phase equiripple: Ripples in group delay to go beyond Bessel filters.
1973. DSL filters allow POTS below 3400 Hz & block digital data above 25 kHz.
1974. Switched capacitor: Faking a resistor with a capacitor and a SPDT switch.

      Fantasy Engineering: Just for fun.

1975. Raising the Titanic, with (a lot of) hydrogen.
1976. Gravitational Subway: From here to anywhere on Earth, in 42 minutes.
1977. In a vacuum tube, a drop to the center of the Earth would take 21 minutes.
1978. Detecting a single graviton is an impossible task.
1979. Controled fusion: Why the Tokamak approach can't produce energy.

      Woodworking

1980. Solid wood: Some common lumber and exotic woods.
1981. Lumber (timber). Primary stock obtained from logs.
1982. Composite boards. From paperboard and cardboard to phenolic hardboard.
1983. Carpenter's glues. Physics & chemistry of wood glues. Cyanoacrylate glue.
1984. Clamps. Gluing any wood joint requires adequate clamping.
1985. Axe and adze: Two related tools which date back to the Stone Age.
1986. Drawknife, travisher and spokeshave. Ancient tools predating planes.
1987. Hand planes: The art of removing high-spots, using a sole as reference.
1988. Power planers: Flatness and uniform thickness from jointers and planers.
1989. Joinery: The various types of joints to choose from.
1990. Portable bench, vise & dogs. In praise of the Workmate 425 and others.
1991. Mallets and hammers.
1992. Metal fasteners: Nails, wood screws  and Twinfast screws. Nuts & bolts.
1993. Dowels and treenails Locking reinforced joints in all-wood construction.
1994. Chisels.
1995. Whittling and carving.
1996. Hand saws and their uses.
1997. Table saw. Rip fence and pushblock. Miter gauge or cross-cut sleds.
1998. Other saws with circular blades: Trim saw, miter saw, radial-arm saw.
1999. Power-saws with non-circular blades: Bandsaw, jigsaw, reciprocating saw.
2000. Drills and drill bits. Hand drill. Handheld power drill. Drill press.
2001. Turning wood on a lathe.
2002. Awls and trammel points. Beam compass.
2003. Calipers and dividers.
2004. Files and rasps.
2005. Sanding.
2006. Dust collection. Cyclone or Thien baffle before vacuum dust extractor.
2007. Staining and distressing. Accenting wood grain and tooling marks.
2008. Sealing and priming. Sealing wood pores.
2009. Finishing: Applying tint, paint, lacquer, varnish...

      Metalworking

2010. Extracting metals: Ores. Mining. Smelting. Recycling.
2011. Alloys. Physics and chemistry of alloys. Engineering grades.
2012. Sheet metal. Most metals or single-phase alloys can be rolled into sheets.
2013. Welding, soldering and brazing. Joining metal pieces without fasteners.
2014. Measuring. Calipers, micrometer, surface plate, surface gauge.
2015. Machinist vise. Holding metal pieces in place while working on them.
2016. Sawing and cutting. Adjustable hacksaw and jeweler's saw.
2017. Filing and grinding.
2018. Drills. Handheld drills and drill presses. Drill bits.
2019. Milling. Making straight or curved cuts.
2020. Lathe. Turning and machining.
2021. Engine turning.
2022. Taps. Threading holes or rods.
2023. Circular champfers. Mandatory external chamfer for off-the-shelf bolts.
2024. Gunsmithing. Some gun parts are illegal to make without a license.
2025. Finishing. Polishing, brushing.
2026. Chemical surface treatments. Blueing steel. Anodizing aluminum.
2027. Painting and coating.
2028. Forging hot iron pieces. A blackmith hammers them on an anvil.
2029. Electrical Discharge Machining (EDM). Electric arcs as cutting tools.

      Steam Engines, Heat Engines

2030. The aeolipile. Ancient steam engine demonstrating jet propulsion.
2031. Edward Somerset of Worcester (1601-1667): Steam fountain blueprint.
2032. Denis Papin (1647-1714): Pressure cooking and the first piston engine.
2033. Thomas Savery (c.1650-1715): Two pistons and an independent boiler.
2034. Thomas Newcomen (1663-1729) & John Calley: Atmospheric engine.
2035. Nicolas-Joseph Cugnot (1725-1804): The first automobile (October 1769).
2036. James Watt (1736-1819): Steam condenser and Watt governor.
2037. Richard Trevithick (1771-1833) and the first railroad locomotives.
2038. Sadi Carnot (1796-1832): Carnot's cycle. The theoretical efficiency limit.
2039. Sir Charles Parsons (1854-1931): The modern steam turbine (1884).
2040. Drinking Bird: Room-temperature engine based on evaporative cooling.

      Thermodynamics

2041. Elementary concept of temperature. The zeroth law of thermodynamics. Lord Kelvin 1824-1907 Hermann von Helmholtz 1821-1894
2042. Conservation of energy: The first law of thermodynamics.
2043. Increase of Entropy: The second law of thermodynamics.
2044. State variables: Extensive and intensive quantities.
2045. Entropy is missing information, a measure of disorder.
2046. Nernst Principle (third law): Entropy is zero at zero temperature.
2047. Thermodynamic potentials are convenient alternatives to internal energy.
2048. Calorimetric coefficients, adiabatic coefficient (g) heat capacities, etc.
2049. Relations between isothermal and isentropic coefficients.
2050. The thermal Grüneisen parameter.
2051. Entropy of a Van der Waals fluid as derived from its equation of state.
2052. Dulong-Petit Law (1819). The molar heat capacity of a metal is about 3 R.
2053. Thermal effects of molecular vibrations at moderate temperatures.
2054. Latent heat (L) is the heat transferred in a change of phase.
2055. Van 't Hoff's equation. How an equilibrium changes as temperature varies.
2056. Cryogenic coefficients: Lower temperature with an isenthalpic expansion.
2057. Peltier effect: Electrical cooling at a junction between dissimilar materials.
2058. Relativistic Thermodynamics: A moving body appears cooler.
2059. Inertia of energy for an object at nonzero temperature.
2060. Stefan's Law: A black body radiates as the fourth power of its temperature.
2061. The "Fourth Law": Is there really an upper bound to temperature?
2062. Hawking radiation: On the entropy and temperature of a black hole.
2063. Partition function: The cornerstone of the statistical approach.

      Thermodynamics and Elasticity

2064. Elastic properties. Reversible deformations in resilient materials.
2065. Hysteresis and resilience. Stored elastic energy is never fully recovered.
2066. Elastomers. Unsaturated rubbers are cured by sulfur vulcanization.
2067. Coefficients ot thermal expansion: Cubical scalar and linear tensor.
2068. Invar anomaly: The low thermal expansion of 36% Ni / 86% Fe alloy.
2069. Waves in a solid: P-waves (fastest), S-waves, E-waves (thin rod), SAW...
2070. Thermodynamics of acoustics: Dynamic coefficients and isothermal ones.
2071. Rayleigh Wave: The quintessential surface acoustic wave (SAW).

      Demons of Classical Physics

2072. Laplace's Demon: Deducing past and future from a detailed snapshot.
2073. Maxwell's Demon: Happily trading information for entropy.
2074. Shockley's Ideal Diode Equation: Diodes don't violate the Second Law.
2075. Szilard's engine: Putting a simple-minded Maxwellian demon to work.
2076. Landauer's principle: The unavoidable thermodynamic cost of forgetting.

      Statistical Physics

2077. Lagrange multipliers. One multiplier for each constraint of an optimization.
2078. Microcanonical equilibrium. Isolated system: All states are equiprobable.
2079. Equipartition of energy. Every degree of freedom gets an equal share.
2080. Canonical equilibrium: Boltzmann factor in a heat bath.
2081. Grand-canonical equilibrium when chemical exchanges are possible.
2082. Bose-Einstein statistics: One state may be occupied by many particles.
2083. Fermi-Dirac statistics: One state is occupied by at most one particle.
2084. Boltzmann statistics: The low-occupancy limit (most states unoccupied).
2085. Maxwell-Boltzmann distribution of molecular speeds in an ideal gas.
2086. Partition function: The cornerstone of the statistical approach.
2087. Fock basis for the tensor product of many identical Hilbert spaces.

      Quantum Mechanics

2088. Quantum Logic: The surprising way quantum probabilities are obtained.
2089. Swapping particles either negates the quantum state or leaves it unchanged.
2090. The Measurement Dilemma: What makes Schrödinger's cat so special?
2091. Matrix Mechanics: Like measurements, matrices don't commute.
2092. Schrödinger's Equation: Nonrelativistic quantum particle in a classical field.
2093. Noether's Theorem: Conservation laws express the symmetries of physics.
      Quantum Formalism :
2094. Kets are Hilbert vectors (duals of bras) on which observables operate.
2095. Hilbert space for a composite system. Tensor product of Hilbert spaces.
2096. Commutators give observables the structure of a Lie algebra.
2097. Some observables are associated with classical physical quantities.
2098. Uncertainty relations hold whenever the commutator does not vanish.
2099. Transverse certainties. Measuring two non-conjugate observables.
2100. Evolution with time of quantum states and average values.
2101. The time-energy uncertainty relation in nonrelativistic quantum theory.
2102. Spin is a form of angular momentum without a classical equivalent.
2103. Pauli matrices: Three 2 by 2 matrices with eigenvalues +1 and -1.
2104. Quantum Entanglement: The singlet and triplet states of two electrons.
2105. Bell's inequality is violated for the singlet state of two electron spins.
2106. KS theorem: Any definite values would violate physical relations (1967).
2107. Mach-Zehnder interferometer by Zehnder (1891) & Ludwig Mach (1892).
2108. Generalizations of Pauli matrices beyond spin ½.
2109. Density operators are quantum representations of imperfectly known states.
2110. Kubo-Martin-Schwinger (KMS) condition.
2111. Quantum stability of ordinary matter depends on electrons being fermions.

      The Schrödinger Equation

2112. Hamilton's analogy equates the principles of Fermat and Maupertuis.
2113. Box confinement by a finite potential in one dimension and 3 dimensions.
2114. Rotator: Quantization of the angular momentum.
2115. Harmonic oscillator.
2116. Coulomb potential: Classification of chemical orbitals.
2117. Wallis formula for p (1655). A quantum-mechanical derivation (2015).
2118. Any second-order linear equation reduces to Schrödinger's equation.
2119. Perturbative solution entails a divergent asymptotic series.

      Counting Photons Nondestructively (Haroche Experiment)

2120. Biographical facts about Serge Haroche: Genealogy, family and career.
2121. The Einstein box: A thought-experiment discussed by Einstein and Bohr.
2122. Micromaser: A closed cavity doesn't require perfect mirrors.
2123. Purcell effect: Enhancement of spontaneous fluorescence in a cavity.
2124. Quantum nondemolition measurements (QND) are almost classical.
2125. Circular Rydberg atom. The outer electron is in orbit at a large distance.
2126. Rydberg clock: Two superposed Rydberg states look like a rotating dipole.
2127. Jaynes-Cummings model (1963): 2-level atom in an optical cavity.
2128. Fabry-Pérot cavity.
2129. Superconductivity of Niobium.
2130. Fock basis. Quantum states where the number of photons is well-defined.
2131. The electromagnetic phase is the conjugate of the number of photons.
2132. Coherent states in electromagnetism (Roy J. Glauber, 1963).
2133. Light-shift effect (Claude Cohen-Tannoudji, 1961).
2134. Rabi escillations at 50 kHz (I.I. Rabi).
2135. Ramsey Interferometer (Norman Ramsey, 1949).
2136. Circuit QED. A solid-state counterpart of Cavity QED.
2137. Questions and Answers. Discussions related to the Haroche experiments.

      Quantum Field Theory (QFT)

2138. The Lamb shift. The original motivation for renormalization.
2139. Quantum Electrodynamics (QED) is the simplest quantum field theory.
2140. Second Quantization: Particles are modes of a quantized field.
2141. Elementary particles: Quarks and leptons. Vector bosons and graviton.
2142. Composite hadrons: Zoo of mesons and baryons begotten by QCD.
2143. Bethe-Salpeter Equation: A relativistic equation for bound-state problems.
2144. Path-integral formulation.
2145. Renormalization. The renormalizability of a theory is a key requirement.
2146. S-matrix: The unitary transformation chaacterizing a scaterring process.
2147. Constructive quantum field theory. Making quantum theory relativistic.
2148. Wightman axioms. Traditional basis for constructive quantum field theory.
2149. Mass gap. Infimum of the energies of states orthogonal to the vacuum.
2150. Coleman-Mandula theorem (1967).
2151. Haag-Lopuszanski-Sohnius theorem (1975).
2152. The Higgs mechanism gives elementary particles an intrinsinc mass.
2153. Yukawa interactions mediated by massive spinless particles.

      Matter and Antimatter

2154. Klein-Gordon equation (1926). A second-order relativistic wave equation.
2155. Dirac wave equation (1928). This first-order equation predicts antimatter.
2156. Discovery of the positron by Carl Anderson (1930). The first antiparticle.
2157. Crossing symmetry: Antimatter is like matter going backward in time.
2158. Excess of matter over antimatter in the early Universe.

      From Ancient Alchemical Recipes to Modern Chemistry

2159. Measuring chemical stuff in moles (mol) makes stoichiometry obvious.
2160. Modern distillation (alembic = still-head) is due to Mary the Jewess.
2161. The retort was a prominent tool of alchemists and chemists for centuries.
2162. Production and distillation of alcohol. Its origins and limitations.
2163. Black powder: An ancient explosive, still used as a propellant (gunpowder).
2164. Predicting explosive reactions: A useful but oversimplified rule of thumb.
2165. Thermite generates temperatures hot enough to weld iron.
2166. Enthalpy of Formation: The tabulated data which gives energy balances.
2167. Exothermic crystallization of sodium acetate trihydrate ("hot ice").
2168. Gibbs Function (free energy): Its sign tells the direction of spontaneity.
2169. Berthollet's Law of Mass Action governs every chemical equilibrium.
2170. Labile is not quite the same as unstable.
2171. Inks: India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.
2172. Traditional pigments: Carbon black, vermillion, brazilin, malachite, etc.
2173. Beeswax is dominated by a long-chain ester (a "wax") called mycerin.
2174. Pine pitch & cedar pitch: Two similar products with different properties.
2175. Gum Arabic: The magic bullet of ancient chemistry.
2176. Ancient acids: From vinegar and lemon juice to vitriolic acid and more.
2177. Gold Chemistry: Aqua regia ("Royal Water") dissolves gold and platinum.
2178. Who was the "father" of modern chemistry?

      Acids & Bases

2179. History of acidity: Mineral acids and organic acids.
2180. Muriatic acid or hydrochloric acid (HCl) is a strong acid.
2181. Sulfuric acid: Strong diprotic acid with weak second dissociation.
2182. Nitric acid ( aqua fortis ).
2183. Carboxylic acids: The weak organic acids (formic, acetic, etc.).
2184. Picric acid.
2185. Sulfonic acids are stronger than their carboxylic counterparts.
2186. Sulfamic acid.
2187. Bronsted acids: are donors of protons (1923).
2188. Lewis acids are acceptors of electron-pairs (1923).
2189. Hammett acidity function.
2190. Superacids: Stronger than pure sulfuric acid.
2191. Superbases have greater affinities for protons than the hydroxide ion.
2192. What is the strongest acid?

      Water & Aqueous Solutions

2193. Pure liquid water includes hydronium and hydroxide ions.
2194. In dilute solutions, the activity of water molecules is nearly constant.
2195. Arrhenius acids protonate water (Arrhenius, 1884).
2196. pH = -log( [H3O+] ) was introduced by Søren P.L. Sørensen in 1909.
2197. Diprotic acids have the same pH effect via either protonation.
2198. Polyprotic acids and bases and their associated dissociation polynomials.
2199. Titration curve: Sharp pH change at an acid-base neutralization point.
2200. Buffer solutions.
2201. Sulfurous acid: Sulfur dioxide solution, containing bisulfite and sulfite ions.

      Organic Chemistry

2202. Birth of organic chemistry: Urea was first made chemically in 1828.
2203. Aliphatic saturated hydrocarbons are called alkanes.
2204. Unsaturated hydrocarbons feature some carbons tied by multiple bonds.
2205. Acetylene and alkynes feature carbon atoms linked by a triple bond.
2206. Aomatic compounds feature 6 coplanar carbons in a ring.
2207. Carbohydrates (glucids) can be decomposed into carbon and water.
2208. Functional groups determine the basic reactions of organic chemistry.
2209. Oxocarbons (oxides of carbon). Organic chemistry without hydrogen.

      Ionization, Oxidation-Reduction and Electrochemistry

2210. The oxidation number increases by oxidation and decreases by reduction.
2211. Salt bridges put solutions in electrical contact but prevent transfers of ions.
2212. Nernst equation: The voltage induced by different concentrations.
2213. Redox Reactions: Oxidizers are reduced by accepting electrons...

      Hands-on Chemistry

2214. Chemistry set from a bygone era (if memory serves).
2215. Basic glassware: Flasks, funnels, tubes, bulbs, condensers, etc.
2216. PTFE = Polytetrafluoroethylene = Teflon®.
2217. Ground-glass joints: Glass-to-glass conical joints have a 1:10 taper.
2218. Titration. Measuring the concentration of a reactant.
2219. Acidity color indicators. From litmus (1300) to universal indicators (1933).
2220. Methylene blue as a redox indicator. The blue bottle experiment.
2221. Sugar dehydration with sulfuric acid. Black snake experiment.
2222. Waterlock: 1 g of sodium polyacetate can hold 825 mL of water.
2223. Negative-X: Water ignites a mixture of zinc and ammonium nitrate.
2224. Nitrogen triiodide Is an extremely unstable explosive when dry.

      Medicine by the Numbers

2225. The normal body temperature is 37°C (98.6°C) or is it?
2226. Normal blood pressure. Systolic (max.) and diastolic (min.) pressures.
2227. Normal pulse. 1 Hz (one hertz) is 60 beats per minute.
2228. Blood circulation (1628). Discovered by William Harvey (1578-1657).
2229. Respiration is a form of combustion (Lavoisier and Laplace, 1780).
2230. Normal caloric intake. 100 W of power is about 2065 kcal/day.
2231. International Unit (IU). Arbitrarily-defined rating of biological activity.
2232. Concentration is an amount (either mass or moles) per volume.
2233. Glycosylated hemoglobin (HbA1c) relates to average blood glucose (bG).
2234. Human fat. Gaining and losing weight. Metabolism.
2235. Body Mass Index (BMI, in kg/m²). Weight divided by square of height.
2236. Medical abbreviations commonly used in prescriptions and elsewhere.
2237. Mosquito-borne diseases: Malaria, dengue, chikungunya, etc.
2238. Propagation of epidemics. Analytic solution of the basic SIR model.
2239. Pandemic: To save the lives of others, get infected as late as possible...
2240. Caffeine is the most widely consumed psychotropic substance.

      General Relativity

2241. Deflection of starlight by the Sun. What does General Relativity change?
2242. The Harress-Sagnac effect. Observer rotating with an optical loop.
2243. Relativistic rigid motion. Equilibrium modified at the speed of sound.
2244. In the Euclidean plane: Contravariance and covariance.
2245. In the Lorentzian plane: Contravariance and covariance revisited.
2246. Tensors of rank n+1 are linear maps that send a vector to a tensor of rank n.
2247. Signature of the quadratic form defined by a given metric tensor.
2248. Covariant and contravariant coordinates of rank-n tensors, in 4 dimensions.
2249. The metric tensor and its inverse. Lowering and raising indices.
2250. Partial derivatives along contravariant or covariant coordinates.
2251. Christoffel symbols: Coordinates of the partial derivatives of basis vectors.
2252. Covariant derivatives. Absolute differentiation. The nabla operator Ñ.
2253. Contravariant derivatives: The lesser-known flavor of absolute derivatives.
2254. The antisymmetric part of Christoffels symbols form a fundamental tensor.
2255. Totally antisymmetric spacetime torsion is described by a vector field.
2256. Levi-Civita symbols: Antisymmetric with respect to any pair of indices.
2257. Einstein's equivalence principle implies vanishing spacetime torsion.
2258. Ricci's theorem: The covariant derivative of the metric tensor vanishes.
2259. Curvature: The Ricci tensor is a contraction of the Riemann tensor.
2260. The Bianchi identity shows that the Einstein tensor is divergence-free.
2261. The Weyl tensor is the traceless component of the Riemann tensor.
2262. Stress tensor: Flow of energy density is density of [conserved] momentum.
2263. Einstein's Field Equations: 16 equations in covariant form (Einstein, 1915).
2264. Free-falling bodies: Their trajectories are geodesics in curved spacetime.
2265. Gravitational lensing. How gravitation bends light.
2266. The "anomalous" precession of Mercury's perihelion is entirely relativistic.
2267. Schwarzschild metric: The earliest exact solution to Einstein's equations.
2268. van Stockum dust: A metric with closed timelike curves (Lanczos, 1924).
2269. What is mass?
2270. Unruh temperature experienced by an accelerating observer.
2271. Electromagnetism: Covariant expressions, using tensors.
2272. Kaluza-Klein theory of electromagnetism involves a fifth dimension.
2273. Harvard Tower Experiment: The slow clock at the bottom of the tower.
2274. Shapiro time delay: Effect on radar signals of gravitational time dilation.
2275. Warp drive. Traveling faster than light (FTL) ?
2276. Frame dragging. The effect predicted by Lense and Thirring in 1918.
2277. The Poynting-Robertson drag: Dust spirals inward around a bright star.

      Relativistic Gravity & Gravitational Waves

2278. Changes in the quadrupole moment of mass cause gravitational waves.
2279. The energy carried by G-waves first betrayed their actual existence (1974).
2280. PDH technique. Optical stabilization of the frequency of laser light (1983).
2281. LIGO: Laser Interferometer Gravitational-wave Observatory (since 1992).
2282. 2017 Nobel Prize for the detection of gravitational waves by LIGO.

      Cosmology 101

2283. Olbers' paradox: The night sky isn't brightly lit. Why?
2284. Mach's principle: Local physics is determined by the whole Universe.
2285. Kant's Island Universes: The Universe is filled with separate galaxies.
2286. The Cosmological Principle: The Universe is homogeneous and isotropic.
2287. Medium-scale structure of the Universe: The foamy distribution of galaxies.
2288. The Big Bang: An idea of Georges Lemaître mocked by Fred Hoyle.
2289. The Cosmic Microwave Background (CMB): Its spectrum and density.
2290. Primordial nuclear soup: Hydrogen, Helium, Deuterium and Lithium.
2291. Cosmic redshift (z) of light from a Universe (1+z) times smaller than now.
2292. Multiple choices and misguided explanations for cosmic redshifts.
2293. Hubble Law relates redshift and distance for comoving points.
2294. Omega (W): The ratio of the density of the Universe to the critical density.
2295. Friedmann universe (1922). Its observed accelerating expansion (1998).
2296. Look-Back Time: The time elapsed since observed light was emitted.
2297. Distance: In a cosmological context, there are several flavors of distances.
2298. Comoving points follow the expansion of the Universe.
2299. The Anthropic Principle: An unsatisfactory type of absolute constraint.
2300. Dark matter & dark energy: Gravity betrays the existence of dark stuff.
2301. The Pioneer Anomaly: The anomalous braking of the Pioneer spaceprobes.

      Galaxies and large-scale structure of the Universe

2302. The Milky Way is the name given to the star system that harbors our Sun.
2303. The local group is dominated by the Milky Way & Andromeda galaxies.
2304. The virgo cluster dominates our corner of the Universe.
2305. Superclusters are the largest objects in the Universe.
2306. Spiral galaxies. Andromeda and the Milky Way are examples..
2307. Ring galaxies a rare type of galaxies, difficult to classify.
2308. Elliptical galaxies Old galaxies. The largest we know is IC 1101.
2309. Supermassive black holes at the core of almost all galaxies...

      Stars and Stellar Objects

2310. Nuclear fusion is what powers the stars.
2311. Brown dwarves glow from gravitational contraction. Fusion isn't ignited.
2312. Red dwarves can burn hydrogen for trillions of years.
2313. The Jeans mass. Above it, gases at temperature T collapse by gravitation.
2314. Main sequence: The evolution of a typical star.
2315. Metallicity (Z) measures the abundance of all elements beyond helium.
2316. The earlest stars (Population III) came from what the Big Bang produced.
2317. Eta Carinae and hypergiants. The most massive stars possible.
2318. Betelgeuse and red supergiants.
2319. Mira (Omicron Ceti). A very peculiar variable star.
2320. Rigel and blue supergiants.
2321. Planetary nebulae: Aftermaths of stellar explosions.
2322. White dwarfs: The ultimate fate of our Sun and other small stars.
2323. Neutron stars: Remnants from the supernova collapse of medium stars.
2324. Remnants of novae and supernovae associated with dated events.
2325. Stellar black holes form when supermassive stars run out of nuclear fuel.
2326. Primordial black holes. Substellar black holes left over from the Big Bang.
2327. Binary stars: Pairs of unlike stars often gravitate around each other.
2328. Binary X-ray source: A small accretor in tight orbit around a donor star.

      The Solar System

2329. Astronomical unit (au). Successive definitions of a standard unit of length.
2330. Mean distance between the Sun and the Earth.nbsp; A tad above 1 au.
2331. Parsec: Triangulating interstellar distances, using the motion of the Earth.
2332. The solar corona is a very hot region of rarefied gas.
2333. The Carrington solar storm started on the 28th of August 1859.
2334. Solar radiation: The Sun has radiated away about 0.03% of its mass.
2335. The Titius-Bode Law: A numerical pattern in solar orbits?
2336. The 4 inner rocky planets: Mercury, Venus, Earth, Mars.
2337. Earth and Moon: This is home.
2338. The asteroid belt: Planetoids and bolids between Mars and Jupiter.
2339. The 4 outer giant gaseous planets: Jupiter, Saturn, Uranus, Neptune.
2340. Chiron and other centaurs have decaying orbits between the giant planets.
2341. Discovery of Neptune: Urbain Le Verrier scooped John Couch Adams.
2342. Pluto and other Kuiper Belt Objects (KBO).
2343. Sedna and other planetoids beyond the Kuiper Belt.
2344. What's a planet? The latest definition excludes Pluto.
2345. Heliosphere and Heliopause: The region affected by solar wind.
2346. Oort's Cloud is a cometary reservoir at the fringe of the Solar System.
2347. 'Oummuamua and Borisov. Interstellar objects visiting the Solar system.

      Chaos Theory

2348. Long-term stability of the Solar system.
2349. The Butterfly Effect in Meteorology. (Edward N. Lorenz, 1962).
2350. The Feigenbaum constants: Bifurcation structure in the onset of chaos.

      Practical Formulas

2351. Easy conversion between Fahrenheit and Celsius scales: F+40 = 1.8 (C+40)
      Automotive :
2352. Fuel efficiency: (miles per gallon) × (liters per 100 km) = 235.21458333...
2353. Car speed is proportional to tire size & engine rpm, divided by gear ratio.
2354. 0 to 60 mph in 4.59 s, may not always mean 201.96 feet.
2355. Car acceleration. Guessing the curve from standard data.
2356. "0 to 60 mph" time, obtained from vehicle mass and actual average power.
2357. Thrust is the ratio of power to speed [measured along direction of thrust].
2358. Power as a function of chamber size for internal combustion engines.
2359. Optimal gear ratio to maximize top speed on a flat road (no wind).
      Surface Areas :
2360. Heron's formula (for the area of a triangle) is related to the Law of Cosines.
2361. Archimedes' formula predates Heron by three centuries.
2362. L'Huilier's formula: Generalizing Heron's formula to spherical triangles.
2363. Brahmagupta's formula gives the area of a quadrilateral (cyclic or not).
2364. Bretschneider's formula for a quadrilateral of given sides and diagonals.
2365. Vectorial area of a quadrilateral: Half the cross-product of its diagonals.
2366. Parabolic segment: 2/3 the area of circumscribed parallelogram or triangle.
      Volumes :
2367. Content of an horizontal cylindrical tank. given the height of the liquid in it.
2368. Volume of a spherical cap, or content of an elliptical vessel, at given level.
2369. Content of a cistern (cylindrical with elliptical ends), at given fluid level.
2370. Volume of a cylinder or prism, possibly with tilted [nonparallel] bases.
2371. Volume of a conical frustum: Formerly a staple of elementary education...
2372. Volume of any frustum when horizontal area is a cubic function of height.
2373. Volume of a sphere... obtained by subtracting a cone from a cylinder !
2374. Cavalieri's quadrature formula for the area under a power curve.
2375. The volume of a tetrahedron is the determinant of three edges, divided by 6.
2376. Volume of a wedge of a cone.
      Averages :
2377. Filling a cistern with several canals. Problem 26 in the Jiuzhang Suanshu.
2378. Splitting a job evenly between two unlike workers.
2379. Splitting a job unevenly between two unlike workers.
2380. Mixing solutions to obtain a predetermined intermediate rating.
2381. Alcohol solutions are rated by volume not by mass.
2382. Mixing alcohol solutions to obtain an exact rating by volume (ABV).
2383. Vinegar is compared to a volumetric mix of pure acetic acid and water.
2384. Preparing solutions with hydrate salts. Example of citric acid.
2385. Special averages: harmonic (for speeds), geometric (for rates), etc.
2386. Mean Gregorian month: either 30.436875 days, or 30.4587294742534...
2387. The arithmetic-geometric mean is related to a complete elliptic integral.
      Geodesy and Astronomy :
2388. Distance to the horizon is proportional to the square root of your altitude.
2389. Superior mirages. How objects beyond the horizon are brought into view.
2390. Distance between two points on a great circle at the surface of the Earth.
2391. Euclidean distance between two cities, along a line through the Earth.
2392. Geodetic coordinates: Point of elevation h at latititude j and longitude q.
2393. The figure of the Earth. Geodetic and geocentric latitudes.
2394. Area of a spherical polygon. How to apply Girard's formula.
2395. Kepler's Third Law: The relation between orbital period and orbit size.

      Philosophy and Science

2396. Creation and Discovery in Science.
2397. Search for Extraterrestrial Intelligence. If we listen, we must talk.
2398. The Anthropic Principle: The laws of physics must allow human life.
2399. Science and Politics: Political support for Science makes a society worthy.
2400. What's Mathematics anyway? The groundwork of scientific knowledge.

      Evolution of Life on Earth

2401. Life: The mysteries of evolution.
2402. Hominid speciess. The human family tree.
2403. Fossil calendars: 420 million years ago, a month was only 9 short days.
2404. LUCA: Last Universal Common Ancestor. A deep-sea creature.
2405. Geologic Time Scale (GTS): Beyond all human calendars.
      

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      Below are topics not yet integrated with the rest of this site's navigation.

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      The Unexplained

2406. The Magnetic Field of the Earth.
2407. Life: The origins of life on Earth.
2408. Extraterrestrial life Is there intelligence out there?
2409. Nemesis: A distant companion of the Sun could cause periodical extinctions.
2410. Current Challenges to established dogma.
2411. Unexplained artifacts and sightings.

      Open Questions (or tough answers)

2412. The Riemann Hypothesis: {Re(s) > 0 & z(s) = 0} Þ {Re(s) = ½}.
2413. Twin prime conjecture: One of the oldest open mathematical questions.
2414. Goldbach conjecture (1742): Any even number >2 is a sum of two primes.
2415. Legendre's conjecture There are primes between squares.
2416. Landau's fourth problem; Infinitely many primes follow a square.
2417. Schnizel's hypothesis H. Polynomials simultaneously prime infinitely often.
2418. P = NP ? Can we find in polynomial time what we can check that fast?
2419. Collatz sequences go from n to n/2 or (3n+1)/2. Do they all lead to 1?
2420. The Poincaré Conjecture (1904). Proven by Grisha Perelman in 2002.
2421. Fermat's Last Theorem (1637). Proven by Andrew Wiles in 1995.
2422. The ABC conjecture (1985) of Joseph Oesterlé and David Masser.
2423. The union-closed sets conjecture (1979) of Péter Frankl.
2424. The Hadwiger conjecture in graph theory (1943).
2425. The Egyptian conjecture (1948) of Straus and Erdös.
2426. Birch and Swinnerton-Dyer conjecture about the rank of an elliptic curve.
2427. Gilbreath's conjecture was first formulated by François Proth in 1878.
2428. g-conjecture of McMullen (1970). Proved by Karim Adiprasito in 2018.
2429. Positive C2-diffeomorphisms on the circle. Do they form a simple group?
2430. Hadwiger-Nelson colorings are such that 2 like points are never 1 unit apart.

      Mathematical Miracles

2431. The only magic hexagon.
2432. The law of small numbers applied to conversion factors.
2433. Quadratic formulas yielding long sequences of prime numbers.
2434. The area under a Gaussian curve involves the square root of p
2435. Exceptional simple Lie groups.
2436. Monstrous Moonshine in Number Theory.

      Trivia

2437. Oldest open mathematical problem: Are there any odd perfect numbers?
2438. Magnetic field of the Earth: South side is near the geographic north pole.
2439. From the north side, a counterclockwise angle is positive by definition.
2440. What initiates the wind? Well, primitive answers were not so wrong...
2441. Why "m" for the slope of a linear function y = m x + b ? [in US textbooks]
2442. The diamond mark on US tape measures corresponds to 8/5 of a foot.
2443. Naming the largest possible number, in n keystrokes or less (Excel syntax).
2444. The "odds in favor" of poker hands: A popular way to express probabilities.
2445. Reverse number sequence(s) on the verso of a book's title page.
2446. Living species: About 1400 000 have been named, but there are many more.
2447. Dimes and pennies: The masses of all current US coins.
2448. Pound of pennies: The dollar equivalent of a pound of pennies is increasing!
2449. Nickels per gallon: Packing more than 5252 coins per gallon of space.

      Geography, Geographical Trivia

2450. Geodetic coordinates, based on the Reference Ellipsoid defined in 1980.
2451. Geocentric coordinates are almost never used in geography or astronomy.
2452. Distance from the center of the Earth to points located at the surface.
2453. The volume of the Grand Canyon: 2 cm (3/4") over the entire Earth.
2454. The Oldest City in the World: Damascus or Jericho?
2455. USA (States & Territories): Postal and area codes, capitals, statehoods, etc.

      Money, Currency, Precious Metals

2456. Inventing Money: Brass in China, electrum in Lydia, gold and silver staters.
2457. Prices of Precious Metals: Current market values (Gold, Silver. Pt, Pd, Rh).
2458. Medieval system: 12 deniers to a sol.
2459. Ancien Régime French monetary system.
2460. British coinage before decimalization.
2461. Exchange rates when the euro was born.
2462. Worldwide circulation of currencies.

      Bitcoin (BTC) and other crypto-currencies

2463. What are bitcoins?
2464. Who invented bitcoins? Who is Satoshi Nakamoto ?
2465. Blockchain, hashpuzzles, Merkle trees, public/private-key cryptography.

      The Counterfeit Penny Problem

2466. Counterfeit Coin: In 3 weighings, find an odd object among 12, 13 or 14.
2467. Counterfeit Penny Problem: Find an odd object in the fewest weighings.
2468. Explicit tables for detecting one odd marble among 41, in 4 weighings.
2469. Find-a-birthday: Detect an odd marble among 365, in 6 weighings.
2470. Error-correcting codes for ternary numeration.
2471. If the counterfeit is known to be heavier, fewer weighings may be sufficient.

      Calendars & Chronology

2472. Calendrical ratios and their slow evolution with time.
2473. Julian Day Number (JDN). Counting days in the simplest of all calendars.
2474. The Week has not always been a period of seven days.
2475. Egyptian year of 365 days: Back to the same season after over 1500 years.
2476. Heliacal rising of Sirius: Sothic dating.
2477. Coptic Calendar: Reformed Egyptian calendar based on the Julian year.
2478. The Julian Calendar: Year starts March 25. Every fourth year is a leap year.
2479. Anno Domini: Counting roughly from the birth of Jesus Christ.
2480. Gregorian Calendar: Multiples of 100 not divisible by 400 aren't leap years.
2481. Counting the days between dates, with a simple formula for month numbers.
2482. Age of the Moon, based on a mean synodic month of 29.530588853 days.
2483. American names of full moons are inherited from Algonquian tribes.
2484. Blue Moon is a term which has been defined in two different ways.
2485. Easter Sunday is defined as the first Sunday after the Paschal full moon.
2486. The Muslim Calendar: The Islamic (Hijri) Calendar (AH = Anno Hegirae).
2487. The Jewish Calendar: An accurate lunisolar calendar, set down by Hillel II.
2488. Zoroastrian Calendar.
2489. The Zodiac: Zodiacal signs and constellations. Precession of equinoxes.
2490. The Iranian Calendar. Solar Hejri [SH] or Anno Persarum [AP].
2491. Enkutatash (Sept. 11 or 12) of Julian year N begins Ethiopian year N-7.
2492. The Chinese Calendar.
2493. The Japanese Calendar.
2494. Mayan System(s): Haab (365), Tzolkin (260), Round (18980), Long Count.
2495. Indian Calendar: The Sun goes through a zodiacal sign in a solar month.
2496. Post-Gregorian Calendars: Painless improvements to the secular calendar.

      Roman Numerals (Archaic, Classic and Medieval)

2497. Roman Numeration: Ancient rules and medieval ones.
2498. An easy conversion table for numbers up to 9999.
2499. Larger Numbers, like 18034...
2500. Extending the Roman system.
2501. The longest year so far, in terms of Roman numerals.
2502. IIS (or HS) is for sesterce (originally, 2½ asses, "unus et unus et semis").
2503. Roman fractions. A rudimentary duodecimal system.

      Humor

2504. Standard jokes. Smile!
2505. Limericks.
2506. Proper credit may not always be possible.
2507. Trick questions can be legitimate ones.
2508. Ignorance is bliss: Why not read all that mathematical stuff faster ?
2509. Silly answers to funny questions.
2510. Why did the chicken cross the road? Scientific and other explanations.
2511. Humorous or inspirational quotations by famous scientists and others.
2512. One great quote to be translated into as many languages as possible.
2513. Famous Last Words: Proofs that the guesses of experts are just guesses.
2514. Famous anecdotes.
2515. Parodies, hoaxes, and practical jokes.
2516. Omnia vulnerant, ultima necat: The day of reckoning.
2517. Funny Units: A millihelen is the amount of beauty that launches one ship.
2518. Funny Prefixes: A lottagram is many grams; an electron is 0.91 lottogram.
2519. The Lamppost Theory: Only look where there's enough light.
2520. Is it insanity or just a viable alternative to orthodoxy?
2521. Anagrams: Rearranging letters may reveal hidden meanings. ;-)
2522. Mnemonics: Remembering things and/or making fun of them.
2523. Acronyms: Funny ones and/or alternate interpretations of serious ones.
2524. Usenet Acronyms: If you can't beat them, join them (and HF, LOL).

      Scientific Symbols and Icons

2525. Adobe's Symbol font: Endangered standard HTML mathematical symbols.
2526. The equality symbol ( = ). The "equal sign" dates back to the 16th century.
2527. The double-harpoon symbol denotes chemical equilibrium.
2528. Line components: Vinculum, bar, solidus, virgule, slash, macron, etc.
2529. The infinity symbol ( ¥ ) introduced in 1655 by John Wallis (1616-1703).
2530. Transfinite numbers and the many faces of mathematical infinity.
2531. Chrevron symbols: Intersection (highest below) or union (lowest above).
2532. Disjoint union. Square "U" or inverted p symbol.
2533. Blackboard bold: Doublestruck characters denote sets of numbers.
2534. The integration sign ( ò ) introduced by Leibniz at the dawn of Calculus.
2535. Evaluation bar. Difference betwen the values of an expression at two points.
2536. The end-of-proof box (or tombstone) is called a halmos symbol (QED).
2537. Two "del" symbols: ¶ for partial derivatives, and Ñ for Hamilton's nabla.
2538. A strange lowercase p (Ã) used only for Weierstrass elliptic functions.
2539. The rod of Asclepius: Medicine and the 13th zodiacal constellation.
2540. The Caduceus: Scepter of Hermes, symbol of commerce (not medicine).
2541. Tetractys: Mystical Pythagorean symbol, "source of everflowing Nature".
2542. The Borromean Rings: Three interwoven rings which are pairwise separate.
2543. The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.
2544. Dangerous-bend symbol: Introduced by Bourbaki, popularized by Knuth.
      
      Gerard Michon
      

      ---

      Monographs and Complements

      ---

2545. About Zero.
2546. Wilson's Theorem.
2547. Counting Polyhedra: A tally of polyhedra with n faces and k edges.
2548. Sagan's number: The number of stars, compared to earthly grains of sand.
2549. The Sand Reckoner: Archimedes fills the cosmos with grains of sand.
      
      Gerard Michon
      

      ---

      Numericana Hall of Fame

      ---

2550. Numericana's list of distinguished Web authors in Science...
2551. Giants of Science: Towering characters in Science history.
2552. Two Solvay conferences helped define modern physics, in 1911 and 1927.
2553. Physical Units: A tribute to the late physicist Richard P. Feynman.
2554. The many faces of Nicolas Bourbaki (b. January 14, 1935).
2555. Lucien Refleu (1920-2005). "Papa" of 600 mathematicians. [ In French ]
2556. Taupe Laplace. [ In French ].
2557. Roger Apéry (1916-1994) and the irrationality of z(3).
2558. Hergé (1907-1983): Tintin and the Science of Jules Verne (1828-1905).
2559. Other biographies: Dulong, Galois, Tannery, Vessiot, Drach, Glénisson...
2560. Escutcheons of Science (Armorial): Coats-of-arms of illustrious scientists.
      • Supplementary Armorial for scientists, engineers, explorers...
      
      Gerard Michon
      

      ---

      In-Depth Reviews of Great Products

      ---

      Basic Stamp from Parallax, Inc.

2561. BASIC Stamp® HomeWork Board™ USB: Review and first baby steps.
2562. Decoding a digital PWM signal with a low-pass filter.
2563. Volume control obtained by attenuating a PWM signal before filtering.
2564. Using an LCD display (HD44780) without built-in PBASIC commands.
2565. A keypad for independence: Cutting off the umbilical USB cord.
2566. Cheapest sonar sensor uses two I/O pins (where one would suffice).
2567. The I2C bus. Example: Interface with a real-time clock (RTC).

      Arduino Mictocontroller Boards.

2568. Genuine Arduino starter kit: All you need for a hands-on guided tour.
2569. The Arduino Uno: The unit which started it all (now in Revision 3 ).
2570. Arduino Leonardo & Micro don't use separate processors for USB.
2571. Chameleon: An Arduino clone and a Parallax Propeller on one board.
2572. Pro and cons of Arduino clones: Avoid them as a first purchase.

      HP Prime Graphing Calculator. Hewlett-Packard's flagship:

2573. Buyer beware: G8A92AA has wireless connectivity, NW80AA doesn't.
2574. Setup: Language, unique name, numerical and calendrical formats.
2575. Connectivity. How HP Prime calculators communicate with PCs.
2576. HP StreamSmart. Four-port data streamer capable of 5700 samples / s.
2577. Hacking the HP Prime. Possible DIY hardware modifications.
2578. Two calculators in one. A numerical workspace next to a separate CAS.
2579. Reverse Polish Notation (RPN) is only an option in the "Home" mode.
2580. The RPN stack is more than a read-only record of previous entries & results.
2581. Objet-oriented functions behave according to the nature of their arguments.
2582. CAS. The Computer Algebra System of Bernard Parisse.
2583. Apps. Blue icons denote graphical applications with similar structures.
2584. User functions. A single-expression definition may involve conditionals.
2585. Programming, using the HP Prime Programming Language (HP PPL).
2586. User-defined apps.
2587. Unit conversions. The HP Prime features 167 basic units and 20 prefixes.
2588. Constants of Nature. 23 built-in physical constants and many others.
2589. Lists and list-processing primitives. Not all list manipulations are possible.
2590. Vectors and matrices. Multidimensional linear algebra.
2591. Polynomials. Algebra on polynomials & special predefined polynomials.
2592. Calendrical functions. Dates are encoded as decimal numbers: yyyy.mmdd
2593. Special Functions. Beyond trigonometric and hyperbolic functions.
2594. Galois Fields. An almost undocumented buggy feature of the HP Prime.
2595. Easter Eggs. "Visit Plot Gallery".
2596. Kudos and Gripes. Likes and Dislikes. Deal-breakers. Bug reports.
2597. Fixed bugs and new kudos.

      Top HP Programmable Calculators: HP-49g+ and HP-50g

2598. Printer : The HP 82240B thermal printer has been standard since 1989.
2599. Modifier keys. Lesser-used functions require several keystrokes.
2600. Infinity: Unsigned algebraic infinity and signed topological infinities.
2601. Physical units: A built-in feature inherited from the HP-28 (1986).
2602. Bug reports: Severe problems and minor ones.
2603. Complex functions: Complex values & arguments. Complex variables.
2604. RPL programming ("Reverse Polish LISP") originated with the HP-28.
2605. Easter eggs: Unofficial features, play Tetris® or MineHunt.

      HP-35s: Released on the 35^th birthday of the HP-35

2606. Modifier keys. Lesser-used functions require several keystrokes.
2607. Reverse Polish Nptation (RPN) was invented by Jan Lukasiewicz in 1924.
2608. Unit conversions: °F/°C | HMS | °/rad | lb/kg | mi/km | in/cm | gal/L.
2609. 40 physical constants (and one mathematical constant) in a single menu.
2610. Bug reports: Severe problems and minor ones.
2611. Complex functions: Complex values & arguments. Complex variables.
2612. Programming: Recorded keystroke sequences. Tests, loops & subroutines.

      Great TI Calculators: TI-92, TI-92+, TI-89, Voyage 200

2613. Modifier keys. Lesser-used functions require several keystrokes.
2614. Physical units: A very nice afterthought, with a few rough edges.
2615. Analytical functions may present discontinuity cliffs in the complex realm.
2616. Wrong! 0 to the 0th power should be 1. ¥ and +¥ shouldn't be equal.
2617. 68000 Assembly Programming: A primer without the help of an assembler.
2618. The clock frequency of your calculator measured with 0.1% accuracy.
2619. TI's BASIC. A built-in interpreted language not designed for speed.
2620. Pretty 2D algebraic displays can only be edited in their 1D version.

      Texas-Instruments Scientific Calculator: TI-36X Pro

2621. The keypad: One shift-key suffices with the introduction of multi-tap.
2622. Integer arithmetic. Numbers with more than 6 digits cannot be factorized.
2623. 20 pairs of unit conversions... and not a single inaccuracy. (That's rare!)
2624. 20 physical constants listed by name, with their units. 9 are numbered.
2625. Bug reports: From minor gripes to more severe flaws.

      Affordable Casio Calculators: fx-991ex vs. fx-115es PLUS

2626. Side-by-side comparison of the fx-991EX and fx-991ES PLUS models.
2627. fx-991ex. The EX series is an improvement on the best-selling ES series.
2628. History of the "natural" Casio scientific calculator ES series.
2629. Mode 4: Hexadecimal or octal arithmetic on 32-bit integers.
2630. Mode 7: Tabulate a function (or a pair of functions with "plus" version).
2631. Scientific constants: Consistent values recommended by CODATA (2010).
2632. Conversion factors between units: A few inaccuracies & one typo.

      Sharp EL-W516 Calculator (EL-W516x, EL-W516xBSL...)

2633. The line of Sharp scientific calculators. What the model numbers mean.
2634. The basics. Color-coded multiple functions.
2635. Entering and exiting special modes.
2636. Solving cubic equations.

      Canon's F-792SGA Calculator (2013) :

2637. The basics.
2638. The good and the bad.

      Nikon DX Photography :

2639. Nikon glossary. Terms, symbols and abbreviations used by Nikon.
2640. Evolution of Nikon DX Digital Cameras, from the D50 to the D500.
2641. Buyer's Guide to all Nikon DX cameras: D3x00, D5x00, D7x00 and D500.
2642. Frames per second: The D5500 can shoot at a top rate of exactly 5 fps.
2643. DX sensors: 23.46 x 15.64 mm (3.91m/p) or 23.49 x 15.66 mm (4.2 m/p).
2644. Back-button focusing. A good customization to make on any camera.
2645. White balance: Compensating for the color of incident light.
2646. Setting up a focus trap.
2647. Commander mode with SB-500 Speedlight, using third-party flash units.
2648. Remote control: Infrared (ML-L3) or cable-release (with timer).
2649. GPS modules communicate using the NMEA 0183 standard protocol.
2650. External power. Connector (AP-5a) & AC adapter (AH-5) are separate.
2651. Tripods & monopods: Basic tripod, versatile travel tripod, hiking stick.
2652. Camera straps.
2653. Camera bags and cases.
2654. A short history of Nikon lenses.
2655. Automatic extension tubes allow an ordinary lens to be used close-up.
2656. Attach a lens backwards to the camera body, or in front of another lens.
2657. Low-light normal lens: Nikon 35mm f/1.8G AF-S DX (54 mm reach).
2658. Fast portrait lens Nikon 50mm f/1.8G AF-S (77 mm DX reach).
2659. Long macro (130 mm DX reach): Nikon 85mm f/3.5G AF-S DX ED VR.
2660. Overview of other portrait lenses. Long lenses with wide apertures.
2661. Superzoom: Nikon 18-300mm f/3.5-6.3G AF-S DX VR (28-460 reach).
2662. Kit lens and compact telephoto zoom tandem: Nikon 18-55 and 55-300mm.
2663. Telephoto zoom AF-S Nikkor 200-500mm f/5.6E ED VR (307-767mm).
2664. Fast (f/2.8) Tokina wide zooms: 11-16mm (17-25) and 11-20mm (17-31).
2665. Ultrawide Samyang fisheye lens: Rokinon 8mm f/3.5 AE (for Nikon).
2666. Bayonet hoods don't interfere with native filters (without step-up rings).
2667. Telescope converter: Turning any F-mount lens into a spotting scope.
2668. Lens pinout: Electrical connections between Nikon bodies and lenses.
2669. Lens identifiers: Database of F-Mount Nikon-compatible digital lenses.
2670. EXIF: The hidden data attached to every JPEG the camera makes.

      Mamiya 645 medium-format film photography (1975).

2671. Mamiya 645 1000S. A complete vintage system on eBay.
2672. Mamiya-Sekor C lenses: Interchangeable lenses for the M645.
2673. Battery: Silver-oxide is usually better than lithium. Alkaline isn't so good.

      Eastman-Kodak's No.2 Brownie Camera (1901-1935).

2674. Meniscus lens. Wollaston landscape lens.
2675. No. 2 Brownie, Model F. The first all-metal Brownie (1924).

      Photographic Film: The soul of the twentieth century.

2676. Ongoing legacy of analog photography: Film is dead, long live Film.
2677. What makes silver so special. The miraculous chemistry of silver.
2678. The latent image. It's true nature remained a mystery for decades.
2679. The magic of photographic emulsions. How silver captures light.
2680. C-41 film processing and proof printing at different professional labs.
2681. "Develop and Scan" services kill the advantage of medium-format film.
2682. Film scanner: A 170ドル flatbed capable of scanning 120-film at 9600 dpi.
2683. Black-and-white film: Ilford, Kodak, Agfa (Rollei) and one Fuji offer.
2684. Rollei infrared film: Processing film based on a flimsy polyester base.
2685. Processing black-and-white film at home. Lots of room for creativity.
2686. Paterson System 4. Tanks, self-loading reels and force film washer.
2687. Developer. The most critical bath in photographic processing.
2688. Stop-bath. An acidic stop-bath puts an abrupt end to the developer's action.
2689. The fixer stabilizes the image by dissolving unused light-sensitive material.
2690. Washing with (bisulfite) salts makes a fresh-water rinse more efficient.
2691. Spotless drying can be achieved with a wetting agent in the final rinse.
2692. History of natural-color reproduction: The path to modern color films.
2693. Color negative films: Kodacolor, Ektar, Agfacolor, Fujicolor.
2694. C-41 processing of color negatives is best left to professional machines.
2695. Color reversal films (slides). Kodachrome (K-14). Ektachrome (E-6).

      Cinematography & Digital Video, with the Lumix GH5.

2696. Movies & Video. Capturing motion with a sequence of images.
2697. Video formats, resolution, color depth and bit rates.
2698. Video Lighting: Color balance, power. LED panel, COB and HMI.
2699. Camera support gear: Monopods and tripods.
2700. 3-axes motorized gimbal mount.
2701. Panasonic Lumix hybrids cameras. The GH5 offers great video features.
2702. Firmware updates and unlocks. GH5 camera, lenses and XLR audio adapter.
2703. Neutral density (ND) filters are essential when shooting video.
2704. Olympus M.Zuiko Digital ED 7-14mm f/2.8 Best MFT wide-zoom lens.
2705. Panasonic 8-18mm f:2.8-4.0. Best MFT lens of 2017, for many reviewers.
2706. Panasonic 12-60mm f:2.8-4.0. Most popular kit lens for the GH5.
2707. Lumix G Leica DG Nocticron 42.5mm f:1.2. Best lens for still portraits.
2708. Panasonic 50-200mm f:2.8-4.0. Best telephoto zoom for the GH5.
2709. Panasonic 100-400mm f:4.0-6.3. Super-telephoto zoom. 800 mm reach.
2710. Proper cinema lenses (no focus breathing). Veydra mini-primes.
2711. Flash strobes. Some TTL flash units for shooting stills with the GH5.
2712. Customizing a GH5 camera cage with the SmallRig system.
2713. DIY no-stich hand strap (with accessory pouch) for camera cages.
2714. Micro four-thirds (MFT or M43). Open standard for mirrorless cameras.
2715. Adapters & converters allow MFT cameras to use reflex-mount lenses.
2716. Serial Digital Interface (SDI). Coaxial cables and locking connectors.
2717. High-Definition Multimedia Interface (HDMI). The consumer standard.
2718. Atomos recorders/monitors: Shogun Inferno, Ninja Inferno, Ninja V.
2719. Monitors and colorimeters to deliver accurate color reproduction.
2720. CODEC standards. COmpression and DECpmpression of video signals.
2721. Look-Up Table (LUT). The ultimate customization of color information.
2722. Editing & post-production. Modifying clips and putting them together.
2723. Montage.
2724. Tips from successful YouTubers.
2725. Teleprompters. The key to smooth on-camera speech delivery.

      Recording Sound for Video (48 kHz / 24-bit)

2726. Microphones: A brief summary of our dedicated page on the topic.
2727. Evolution of sound recording. From soot, tinfoil and wax to 24-bit audio.
2728. Rooms acoustics. Suppressing outside noise. Trapping unwanted reverb.
2729. Reverbs and delays. Natural and synthetic reverberation and echoes.
2730. Pop filters. Pop screens suppress plosives and shield against spit...
2731. Microphone stands & booms. The most common thread mount is 5/8''-27.
2732. Audio connectors: Phone jacks (¼'', 3.5mm, 2.5mm) and balanced XLR.
2733. Microphone cable: Balanced line formed by a shielded twisted pair.
2734. Volume: Sound level, normalized to current broadcast standards.
2735. Equalization (EQ). Adjusting the frequency response for fidelity or effect.
2736. AGC, limiters and compressors: Best use of a limited dynamic range.
2737. Preamplifiers. Converting a microphone output into a usable line signal.
2738. Analog low-pass filtering is paramount before digitization of any signal.
2739. Analog-to-digital conversion (ADC).
2740. Digital-to-analog conversion (DAC).
2741. Generation loss. Signal degradation due to each ADC/DAC pair.
2742. Digital wireless systems: Sennheiser, Røde, Sony, Saramonic, Azden.
2743. The DMW-XLR1 provides the GH5 camera with two 24-bit audio channels.
2744. Analog mixer. Mixing sound from several audio sources in real-time.
2745. Direct insertion (DI) boxes intercept a clean signal between guitar and amp.
2746. Audio impedance transformers. Step-up to use a mic with a guitar amp.
2747. Match an output impedance with a load equal to its complex conjugate.
2748. Handy audio recorders: Zoom H1, H1n, H2n, H4n Pro, H5, H6 (and F1).
2749. Zoom H1 ultra-portable recorder: Single-channel and built-in stereo mic.
2750. Zoom H5 four-channel recorder: XLR inputs & interchangeable capsules.
2751. Audio interfaces between various audio signals and a digital computer.
2752. The Art of Slating: Clapboard conventions and etiquette.
2753. Soundtrack. Affordable licensing of music for amateurs.
2754. Foley, sound effects and sound design.
2755. Voice-over. Crafting spoken words off-camera. Dog-clicker editing.
2756. Digital audio workstation (DAW). Putting it all together...

      Microphones : Fundamental Issues and Buyer's Guide

2757. Sound What pressure waves entail. Thermodynamics of acoustics.
2758. Classifying microphones according to which physical quantity they exploit.
2759. Unified theory of inductive (dynamic) and condenser microphones.
2760. Noise-floor: What's left when the interesting stuff is gone.
2761. Microphone sensitivity (mV/Pa): voltage output to sound pressure input.
2762. Pickup patterns: Omnidirectional, figure-8, cardioid, and beyond...
2763. Dual-diaphragm microphones allow switchable or variable pickup patterns.
2764. Compound microphones with adjustable frequency response..
2765. Calibrated measurement microphones. Testing acoustics and audio gear.
2766. Attenuation pad switch allows a microphone to record louder sounds.
2767. High-pass filter (HPF). A switchable option built into many microphones.
2768. Characteristics of full-size wired microphones. Capacitive or inductive.
2769. Acoustic properties of large diaphragms. Resonances and pickup patterns.
2770. Large-diaphragm condenser microphones (LDC). Studio microphones.
2771. Dynamic microphones. Rugged stage microphones with limited bandwidth.
2772. Ribbon microphones. Special dynamic type, with output transformers.
2773. Small-diaphragm pencil microphones are often available in matched pairs.
2774. Shotgun microphones. Highly-directive narrow condenser microphones.
2775. Lavalier microphones (lapel mics) isolate the speaker from ambient sound.

Noted Numericana fans (and/or contributors) in alphabetical order:

• Max Alekseyev, Silent Circles.
• Stuart Errol Anderson, Enumeration of Polyhedra.
• Gottfried Barthel, Generic Carmichael Numbers.
• David Cantrell, Perimeter of an Ellipse (2001, 2004, etc.)
• Scott Cram, Magician (Grey Matters).
• Joe Crump, Carmichael Divisors.
• Paul Godfrey, Lanczos Formula.
• Michel Marcus, Hemiperfect Numbers (11/2, 13/2, 15/2, 17/2).
• Ed Pegg, Jr. Recreational Mathematics (MathPuzzle).
• Belisha Price, Polyhedra.
• Knud Thomsen, Surface Area of an Ellipsoid.
• Robin Whitty, Theorems. Jochen Wilke (b. March 19, 1960)
• Jochen Wilke, Escutcheons of Science.

Guest Authors:

• Lanczos Implementation of the Gamma Function by Paul Godfrey.

Public-Domain Texts:

• Preliminary report (1819) on the Weights & Measures reform of 1824 (UK).

• André Adler, dit Bresse (1907-2004) by Alain Dalançon (2020年12月26日).
• Numericana by School and College Listings (before 2020年06月12日).
• Extension of Wilson's theorem by PrimeNumberHobbyist (2019年02月21日).
• Relativistic temperature by my2cts (Physics StackExchange, 2018年07月20日).
• Favorite thin (content-heavy) websites by Tom Campbell (2017年01月07日).
• Mondrian art puzzles - Errata by Ed. Pegg, Jr. (2016年12月02日).
• Hadamard matrix of order 428 by Pradeep Mutalik (2015年12月22日).
• Homer Simpson's physics (h) by Karen Kaplan, LA Times (2015年03月05日). Homer Simpson
• Bachet Squares: Combinatorics, in Italian (before 2015年02月03日).
• Mathematical Tricks and Puzzles. Stanley Eigen, NU (2014年10月19日).
• O que é tonelada bruta? Ana Cristina Carneiro (bef. 2014年09月19日).
• Giants of Science (Facebook) Allan Tate (2014年09月13日).
• Penny problem: 13 coins & good coin. Quora (2014年09月12日).
• Coat-of-arms of Darwin (FB) Family Roots, Inc. (2014年09月10日).
• Kruskal Count (in Greek) by Athanasios Drougas (2014年09月10日).
• What are Gross Tons? by Latoya McGill (2014年08月14日).
• Schrödinger's Wave Equation by The Star Garden (before 2014年08月13日).
• Martin Gardner on Video by Gathering 4 Gardner (before 2014年08月12日).
• Wow, good design, color, workmanship. by Agent007 (2014年08月05日).
• Number 336 in car-speed formula by WVZR-1 (2014年08月03日).
• Primes dividing no Carmichael number? by Dane Bouchie (2014年07月30日).
• Profusion of units of energy by RobPhy (2014年07月30日).
• Approximating the perimeter of an ellipse by Buri (2014年07月10日).
• Niels Bohr's own coat-of-arms design by Clay Statue (bef. 2014年07月10日).
• Gold to silver value ratio (in gaming) by Orfamay Quest (2014年06月03日).
• Nicolas Bourbaki by Khaleed McClain (2014年05月27日).
• Geodesics on Tannery's Pear by David Joyce, Clark U. (2014年04月21日).
• Classifying polyhedra by Prof. Peter Magyar, MSU (2014年04月02日).
• Minesweeper versions: Minehunt on HP calculators (before 2014年03月14日).
• The Sagnac effect by Pancakes3 (2014年03月14日).
• Counting convex polyhedra by Achille Hui (2014年02月12日).
• History of Math Symbols and Heraldry (before 2014年02月07日).
• Relativistic Thermodynamics by Imad Ahmad Barghouthi (2014年02月02日).
• ADSL Filter by AG6QR (2014年01月22日).
• On the Perimeter of an Ellipse (pdf) by Paul Abbott (bef. 2014年01月21日).
• Peter Cameron's Blog: Sidebar (before 2014年01月21日).
• What are Stellated Polyhedra? by Michael Signal (before 2014年01月20日).
• South-East-North triangle puzzle by Bruce Hamilton (2014年01月05日).
• Coat-of-arms from companies (in German) Jochen Wilke (2014年01月04日).
• What's a Jiffy? by Zeal (before 2013年12月08日).
• 1496225352 Tetradecahedra by Joseph Nebus (2013年12月03日).
• Online Community by Allan R. Tate (before 2013年12月01日).
• Three dice to rule them all by Zed Lopez (2013年11月25日).
• Radii of an ellipse of known perimeter by rgesh (2013年11月22日).
• Geometry: A beginner's journey by Mathematical Hacker (2013年11月16日).
• Cartes à jouer (in French) by Jean-Paul Davalan (2013年11月16日).
• Die richtige Einheit des Wirkungsquantums by Roland (2013年10月31日).
• A Wonderful Pendulum Equation by Tom Van Baak (before 2013年10月31日).
• Sagan's Number is the number of stars in the Universe (2013年09月24日).
• Mathematics in Europe: Top 5 inspirations for Sept. (bef. 2013年09月24日).
• Grains of sand from the Sahara by IronWing (2013年09月22日).
• Enigma Card Trick by Scott Cram (2013年08月18日).
• Magia y matemàticas (in Spanish) Kruskal Count (2013年07月26日).
• Who promotes science best? in Czech (2013年07月01日).
• Heinrich Rubens (1865-1922) in Hungarian (before 2013年06月27日).
• Wilson's theorem in Japanese (before 2013年06月23日).
• High-School Mathematics links in Finnish (before 2013年06月23日).
• Niels Bohr's coat-of-arms by Jedi (2013年06月17日).
• Educational Websites by Family iConnect Software (before 2013年06月05日).
• Units of Planck's constant (in German) by Einbaum (2013年06月01日).
• Pierre Deligne's coat-of-arms by Jochen Wilke (2013年05月27日).
• Tetradecahedral dice (D14) by Busman (before 2013年05月21日).
• Arithmetic-Geometric Mean Calculator by Thomas Blankenhorn (2013年05月18日).
• Mendelevium University of Split, Croatia (before 2013年05月14日).
• NIHS gear standards (in Spanish) by Claimsys (2013年05月05日).
• Coats-of-Arms of Scientists by Jean-Marie De Brabandere (bef. 2013年05月04日).
• Game: Dots and Boxes on Public Pad (before 2013年05月02日).
• On the Lanczos approximation by Robert Munafo (bef. 2013年04月27日).
• Coat-of-arms of Emperor Akihito (in Japanese) Sawakimi (2013年04月20日).
• Margaret Thatcher, Iron Scientist by Roxanne Palmer, IBT (2013年04月17日).
• Multiperfect and hemiperfect numbers by Timday (2013年04月16日).
• Margaret Thatcher, RIP (in Dutch) by Jeroen (2013年04月15日).
• Arms of Margaret Thatcher by Patrick Baty (2013年04月09日).
• Margaret Thatcher "Heraldic symbols" by Claire Gibson (2013年04月09日).
• Arms of Margaret Thatcher (1925-2013) Pro Heraldica (2013年04月09日).
• Answers featuring www.numericana.com plaintxt.org (bef. 2013年04月03日).
• Game of Life by Daniel (2013年03月29日).
• Journal of American Science 2013;9 (5) Nazek A. AL-Essa (2013年03月27日).
• Five-card trick by Cotpi (2013年03月18日).
• The Boxer's Puzzle (by Sam Loyd) at Math Munch (2013年03月11日).
• Coat-of-Arms of Ørsted at PopScreen (before 2013年02月14日).
• Weighing puzzle by MrWondeerful (2013年02月13日).
• Portrait of Legendre by Peter M. Lee (before 2013年02月06日).
• Mathematics 3360 at Cornell by Pr. Louis Billera (before 2013年02月05日).
• XLVII & Avocadoes by Florida Division of Plant Industry (2013年01月30日).
• Solvay conferences by Dr. Charlie McAllister (before 2013年01月29日).
• Matematik: Rogers Wiki by Roger Bengtsson (before 2013年01月21日).
• Spanish units of length Realmagick shrine of knowledge (bef. 2013年01月19日).
• GUESSOUS, Abdelmalek... at PagesMaroc (before 2013年01月16日).
• Partitions of an integer... by David RF (2013年01月05日).
• Lords of the Tiny Particles by Terakkifizik (before 2013年01月02日).
• Electromagnetism by Dr. Omed Ghareb Abdullah (before 2012年12月31日).
• Who invented the Btu? by Melissa Sherrard (before 2012年12月22日).
• A man who just doesn't want to reveal himself by Andrien (2012年12月15日).
• Arithmetic-Geometric Mean (agm) by Thoughtful Living (2012年12月04日).
• Adrien-Marie Legendre (in Russian) by Zisl (2012年12月02日).
• Coat-of-arms of scientists (Peru) Asperger's sufferers (2012年11月24日).
• TI-92 Nostalgie (in French) by Moe, Journal du Geek (2012年11月19日).
• School (Pinterest "repin") by Carina Anderson (2012年11月17日).
• Ternary cards (in "Teaching Math") by Monica Pravlik (2012年11月17日).
• Official SI metric prefixes by Daniel B. Sedory (before 2012年11月11日).
• Lycée Malherbe de Caen. Pictures (before 2012年10月21日).
• First Question to the SSSF by B.C. (2012年10月18日).
• Magic Assistant Riddle by Pex (2012年10月10日).
• Mon frère Ainchtaine... (Solvay Conference) by TuniForce (2012年10月07日).
• Mendelevium by Eni Generalic (2012年10月05日).
• Equal sign by Mario Estanislao Cesar Ariet (2012年09月30日).
• Vara de Solomon by CharlieJr (2012年09月28日).
• Zero to the 0th power (is equal to 1) by Christian Bokhove (2012年09月24日).
• Backyard [Solar] Energy by amrlthomas (2012年09月14日).
• Georg Mohr (1640-1697) in Hungarian (bef. 2012年09月12日).
• Coat-of-arms of Paracelsus by Paul Ferguson (2012年09月06日).
• Compton Effect... by Soprano (2012年09月03日).
• Weighing nine balls... by Wizardskills (2012年08月29日).
• Advanced Number Theory by Marion Scheepers, Boise State U. (2012年08月28日).
• Pierre Deligne by Siobhan Roberts (2012年08月26日).
• Johnsonian polytopes by Marek14 (2012年08月24日).
• Complex Gamma Function in Japanese (before 2012年08月24日).
• Roger Apéry on OEISwiki by Daniel Forgues (2012年08月20日).
• Claude-Gaspard Bachet de Méziriac in Hungarian (before 2012年08月14日).
• An Eloquent Formula for the Perimeter of an Ellipse by Semjon Adlaj (2012年08月09日).
• Roller Coaster Loops at LiveBinders.com (before 2012年08月04日).
• Pseudo rhombic cuboctahedron by Thomas C. Hales (bef. 2012年08月04日).
• Winning strategy for Nim in Japanese (before 2012年07月31日).
• Symbols at redconceptual.com (before 2012年07月31日).
• Vindication of Goldbach's conjecture by David Bernier (2012年07月24日).
• Final Answers - Science - NUMERICANA, review by Niles (2012年07月24日).
• Perimeter of an Ellipse (in Chinese) by Chang (bef. 2012年07月22日).
• Andrew Trusty's friendfeeed on Combinatorics and... (2012年07月20日 10:42).
• Counting combinations/permutations with polynomials by Kalid (2012年07月19日).
• Topics in Combinatorics and Probability by Dexter Morgan (2012年07月18日).
• On Combinatorics and Probability Topics (Reddit) by swvist (2012年07月18日).
• Combinatorics and Probability Topics (Hacker News) by llambda (2012年07月18日).
• Combinatorics and Probability Friendfeed by Buddhika (2012年07月18日).
• Aire d'un Ellipsoïde (nomograph) by Jean-Paul Davalan (bef. 2012年07月09日).
• Joy of Creative Math Problem Solving (before 2012年07月05日).
• "Ein Mathematiker erklärt..." (7-11 Problem) by Omar (2012年07月04日).
• Pinterest summary of some "pins" about Numericana (before 2012年06月30日).
• La température peut elle créer la gravité ? by Rokag3 (2012年06月25日).
• Mathematical Constants by Stanislav Sýkora (before 2012年06月21日).
• Assemblée des Français de l'étranger (before 2012年06月21日).
• Giants of Science repinned by Nights of the Crusades (2012年06月20日).
• Calcolo Combinatorio (in Italian) by Giancarlo Zilio (bef. 2012年06月20日).
• Leaning Tower of Pisa (in Polish) by Scyth (before 2012年06月20日).
• Giants of Science repinned by Sandro Marco (2012年06月13日).
• Complex analysis by Felix Fischman (before 2012年06月09日).
• Martin David Kruskal (1925-2006) in Greek (2012年05月25日).
• Giants of Science pinned in Loving Science (before 2012年05月25日).
• Islamic creationism by Maripat (2012年05月23日).
• Leaning Tower of Pisa (in Polish) by Scyth (2012年05月15日).
• A Jiffy is an actual unit... by Daven Hiskey (2012年05月03日).
• Ernest Rutherford, 100 NZ $ by Libyaitalia (before 2012年04月20日).
• Perimeter of an ellipse (Lithuanian) by Eugenijus Paliokas (2012年04月14日).
• Alfred de Caston / Antoine Aurifeuille by Léon la Lune (2012年04月12日).
• Heraldry, Shields and Blazoning by Keith J. Davies (2012年04月11日).
• Units for Planck's constant (in German) by Alex (2012年04月11日).
• Numericana's Take on Calendars in Russian (2012年04月10日).
• Escutcheons of Science by Sporkchop (2012年04月09日).
• Magic Roadshow #129 by Rick Carruth (before 2012年03月31日).
• Drehmoment (= torque) at German Metapedia (2012年03月24日).
• The Best Card Trick at copti.com (2012年03月19日).
• Advice for mesh generation in CFD by Stefan (2012年03月16日).
• Numericana: Math and Science EEWeb Engineering Site of the Day (2012年03月01日).
• Kruskal Paths to God by Shchyndrl (2012年02月28日).
• Should your Child Learn Roman Numerals? (2012年02月27日).
• Gutenberg Coat-of-arms Heraldry Research Institute (2012年02月24日).
• Relative speed vs. closing speed: Pseudo-scientific denial (2012年02月24日).
• Doppler effect refutes Special Relativity... not! (2012年02月22日).
• Rediscovery of Biblical Glue by Jerome A. Samounce (2012年02月14日)
• Super Bowl XLVI: Mystery of Roman Numerals, Bob Holt (2012年02月06日).
• Super Bowl XLVI's Roman Numerals by James Johnson (2012年02月05日).
• Every Day is Special : Super Bowl XLVI (2012年02月05日).
• This website might die today... by Autonemesis (2012年02月05日).
• RE: ZGAMMA revisited by Marcel Hendrix (2012年02月04日).
• Super Bowl L by Schweady, BWCA Messageboard (2012年02月04日).
• Super Bowl XLVI ? by Evann Gastaldo, Newser Staff (2012年02月04日).
• Roman numerals Greek to modern youth by Leanne Italie (2012年02月03日).
• SuperBowl and Roman Numerals by Matthaeus (2012年02月03日).
• Roman Numerals Disappearing by Brian K. Finnicum (2012年02月02日).
• Combinatorial calculus in Thai (2012年02月01日).
• XLVI: To Some, it may as well be Greek... by Leanne Italie (2012年01月31日).
• XLVI ? by Leanne Italie at JournalGazette.net (2012年01月31日).
• Super Bowl & Roman Numerals (in Chinese) WorldJournal (2012年01月30日).
• Super Bowl XLVI: Du Chinois pour les jeunes Métro, Montréal (2012年01月30日).
• Super Bowl refresher on Roman numerals [ Fox News ] (2012年01月30日).
  Live links from Verizon entertainment, Associated Press, etc.
• Deciphering the Super Bowl by Leanne Italie, Associated Press (2012年01月30日).
  Post and Courier, Rapid City Journal, Seattle Times (m), Boston Globe...
• Traditional Chinese Medicine apologetics by Sonic (2012年01月20日).
• Transfinite numbers at The Motley Fool by TMFAleph1 (2012年01月19日).
• Nalimov endgame tables by Malbase (2012年01月17日).
• Zibaldone Scientifico in Italian (before 2012年01月14日).
• Ire in Adversa (George Gabriel Stokes) by Stephen Smith (2012年01月09日).
• Coat-of-arms of Leonardo da Vinci in Catalan (2012年01月07日).
• Bézier curves by Zaelia from France (2011年12月19日).
• Ellipse circumference calculator by Thomas Blankenhorn (2011年12月17日).
• 5-card trick of Fitch Cheney by Steven J. Miller (2011年12月13日).
• Le théorème de Wilson (in French) by Mathesisuniversalis (2011年12月12日).
• Solvay Conferences (in Greek) by John Fiorentinos (2011年11月25日).
• Moebius Function by Gabesoft (2011年11月11日).
• Numerical Coincidences (in Swedish) by Emil (2011年11月11日 11:11).
• Kruskal Paths with Dice (in Russian) at Watermelon (2011年11月01日).
• x and y can't both be odd (if x² + y² = z² ) by Henry (2011年10月25日).
• Solvay 1927 in Spanish (2011年10月24日).
• Wilson's theorem (in Japanese) by OTO Suu (2011年10月11日).
• Kruskal Count (in Polish) by Pierro (2011年10月10日).
• Perimeter of an Ellipse (Mathematica 11,2) by Paul Abbott (2011年10月06日).
• What do you always carry? (TI-92+) by Andernerd (2011年10月05日).
• Notation for standard uncertainty (misunderstood) by gue5t (2011年09月22日).
• Mathematical magic trick for the classroom in Thai (2011年09月21日)
• The Magic of Maths at Maths Week, Ireland (before 2011年09月21日).
• An invitation to a funny number system by Brent Yorgey (2011年09月17日).
• Decadics by Brent Yorgey, via Matt Gardner Spencer (2011年09月14日).
• Grey Elephants in Denmark by John Powell (2011年09月08日).
• Photon-photon scattering by Wukunlin (2011年09月02日).
• Bohr & Rutherford by Christopher G. Fletcher (2011年08月30日).
• Legendre and Fourier by Alexandre Eremenko, Purdue (bef. 2011年08月26日).
• Michon du Marais in the genealogy of Paul Jame (before 2011年08月26日).
• Many periodic tables... refresh screen to see another... (bef. 2011年08月24日).
• Why 10-adics are not a field by Citan Uzuki (2011年08月23日).
• Electromagnetism by Dr. Omed Ghareb Abdullah (before 2011年08月15日).
• Symbol font problem by Miller (2011年08月14日).
• Unicode fan against the Symbol font by Andreas Prilop (2011年08月13日).
• Entertaining Heraldry (Russian) by Spamsink, Fremont, CA (2011年08月12日).
• Proposed new Unicode characters by Leo Broukhis (2011年08月12日).
• Putting the Symbol font issue to rest by Leo Broukhis (2011年08月12日).
• The Jiffy is a unit of time... in Friday Thoughts (2011年08月12日).
• Tycho Brahe in altered painting by Arienski Crowley (2011年08月11日).
• Lanczos implementation of the Gamma function by Jack (2011年08月05日).
• Magia y Matemáticas Real Sociedad Matemática Española (2011年07月30日).
• Counting Square-Free Numbers by Jakub Pawlewicz (2011年07月25日).
• Coriolis deviation (in Serbian) by Trifo (2011年07月18日).
• Subtractive principle in Roman numerals (before 2011年07月18日).
• Coat-of-arms of Emperor Akihito in Japanese (before 2011年07月14日).
• Good Stuff on the Web (Swedish) by Roger Bengtsson (bef. 2011年07月10日).
• Matematika-Ku (Indonesian) by Hilman Permana (bef. 2011年07月10日).
• Heliocentrism in the second century BC by Brocke (2011年07月08日).
• Martin Gardner's Gliders by Chris Greaves (2011年07月07日).
• Physics of gases by Takeovers (2011年07月03日).
• July 1 in Chemistry (Te, 1740) by Carmen Giunta (before 2011年07月02日).
• Simple Symbols by TwoLeftFeet on MetaFilter (2011年07月01日).
• Taaffeite (in Russian) by Golodny (2011年06月29日).
• Math Magic Trick (in Thai) by Sri Narong (2011年06月28日).
• Hall of Fame by Harish Reddy (before 2011年06月26日).
• The Domino Effect by Jim Wilder (2011年06月25日).
• Computing the n-th partition number. Keith Matthews (bef. 2011年06月23日).
• Blood Red Ink by The Good Captain (2011年06月22日).
• General Theory of Relativity saved by Srinivazen (2011年06月20日).
• Mathematical Magic Tricks by Lisa Atkinson (before 2011年06月14日).
• Richtige Einheit des Wirkungsquantums by Roland (2011年06月11日).
• Planck's Constant in Russian (before 2011年06月05日).
• Maths magic tricks by Sumedha (2011年05月28日).
• History of mathematical symbols by Jim Loats, Ph.D. (bef. 2011年05月16日).
• Counting Polyhedra by Stuart Errol Anderson (2011年05月15日).
• Programming and continued fractions, in Lithuanian (bef. 2011年05月14日).
• SurfSafely Directory > Education > Science (before 2011年05月14日).
• Google Directory > Science > Math > Recreations (bef. 2011年05月12日).
• Mathematical Magic Tricks by Timon Piccini (before 2011年05月10日).
• Gear (in BEAM robotics wiki) by Clifford L. Boerema Jr. (2011年05月07日).
• Breteuil ring found in French West Indies by Mackaydon (2011年05月06日).
• Fitch Cheney Card Trick Update by Scott Cram (2011年05月05日).
• Wiskundig broeden op een ei (2004) by André Heck (bef. 2011年05月02日).
• Mathematical Magic on Doug Dyment's Portal (before 2011年04月28日).
• The Saga of Mathematics by Lewinter & Widulski (before 2011年04月27日).
• Thanks Internet (Bohr's coat-of-arms) by Duriel (2011年04月25日).
• Set Theory and Logic (Infinity) "FriendFeed" by Janmhil (2011年04月25日).
• Circumference of an ellipse by MathMan (2011年04月25日).
• Leading 50 math sites: Pop chart at CuriousMath.com (bef. 2011年04月24日).
• Links and publications by Stuart Errol Anderson (before 2011年04月24日).
• Definition(s) of the Jiffy? by Stevenal (2011年04月18日).
• Coat-of-arms of Pierre de Fermat by Jochen Wilke (2011年04月17日).
• Partitions d'un entier (French) by René Bastian (2011年04月14日).
• A chevron can mean minimum by Zev Chonoles (2011年04月10日).
• Friday Puzzle and Cicada Math by Shecky Riemann, NC (2011年04月08日).
• The Kruskal count by wwwckq (2011年04月06日).
• What are gross tons? by Latoya McGill, eHow contributor (2011年04月01日).
• Roger Apéry's constant (in French) by Mathesisuniversalis (2011年03月30日).
• Amazing Website with Amazing People in Tornado (2011年03月26日).
• 12-Marble weighing conundrum by Darkapothem2000 (2011年03月22日).
• Perimeter of an ellipse by Rothrock (2011年03月21日).
• Magic Maths by Miss Pollock (2011年03月06日).
• Escut d'armes de Leonardo da Vinci. Institut Nova Historia (2011年03月01日).
• Einheit des Wirkungsquantums (in German) by Newton (2011年02月22日).
• Gum Arabic by Hidden Lethe (2011年02月12日).
• More than I wanted to know about ink by Moonflowerjewelry (2011年02月11日).
• Enlaces para futuros educadores matemáticos (before 2011年02月07日).
• Art of Problem Solving by Lindlar (2011年02月04日).
• Units for Planck's constant (in German) by Überlebender (2011年02月03日).
• Polyhedra in loop quantum gravity. Bianchi, Dona, Speziale (2011年02月01日).
• Integer Partition and Ferrers Diagrams by chinmaylokesh (2011年01月26日).
• Why it's XLV and not VL by Sarissa (2011年01月24日).
• Encoding Partitions by martin-boundary (2011年01月22日).
• Ken Ono's partition function newsbreak by Robin Whitty (2011年01月22日).
• Super Bowl XLV (not VL) by GradyTheBadger (2011年01月18日).
• Coat-of-arms of Niels Bohr by Michael van de Gaer (2011年01月18日).
• Auer von Welsbach Museum: Austria & Germany (before 2011年01月17日).
• Number of decahedra in Fuzzy Percentile Dice by Windell (2011年01月11日).
• How to do a Magic Trick (before 2010年12月31日).
• Grey Elephants in Denmark by Bernard Kwan (2010年12月29日).
• Vertices, Polyhedra, Polytopes at refertus.info (before 2010年12月29日).
• Planck's constant in Arabic (before 2010年12月28日).
• Isaac Newton's crossbones by ResponsibleJoe (2010年12月28日).
• Die Einheit des Planckschen Wirkungsquantums by Noanna (2010年12月27日).
• Isaac Newton, a pirate? by Catalintenita from Bucharest (2010年12月25日).
• Escutcheons of Science by De Elia Artista (2010年12月24日).
• Niels Bohr & the Order of the Elephant by Jessamyn West (2010年12月23日).
• Opposites are Complementary by Matthew Battles (2010年12月23日).
• Coat-of-arms of Aki Hito by Robert Witting (2010年12月23日).
• Wrong portrait of Legendre on reddit by d00kd00kd00k (2010年12月20日).
• Arc d'une ellipse (in French) by Sylvainc2 (2010年12月12日).
• Calculating the perimeter of an oval by Joan Whetzel (2010年12月10日).
• Math Magic by Rip Skidmark (before 2010年12月06日).
• Last to be chosen by Scott Cram (2010年12月02日).
• Surface area of an ellipsoid (in German) by Fremdgaenger (2010年11月27日).
• Partition function by Alphachapmtl (2010年11月26日).
• Zeno [misguided & insulting] by doronshadmi (2010年11月11日).
• Definition of a scientific theory by JMR (2010年11月08日).
• Why a diode can't produce free energy by CWatters (2010年11月05日).
• Scientific theories by Natepro (2010年10月22日).
• Infinity (you asked for it) by Odilon (2010年10月18日).
• Mathematics Resources on the Web by Robin Whitty (bef. 2010年10月16日).
• Perpetual motion with a diode? by Snowhare (2010年10月13日).
• Richard Feynman on physical units by SponsoredWalk (2010年10月10日).
• Carmichael numbers (in Greek) by Achilleas (2010年10月10日).
• Conway's Nimbers by Alasdair McAndrew of Melbourne (2010年10月09日).
• Food for Geeks (Feigenbaum's constant) by yor_on (2010年10月06日).
• Mixed blessings of Fermat. Block Notes Mathematico (2010年10月06日).
• Partition Function in the thesis of Marcus Vollmer (2010年10月04日).
• Plaques and notices in Cambridge. Shields of scientists, 1886 (2010年10月01日).
• How much beer in a half-keg? by Michael Roennevig (2010年09月29日).
• Niels Bohr's coat-of-arms by Skydivephil (2010年09月29日).
• Glen Elert (physics teacher) by Ej Hill (2010年09月16日).
• Table of Polyhedra Enumerations by Ed Pegg, Jr. (2010年09月06日).
• International Heraldry by Stephen Plowman (bef. 2010年09月03日).
• Genesis (in Greek) by Patroklos (2010年09月02日).
• Stats 721 by Rachel Fewster, University of Auckland (bef. 2010年08月26日).
• First scientist to be knighted by Turenne (2010年08月24日).
• Tables of Dirichlet Characters by David Ireland (bef. 2010年08月19日).
• Creating a Novelty of Fact Invention by Leslie R. Pastor (2010年08月17日).
• L'année dernière à Marienbad [ m ] in Japanese (2010年08月16日).
• Tautology about Carmichael multiples by Richard Mathar (2010年08月14日).
• Curves of Constant Width by Michael de Villiers (2010年08月11日).
• Homage to Martin Gardner by Ralph Dumain (before 2010年08月08日).
• Recreational Math & Physics at SMILE, by Roy Coleman (2010年08月05日).
• 7-11 Puzzle (in Italian) by Cece (2010年07月29日).
• Coat-of-arms of Niels Bohr (2010年07月27日).
• Answer to Richard Wiseman's Friday Puzzle by Béranger (2010年07月26日).
• Infinity Symbol by catfi9ht (before 2010年07月25日).
• Math Open Reference by John Page (2010年07月24日).
• Polyhedra with large number of edges by Andreas Rüdinger (2010年07月24日).
• How to calculate dB gain at eHow.com (before 2010年07月24日).
• Ink made to last... by Mark J. Anderson (2010年07月20日).
• Partition von Zahlen (in German) by Mystic (2010年07月20日).
• Value of 19 pounds of pennies by Cait D (2010年07月19日).
• Ramsey Theory by Doctordick (2010年07月16日).
• Lab Report by Bahinting et al. at Mindanao State U. (2010年07月15日).
• billions. Intro C: Mini-FAQ on Words and Phrases (before 2010年07月12日).
• Why Does M Mean Slope? by JillB (2010年07月10日).
• July 4th Brain Fun by Scott Cram (2010年07月04日).
• The Martin Gardner Thread by Greymatters (2010年06月27日).
• Dudeney's Kayles by Scott Cram (2010年06月27日).
• Ellissoide (in Italian) by Mizarino (2010年06月18日).
• Infinite and Unreal Numbers by Akash (2010年06月16日).
• Cantor and Set Theory by Kanishka (2010年06月16日).
• In the wake of Entropy (Science Forums) by Forufes (2010年06月09日).
• Bombieri's Napkin Problem discussed on Math Overflow (2010年06月08日).
• Ternary cards (in Spanish) by Pedro Alegría (2010年06月07日).
• Who is Kevin Brown? (Physicsforum) by JesseM (2010年06月05日).
• Bohr's coat-of-arms (in Chinese) by Skyhiker (2010年06月04日).
• Newton's coat-of-arms. Don't mess with a genius (2010年06月04日).
• How to Learn Math Through Magic by Steph Ellen (2010年06月02日).
• Martin Gardner (1914-2010) by Liz E. (2010年05月31日).
• Ask a Mathematician by Anthony Iachini (bef. 2010年05月27日).
• Martin Gardner, 1914-2010 (in Russian) by Yggaz (2010年05月23日).
• Martin Gardner is dead (in Russian) by Robot (2010年05月23日).
• In memory of Martin Gardner (in Russian) by Robot (2010年05月23日).
• Martin Gardner R.I.P. (in Russian) [ 1 ] Constantin Knop (2010年05月23日).
• Isaac Newton's coat-of-arms by Rayscan (2010年05月11日).
• How much money is a pound of pennies worth? by Jo G (2010年05月08日).
• High-Energy Physics (Outreach) by Pr. Nikos Varelas (bef. 2010年05月06日).
• Measurements and Units Defined by Kristin Harley (2010年05月02日).
• Megas 2 by JNS (before 2010年04月18日).
• Permeability units by Chris Horn (2010年04月16日).
• April 13 in Chemistry (v. Reichenstein) by Carmen Giunta (2010年04月13日).
• Math Curriculum (OrangeMath) by Dennis Ashendorf (bef. 2010年04月13日).
• British Thermal Unit (BTU) by Chuck Wheeler (2010年04月13日).
• 100 Bars of Gold by Gronkie (2010年04月12日).
• Escutcheons... by Rolando Julio José de YÑIGO y GENIO (2010年04月12日).
• Old Brain Teaser (Weighing) by gronkie (2010年04月12日).
• What are the first 18 Roman numerals? by Tyler D. (2010年04月08日).
• Elliptic integral & AG mean Wapedia Mobile Encyclopedia (2010年04月07日).
• Arms of Abel & Symbols by Carlos Galindo (2010年04月05日).
• Cross of St. Boniface in Log24 by Steven H. Cullinane (2010年04月04日).
• Tons in "PDRacer: Puddle Duck Sailing" by Andrew Linn (2010年04月03日).
• Arithmetic of Medicine "All about International Unit (IU)" (2010年04月02日).
• Numericana (Facebook group) by Gérard P. Michon (2010年03月31日).
• Plank's Constant (in Russian) in Traditio Wiki (before 2010年03月31日).
• CONSTANTINOPLE [ & mobile version ] by TheoGB (2010年03月29日).
• Jan Amos Comenius and blackpowder (in Czech) by Luba (2010年03月29日).
• Recreational Maths: Year 7 Mathematics at Echuca College (2010年03月28日).
• Critical state of Nitrogen (in Persian) by Meytim (2010年03月28日).
• Coat-of-arms of Emperor Akihito [Ceron.jp] by Nageniku (2010年03月27日).
• Numericana's dashboard by Gérard P. Michon (2010年03月25日).
• Euler Problem 76 (in Bosnian) by Bahrudin Hrnjica (2010年03月24日).
• Recreational Mathematics by Subadra (2010年03月24日).
• Mantling (Heraldry) in Japanese Wikipedia (2010年03月24日).
• Isaac Newton's coat of arms by ParallelBotany (2010年03月22日).
• Why is a standard g exactly 9.80665m/s/s? [ 1 ] Fume Troll (2010年03月22日).
• Useful Links by Kamer Kaya of CERFACS (before 2010年03月14日).
• Epilogue on Legendre Portraits [AMS, 57, 6, 203] P. Duren (2010年03月01日).
• Educational Links by Math Lady (2010年02月13日).
• Perimeter of an ellipse (in Chinese) by Zhangjiali (2010年02月04日).
• Lewis Carroll's Monkey Puzzle by Bigtop (2010年02月04日).
• Math is Math... by Tom Nalevanko (2010年02月03日).
• Anecdotes scientifiques by Tropique (2010年02月01日).
• Escutcheons of Science by Peter Meister (2010年02月01日).
• Arms of Galileo by Bernhard Peter = Herold-vom-Rhein (2010年01月28日).
• Ruggero Gabbrielli's Foam by Giacomo (2010年01月27日).
• Euler-Descartes equation by 19106 (2010年01月21日).
• 10 Great Sites for Geeking Out by Pi Guy (2010年01月17日).
• Distance from Alexandria to Aswan: 5000 stadia? (bef. 2010年01月17日).
• News report #6 (Legendre's portrait) by Tom Koornwinder (2010年01月15日)
• Rediscovering an ancient glue [2] by Jerome A. Samounce (2010年01月11日).
• An eclectic collection of units by Stephen Lower (before 2010年01月11日).
• The amazing mouth of Kreskin and Math Magic (2010年01月03日).
• Obsolete inches (USMA) by John Steele & Pat Naughtin (2009年12月28日).
• Black diamonds on a tape measure by Woodeneye (2009年12月22日).
• Welke Legendre? (in Dutch) by Jeanine (2009年12月17日).
• The polyominoes of Sol Golomb by Travis Hale (2009年12月15日).
• ¿ Por qué no es primo el 1 ? FAQ at MaTeTam (before 2009年12月12日).
• Thomsen's formula by Cavarretta, O'Sullivan & Coop (bef. 2009年12月08日).
• Ternary Magic Cards at Oakland/East Bay Math Circle (bef. 2009年12月08日).
• Inclusion-Exclusion (with short URL) by Jordan Cole (2009年12月06日).
• Thomsen's formula by Richard F. Burton (2009年12月01日).
• Cooling of ALH84001 by Kathie Thomas-Keprta & al. (2009年11月25日).
• Blason de François Viète (in French) by Jean de Parthenay (2009年11月25日).
• Symbol for the set of prime numbers by APcalculus (2009年11月24日).
• Circumference of an Ellipse by David Biddulph (2009年11月24日).
• On the Perimeter of an Ellipse [ pdf ] by Paul Abbott (bef. 2009年11月23日).
• Cycling Performance Simplified by Bob Fugett (2009年11月22日).
• Legendre who? (Division by Zero) by Dave Richeson (2009年11月19日).
• The scary face of a great mathematician in Korean (2009年11月17日)
• Legendre != Legendre Programa de Educação Tutorial (2009年11月16日)
• Wow... I must buy this book!!! by Peter Theoh (2009年11月16日)
• Pythagoras Constant at Barcodes, Inc. (before 2009年11月16日)
• The Orbode Company links by Angelo (before 2009年11月16日)
• Theoretische Elektrotechnik by Karl Küpfmüller (before 2009年11月16日)
• Legendre, we hardly knew ye! in Ars Mathematica (2009年11月15日)
• Legendre on Wikimedia Commons thanks to Pieter Kuiper (2009年11月14日)
• Mistaken portrait of Legendre (AMS) by Peter Duren (2009年11月11日).
• Gray Elephants in Denmark by Buckwheat469 (2009年11月10日).
• About Abundancy by Walter Nissen & Michel Marcus (bef. 2009年11月08日).
• Superabundant numbers, etc. (in French) by mam (2009年11月06日).
• Scientific biographies by Robin Whitty (bef. 2009年11月02日).
• Stars in Galaxy, galaxies in Universe at Ecolution Trail (bef. 2009年10月24日).
• Tycho Brahe (A Universe in Time) by Ken R. Pinkela (bef. 2009年10月21日).
• BENG 221 Math. Methods in Bioengineering, at UCSD (bef. 2009年10月19日).
• Crazy about numbers by himanshupratap (2009年10月18日).
• Portrait of Legendre (in Russian) by iv6 (2009年10月17日).
• Bohr's coat-of-arms (in Dutch) by Michelin (2009年10月11日). Leonardo da Vinci
• Arms of Leonardo da Vinci (in Catalan) by JM (2009年10月08日).
• Mathematics Help by Dr. Bruce W. Coletti (before 2009年10月04日).
• What is the circumference of an ellipse? WikiAnswers (bef. 2009年10月03日).
• How to simulate a Poisson process by Harry Kim (2009年09月30日).
• Legendre & Fourier caricatures by Stpasha & Infofiltrage (2009年09月29日).
• Broken calculator with 6 keys in Chinese (before 2009年09月28日).
• SAM 206 - Septenber 2009 by Opie Houston (before 2009年09月22日).
• Knight & Day (Niels Bohr) by Elizabeth Reninger (2009年09月23日).
• Lewis Carrol's Monkey Puzzle in Japanese (2009年09月17日).
• The Magic of 9 by Pi Guy (2009年09月10日).
• Mathematics Help on the Internet in Finnish (before 2009年09月09日).
• Erroneous portrait of Legendre Wikipedia deletion request (2009年09月07日).
• Multiplication of complex numbers by David (2009年09月04日).
• Nautical Leagues by Valentine (2009年09月01日).
• First Solvay Conference (1911) in CosmoLearning (bef. 2009年08月31日).
• Arms of August Wilhelm von Hofmann by Jochen Wilke (2009年08月29日).
• Militant blogs (in Czech) by Satori (2009年08月27日).
• Kevin S. Brown & MathPages (in Polish) by 5-Grid (2009年08月14日).
• Mathematik (in German) by Frank Dreßler (2009年08月06日).
• Numericana's hall of fame (in Swedish) by Tonyf (2009年08月03日).
• Perimeter of an ellipse, in Japanese (2009年08月01日).
• M. Pfeifer = Le Bicou Taupe Ampère, Lycée Thiers (bef. 2009年07月31日).
• Along Came Leonhard Euler by HumblyTheirs (2009年07月30日).
• Ramanujan's number by Graham D (2009年07月30日).
• Sum of reciprocals of Catalan numbers by Juan Márquez (2009年07月29日).
• About Numericana (in Russian) ... and the Symbol font (2009年07月26日).
• Coats-of-Arms of Nobel Laureates by Jochen Wilke (2009年07月14日).
• Tetragonal Antiprism (in German) by Jürgen Köller (before 2009年07月01日).
• Graduation speech commented by Jonathan Vos Post (2009年06月29日).
• On multiple-choice questions in Terry Tao's blog (2009年06月29日).
• 50th birthday of Glénisson's formulas (in French) by Doris (2009年06月28日).
• Zillions by JimmyG at AnswerBag.com (2009年06月27日).
• 24 Tintin albums by @Li3n (2009年06月26日).
• Wythoff's game on the quarter-plane by Aleph_One (2009年06月24日).
• Math Magic by Mrs. Schlueck, Hillside Grade School (bef. 2009年06月23日).
• Who wrote Reflections on Relativity ? by Blackhead (2009年06月15日).
• Diodes obey second law of thermodynamics by mrflora (2009年06月14日).
• Quadratic Reciprocity by Graeme McRae (before 2009年06月09日).
• Ellipsoid (in German) by Jürgen Köller (before 2009年06月08日).
• Flattening a sphere by Yotch (2009年06月07日).
• Juan Pablo Muñoz's Library at diigo.com (2009年06月04日).
• Planar Curves in Community College Calculus by Vlorbik (2009年05月11日).
• Arms of Johan Gadolin (1760-1852) by Jochen Wilke (2009年05月08日).
• Arms of Pierre Deligne (b. 1944) by Jochen Wilke (2009年05月07日).
• Voyage 200 bookmarks by carlosvicunav (before 2009年05月06日).
• About Edouard Herzen at Lexilogia by Stathis (2009年05月06日).
• Continued Fractions (book review) by A. Ya. Khinchin (bef. 2009年05月06日).
• Icosahedral die ("scattergories cube") by Skateandscoot (2009年05月02日).
• Look-back time by Dave Mitsky (2009年05月01日).
• Pierre Deligne's coat-of-arms [ 1 ] by Jordan S. Ellenberg (2009年04月29日).
• Notes about math trick by Opieh in CHAT XXXIV (2009年04月23日).
• Circumference of an Ellipse by Shahram Zafary (2009年04月21日).
• Enumeration of polyhedra (in German) by PhotonX (2009年04月17日).
• The Kruskal Principle by Scott Cram (2009年04月09日).
• History of Logarithms by Ms. Fleming (before 2009年04月05日).
• Heraldry Links by Marek Nienaltowski (before 2009年04月05日).
• Perimeter of an Ellipse (in Dutch) by Kjell (2009年04月04日).
• The constants of Mathematics by Chuck Knepfle (before 2009年04月02日).
• Numericana, Enciclopedia dei Numeri by Lorenzo Zaninetti (2009年03月30日).
• La aritmética p-ádica (in Spanish) by Remolon (2009年03月27日).
• Mathematical Puzzles [+ German ] by Thomas Risse (bef. 2009年03月26日).
• Malo Mori Quam Foedari by Linda_J (2009年03月09日).
• Coat-of-arms of Hans Christian Ørsted by Jochen Wilke (2009年02月21日).
• Gear Design and Rack Radius by BabyBear (2009年02月19日).
• Law of Gravity. Long quotation by indy_jh (2009年02月18日).
• Feynman Apologetics by Viren Fernandes (before 2009年02月15日).
• Arms of Scientists and Explorers by Jochen Wilke (2009年02月14日).
• Coat-of-arms of Niels Bohr (in Czech) by Faleristika (2009年02月14日).
• The 1089 force by fkace63 (2009年02月14日).
• Introducing Numericana at www.hazemsakeek.com (2009年02月09日).
• Advice about ellipse formula by Noone (2009年02月06日).
• NationMaster > Encyclopedia > Benjamin Thompson (before 2009年02月05日).
• Units of Energy & Power at Instituto Superior Técnico (2009年02月04日).
• Brain Teasers at The Crafty Puzzles Company (2009年02月03日).
• Good On-Line Armorials by David B. Appleton (before 2009年02月03日).
• Doplnky o grupach (in Slovakian) by Tomas Richta (bef. 2009年01月29日).
• Podrobneji o pologrupach by Tomas Richta (before 2009年01月29日).
• Composite and Prime Numbers... by Ulrich H. Kurzweg (2009年01月27日).
• Demon of Laplace in The PC Engine Software Bible (bef. 2009年01月26日).
• Tycho Brahe and Hamlet by MinnesotaStan (2009年01月24日).
• The color of Tintin's hair by SheSaidDestroy (2009年01月23日).
• Coat-of-arms of Emperor Akihito in Out of Eden (2009年01月21日).
• Mathematical Symbols by Robin Whitty (2009年01月20日).
• Physics at Numericana (in Japanese) by Shimota-79 (2009年01月17日).
• Planck Length & Planck Time (in German) by Ungläubige (2009年01月17日).
• Enumeration of Polyhedra (in German) by PhotonX (2009年01月17日).
• Niels Bohr's coat-of-arms (in German) by DrumDub (2009年01月16日).
• Differential Question (Camaro automobile) by 69 rs/ss (2009年01月13日).
• Dead Sea Scrolls and Iron Gall Ink by Count Iblis (2009年01月12日).
• Ideal Gases by professor Ruslan Ozerov (before 2009年01月12日).
• Balance Problem by Deepika (2009年01月09日).
• Perimeter of an ellipse by BadBoy (2009年01月08日).
• Languages in Images and Paintings by Imelda Almqvist (2009年01月08日).
• ADSL filters and Sky Digibox by Sam Radford (before 2009年01月08日).
• Scientific Symbols and Icons... at StumbleUpon (before 2009年01月06日).
• ADSL filters by Sam Radford (2009年01月05日).
• Kruskal Paths to God by Scott Cram (2009年01月01日).
• Tombstone Symbol in WikiAnswers, by Hisoftsol (before 2008年12月30日).
• Last Year at Marienbad. Numerical methods, in Polish (bef. 2008年12月23日).
• A Little Physics Conundrum by Thoughtfully (2008年12月15日).
• Heraldic links (in German) at GenWiki (before 2008年12月13日).
• Counterfeit coin problem by Vikas (2008年12月07日).
• Taoism and Bohr's Coat-of-Arms by NonTien (2008年12月04日).
• Einstein and Intellectual Property by Noah Callaway (2008年12月02日).
• Richard Feynman & Machine Needles by Quilting Kathy (2008年11月29日).
• Tons of fun math stuff (Yahoo! Answers) by Josh S (2008年11月25日).
• Wanted Dead or Alive: Schrödinger's Cat by J (2008年11月23日).
• Bohr's coat-of-arms in Estonian (2008年11月20日).
• Counting Abelian groups by NonCommAlg (2008年11月16日).
• Symbols, equality, yin-yang Mrs. Carrigan's Math page (bef. 2008年11月14日).
• Coat-of-arms of Ferdinand von Zeppelin by Jochen Wilke (2008年11月11日).
• The Lamppost Theory by A.H. Roslan Harahap (before 2008年11月08日).
• Matemáticas y trucos mentales de cálculo in Spanish (2008年11月07日).
• Academic Poker Resources ay Big Edge Poker (before 2008年11月06日).
• Introduction to Combinatorial Analysis by Levent Kandiller (2008年11月03日).
• Rosenkrantz and Guildenstern coats-of-arms by Jerzi Lucki (2008年10月31日).
• Recommending the TI Voyager 200 (in Croatian) by Harry (2008年10月30日).
• Lleonard Vinci (in Catalan) by Joan Calsapeu (2008年10月17日).
• Solvay conferences in links by Lyzanor at delicious.com (2008年10月10日).
• Arms of Leonardo da Vinci (in Catalan) by JM (2008年10月08日).
• Linear Systems at NCTM's Illuminations (before 2008年10月08日).
• Tropentag 2008 at the University of Hohenheim (2008年10月07日).
• El escudo heráldico de Leonardo da Vinci (Catalan) by Pep (2008年10月04日).
• Arms of Leonardo da Vinci (in Catalan) by Jordi Bilbeny (2008年10月03日).
• Maintaining a constant circumference by Bluechip (2008年10月02日).
• Fork in the road by Sanders (2008年10月02日).
• Tycho Brahe and William Shakespeare by Ian (2008年09月29日).
• Exact value of the horsepower (hp) by Jay Belanger (2008年09月25日).
• Pullup assistant bands as ellipses by Cybercop (2008年09月25日).
• Daily Links [ 1 ] on Webliminal by Ernest Ackermann (2008年09月12日).
• Comment calculer le périmètre d'une ellipse? In French. (2008年09月11日).
• Rediscovering Mathematics (MA 143) Shai Simonson (bef. 2008年09月11日).
• Hymne à la Taupe by Geckox (2008年08月22日).
• Wernher von Braun in Catalan [?] (before 2008年08月16日).
• Packing People in the Grand Canyon by Mikabee (2008年08月15日).
• Last Year at Marienbad by AngryCandy (2008年08月09日).
• Szilard's engine (in Japanese) by hiroki_f (2008年08月04日).
• Parabolic arc length by 11rdc11 (2008年08月03日).
• Maxwell Equations & Lorentz Force by Zentara (2008年08月01日).
• The Symbol font problem by David W. Knight (bef. 2008年07月31日).
• Wythoff's Game, M.A. thesis by Kimberly Hirschfeld-Cotton (July 2008).
• Kilotons & megatons. United Nations Cyberschoolbus (bef. 2008年07月30日).
• Polyhedra with large number of edges by Andreas Rüdinger (2008年07月28日).
• Apéry's Constant by Shane (2008年07月28日).
• Ed Pegg, Jr. (anagram) by Rotund Filipino (2008年07月24日).
• Fundamentals of Electronics. Dr. Pawel Niewczas (before 2008年07月22日).
• Smallest recorded measurement by Brian at Answerbag (2008年07月19日).
• Fun Site by Trisha (2008年07月13日).
• Set Theory at GiftedResources.org (before 2008年07月10日).
• Thomsen's formulae by Kingston, Clayton, Priest and Best (2008年07月06日).
• Links on Wikipedia by Robin Whitty "Charleswallingford" (2008年06月25日).
• Quick Snippets (Grey Matters) by Scott Cram (2008年06月05日).
• More Math Magic than you can possibly stand! by J. Kronenberg (2008-05).
• How many elephants are there in Denmark? by Ned Kelly (2008年05月31日).
• Visit to the 7-11 by ChuckJerry (2008年05月29日).
• Mathematical Brooding over an Egg [1] by André Heck (bef. 2008年05月27日).
• The diamond mark at every 19.2" by kpepin (2008年05月21日).
• Coriolis deviation in a Tower of Pisa drop by Markogts (2008年05月21日).
• Meaning of h-bar symbol by Starman (2008年05月19日).
• Number Theory Education Resources by Mike at Mahalo (2008年05月13日).
• Escutcheons of Science at Forum-Rycerstwa (before 2008年05月13日).
• Billion on The Timber Yard by Noddy330 (2008年05月04日).
• Périmètre d'une ellipse (in French) by Valeri Astanoff (2008年05月04日).
• Escudos de Científicos by Jose Juan Carrion Rangel (2008年04月27日).
• Iron-gall ink in a fountain pen by Ondina (2008年04月14日).
• Funny units by bp56 (2008年04月13日).
• Wilson's Theorem by Balan (2008年04月09日).
• Perimeter of an Ellipse by Hellburner at ars technica (2008年04月09日).
• Numericana Hall of Fame by Seth Tyler at ISH (before 2008年04月01日).
• Bohr's coat-of-arms (in Latvian) by Ierindnieks (2008年03月28日).
• Naming Polyhedra [ 1 | 2 ] (German) by Hugo Pfoertner (2008年03月26日).
• Demons of Physics by Ughaibu (2008年03月23日).
• Library Links Metropolitan Community College, MO (bef. 2008年03月22日).
• Steven Weinberg by Pentcho Valev (2008年03月21日).
• Feynman on Physical Units by Henry L. Neal at CAU (bef. 2008年03月18日).
• Famous Scientists: Archimedes by Marie P., 4th Grade (2008年03月10日).
• Student learns to deadpan by Gregg Chenoweth (2008年02月25日).
• C183 and C209 by Rogue, updated by Fivemack (2008年02月14日).
• Wieferich-related factoring (C183, C209) [ 1 ] Zeta-Flux (2008年02月13日).
• Circumference of an ellipse in QBasic and Lisp, by tcebob (2008年02月11日).
• The Natural Number e by Stirling Stan (2008年02月10日).
• Perimeter of an ellipse by John Nimphius (2008年02月03日).
• Heraldik/Linkliste (in German) Internationale Seiten (2008年02月01日).
• Borromean rings by Emlaabs (2008年01月30日).
• Blason de Adrien-Marie Legendre by Infofiltrage (2008年01月28日).
• Mathematics at 2000clicks.com (2008年01月26日).
• Multiplying Nimbers by Joshua Zucker (2008年01月17日).
• Hacks (Funny Units) by Dr. Jeremy Frank of NASA (2008年01月15日).
• Cool math-magic sites by John Dougherty (before 2008年01月11日).
• Math Symbol (Equal sign) by Lori Carrigan (before 2008年01月11日).
• Math Symbol (Equal sign) by Donette Carter (before 2008年01月08日).
• The elusive Kevin S. Brown (in Russian) Z.K. Silagadze (2008年01月06日).
• Archimedes Spiral by David McCombs. (before 2008年01月03日).
• Carmichael Multiples by Joe Crump. (2008年01月02日).
• The Tai-Chi or Yin-Yang symbol by Barbara (2007年12月31日).
• Pétrole, gaz et les autres énergies by Albert Legault (bef. 2007年12月31日).
• Surfing Note in Mathematics Teacher by Derek Smith (bef. 2007年12月31日).
• Abell 1835 IR1916 (in Spanish) by Horacio C. (2007年12月30日).
• Area of an Ellipsoid (in French) by Cauchy (2007年12月25日).
• Length of an Ellipse in Russian (2007年12月24日).
• Area of an Ellipsoid (in French) by Zapple (2007年12月23日).
• Contraria Sunt Complementa by Kyle Matthew Oliver (2007年12月17日).
• Tycho Brahe by Lupo LeBoucher (2007年12月10日).
• Surface area of an elliptical dish head by Oil Field Trash (2007年12月09日).
• The experiment Galileo didn't do by Gideon Reich (2007年12月01日).
• Computing the partition function (in Polish) by Samolot (2007年11月25日).
• Millikan's experiments (Polish) by Arkadiusz Jadczyk (2007年11月24日).
• Huge numbers in Roman nnumerals by Maryt/theteach (2007年11月18日).
• Arc length of a parabola (U. of Regina) Harley Weston (bef. 2007年11月15日).
• Fun with math?! (true nature of magic) by Neutronjockey (2007年11月14日).
• From Slugs to Kegs by Snatirup (2007年11月12日).
• Grave of Count Rumford by John A. Robertson (before 2007年11月11日).
• Divisors of zero (Math 651, Ohio State) by G.A. Edgar (bef. 2007年11月08日).
• Prominent Scientists. Amman Baccalaureate School, Jordan (2007年11月07日).
• Epitaph of Roger Apéry by Bertrand Beyern (before 2007年11月06日).
• How many stars in the Andromeda galaxy? Chris Wagoner (2007年11月04日).
• Arms of Adrien-Marie Legendre by Infofiltrage (2007年10月31日).
• Squuzuu = Partition Function by Abbas (2007年10月28日).
• Circumference of an ellipse by Devrath (2007年10月28日).
• Historical Armorials at The Heraldry Society (before 2007年10月27日).
• Useful Stuff by J. Paul Gustafson at Messiah College (bef. 2007年10月25日).
• Surface area of an ellipsoid by Raxit (2007年10月24日).
• Planar Curves by Bryan Bishop (before 2007年10月24日).
• Introduction to Group Theory by Anton Gorodetski (before 2007年10月24日).
• Sources / Fonti for the libconic mathematical project (bef. 2007年10月21日).
• Credits / Meritii : Perimeter of an ellipse for libconic (before 2007年10月21日).
• DCMI Date Working Group Dublin Core Metadata Initiative (bef. 2007年10月21日).
• 0^0 = 1 in "The Student Room" by FusionSKD (2007年10月20日).
• Références mathématiques [ 1 ] by MC (before 2007年10月20日).
• The Superfund Three-card Monte by TMFMarathonMan (2007年10月19日).
• Favorite Math Links (in Finnish) by Janne Aukia (2007年10月18日).
• Favorite Links (in Finnish) by Jannep (2007年10月18日).
• Miracle de la Monarchie by JesusFranco on zentropa.info (2007年10月17日).
• Best of the Web on hexahedra, polyhedra & polytopes (bef. 2007年10月17日).
• Infinity (clippings) Concord Middle School (bef. 2007年10月14日).
• Baltzar von Platen by Jeremy (2007年10月11日).
• Carmichael Zahlen in Meyers Lexikon (before 2007年10月08日).
• Taaffe Viscount (in German) by Frank Martinoff (2007年10月01日).
• Two odd balls (found by weighing) by Opalg (2007年09月30日).
• Arms of scientists by Stephen J.F. Plowman, TD FSA Scot (2007年09月30日).
• Volume of a Spheroid by LearnExcel (2007年09月27日).
• Mathematics & Physics [2, 3] in Bulgarian. B.B. Bantchev (bef. 2007年09月26日).
• Aurifeuillian Factorization by James Wanless (2007年09月23日).
• Stars vs. Sand Recounted in RASNZ newsletter, by Laintal (2007年09月23日).
• Intégrales elliptiques de deuxième espèce by Framboise (2007年09月22日).
• Infinity symbol for Mrs. Martin's Fourth Grade (before 2007年09月18日).
• Fattest tetragonal antiwedge? by Erimo (2007年09月17日).
• Unit(s) for Planck's constant (in German) by Martin Müller (2007年09月16日).
• Tetragonal antiwedge in Erimo's Journal (2007年09月15日).
• A Mecânica do Século XVII in História da Ciência (2007年09月04日).
• Una vaschetta origami by Giorgio Vecchi (2007年09月04日).
• Really cool maths teasers in Orange Sunset (2007年09月03日).
• Prime Number in Conservapedia (before 2007年09月01日).
• 146 old pennies to a pound by Mitch Ross (2007年08月29日).
• Sir George Gabriel Stokes: August 13 in Science History (bef. 2007年08月28日).
• Linki do ciekawych stron Escutcheons of Science (before 2007年08月22日).
• Volume of the Grand Canyon by Orbinspace (2007年08月21日).
• EDC 5V95 Syllabus by Trena L. Wilkerson at Baylor (bef. 2007年08月20日).
• Wernher von Braun at www.spock.com (before 2007年08月20日).
• Perimeter of an Ellipse by WesleySonofCornelius (2007年08月16日).
• Trigonometry Class in "The Classroom" (before 2007年08月16日).
• Useful Reference Web Sites by David G. Simpson (before 2007年08月16日).
• Another page Jim might like by Rybo on GeoJourney (2007年08月05日).
• Michon's conjecture on Carmichael divisors by Maxal (2007年08月05日).
• Fundamentals of Electronics by Dr. Stephen McArthur (bef. 2007年08月04日).
• Good Websites by Forever May Not (2007年08月01日).
• Foundations of Mathematics at University of Brighton (bef. 2007年07月31日).
• Properties of Specific Numbers by Robert P. Munafo (before 2007年07月31日).
• Mathématiques ludiques (in French) by Tropique (2007年07月27日).
• Another nice site by Galactus at MathHelpForum (2007年07月25日).
• Arni's Mindgames by Chandru Arni (before 2007年07月23日).
• Wie das rad richtig benutzt wird (in German) by Moriaan (2007年07月21日).
• Plankschen Konstante (in German) by King_of_Loss (2007年07月20日).
• Comedy Shtick and Counterfeit Penny by D. Frisch (2007年07月19日).
• Grey Elephants in Denmark by Mark Runge (2007年07月19日).
• Niste rezolvari (in Romanian) by Constantin Obreja (2007年07月17日).
• How to become a theoretical physicist. Bryan Bishop (bef. 2007年07月16日).
• German horsepower by Djmatty (2007年07月14日).
• Vector Calculus by Michael Jansson (before 2007年07月13日).
• Verizon gets out of the copper business [1] Mark Ontkush (2007年07月12日).
• Unit for Planck's Constant (in German) by Alex (2007年07月06日).
• Die Heisenbergsche Unschärferelation (German) pgz (bef. 2007年07月06日).
• Perimeter of an ellipse (using Adobe Illustrator) by Amy (2007年07月05日).
• Bohr coat of arms by Diane Kulisek (2007年07月04日).
• Counting without knowing how to count by Mathmate (2007年07月02日).
• Patt Hawkey's Math Links. Parks Junior High School (bef. 2007年06月29日).
• Counterfeit penny [ mobile ] by Thaltgen (before 2007年06月29日).
• Heraldry Links at ForumRycerstwa hosted by Yahoo! (bef. 2007年06月19日).
• Perimeter of an ellipse by Sam at UK Vintage Radio (2007年06月18日).
• Elliptic arc by Xeriar (2007年06月17日).
• Perfect Numbers, Mersenne Primes by Martin Gelbaum (bef. 2007年06月13日).
• Coat-of-arms of Karl Drais? (in German) by Jochen Wilke (2007年06月12日).
• 1 is not a prime number (in Romanian) by Lendar (2007年06月10日).
• Armigerous Scientists. Book of THoTH. MediaMonkey (2007年06月07日).
• Mathematical Magic by Byrd (2007年06月06日).
• Infamous Measurement Problem by sweatty_black_hole (2007年05月24日).
• Mathematical Humor (update) by Scott Cram (2007年05月24日).
• Liczbach rzymskich: Roman numerals, in Polish (before 2007年05月24日).
• Infinity Symbol Cloth by Lady Tricotine (2007年05月21日).
• Etymology of behen by Scion (before 2007年05月21日).
• Author of MathPages by Daniel Duparc (2007年05月20日).
• Coalescing black-hole binaries by Damour & Nagar (2007年05月17日).
• Infinite loops by Mike (2007年05月15日).
• Car speed (in Romanian) by Victori (2007年05月11日).
• Elliptical orbit by Ilya (2007年05月06日).
• Gamma Function Citizendium Article (2007年05月05日).
• Monkey Gravity by Yucha Yamaguchi (2007年05月03日).
• Centro Virtual de Divulgación de las Matemáticas (before 2007年05月01日).
• Additional Info by David Fawcette at Columbia State (bef. 2007年05月01日).
• Symmetrical Polynomials by Francisco de Léon-Sotelo (2007年04月29日).
• Nautical mile? by smci at "Yahoo! Answers" (2007年04月28日).
• Archimedes [ 2 ] (before 2007年04月25日).
• Mr. Miller makes the Numericana Hall of Fame (2007年04月25日).
• Tycho Brahe's family... by Vacapinta (2007年04月23日).
• Mental Magic at magicpen.com (before 2007年04月19日).
• Review of Math Resources in I Like Math Blog (before 2007年04月19日).
• Arms of Wernher von Braun by Ahnungslos (2007年04月15日).
• António Aniceto Monteiro by Jorge (2007-04-14).
• Stemmi di scienziati by Sebastiano Pasquini (before 2007年04月11日).
• Perimeter of an Ellipse at Math Open Reference (before 2007年04月11日).
• Calculus Symbols at MathRamz.com (2007年04月10日).
• Mensa Sverige Private forum, in Swedish (2007年04月08日).
• Galeria (Polish) by Karolina Alicja Kaszuba (before 2007年04月08日).
• Wappeneintrag etc. by Dr. Frank Joachim Reuther (before 2007年04月07日).
• How many stars are there? in Sergiu M's Blog (2007年04月05日).
• Ships and Horizon by David A. Smith (2007年04月02日).
• Theorem of the Day (related sites) by Robin Whitty (before 2007年03月31日).
• Symbols MAT 22A-1, Spring Quarter 2007 at UC Davis (bef. 2007年03月31日).
• Vapen samlingar på internet (Swedish) by Awdaniec (2007年03月26日).
• The Kruskal Count by Sir Ross (2007年03月26日).
• Bimetallic Ratio by CJS (2007年03月25日).
• Vapen f r berömda/kända/ökända personer by Marcus K (2007年03月20日).
• A Slice of Pi by Alex S. (2007年03月15日).
• Pi Day and noteworthy constants... by Stephen Speicher (2007年03月14日).
• Quart of quarters by Squink (2007年03月11日).
• Stoichiometry of Black Powder by KevFla2001 (2007年03月03日).
• Arms of Joseph Lister by Stephen J F Plowman (2007年02月28日).
• Heraldik för berömda/kända personer by Marcus Karlsson (2007年02月19日).
• Butylic and butyric by Porlock Junior in Pharyngula blog (2007年02月18日).
• Logaritma (in Turkish) by Eddie (2007年02月15日).
• Mathematics for Fun: Eye Opener (before 2007年02月10日).
• Basic Trigonometry by Roy (2007年02月07日).
• Visconti, Sforza, Alfa-Romeo (in Swedish) by Awdaniec (2007年02月01日).
• Vapen för berömda personer (in Swedish) by Marcus K. (2007年01月30日).
• Metric value of a pound (The AnswerBank) by mibn2cweus (2007年01月30日).
• Ryedale's penny sorter by citizenkane (around 2007年01月29日).
• Perimeter of an ellipse by DrLaszloJamf (2007年01月26日).
• Carl von Weizsäcker by Norman Redington (before 2007年01月26日).
• Partition Function by Sam Kong (2007年01月24日).
• Counterfeit Coin Challenge (2007年01月23日).
• Cherish Freedom in Out From Under (2007年01月18日).
• Bereken de omtrek van een ellips in Dutch Wikipedia (bef. 2007年01月17日).
• Mathematical News by Umberto Cerruti (before 2007年01月15日).
• Black truss diamond markings by Schaula359 (2007年01月14日).
• Coat-of-arms of Margaret Thatcher by Sean J. Murphy (2007年01月11日).
• Thomsen's formula for the surface area of an ellipsoid (bef. 2007年01月11日).
• Mathematical Magic Tricks Everyone's bookmarks (2007年01月09日).
• The countability of rational numbers by Krusty (2007年01月08日).
• Horsepower at CycleChaos (before 2007年01月06日).
• Measurements & Units at Uncover the Net (before 2007年01月03日).
• SchoolNotes of Mr. Keeler, Fort Walton Beach HS (before 2007年01月03日).
• Diffusion Quadratic Residues by RFZ_Quest (2006年12月03日).
• Tycho Brahe the Great Dane by Pmgisme (2006年12月03日).
• Perimeter of an Ellipse in Dutch Wikipedia (2006年12月01日).
• Roman Numerals by CCC (2006年11月30日).
• Alchemical substances [ 2 ] by Lokiyn (2006年11月29日).
• Epiphany and Humor by Mohan K.V (2006年11月27日).
• Three Prisoners Paradox by Scott Cram (2006年11月26日).
• Zoroastrian symbol in the Bible? by Seeker of Truth (2006年11月22日).
• Mathematics Links by Thomas Fink (before 2006年11月22日).
• Currency by Gaeleth (2006年11月21日).
• Mrs. Bennett's page of Links Whitefish School District (bef. 2006年11月21日).
• The full 1089 force by Scott Cram (2006年11月19日).
• Mathematical abbreviation ("Resp.") by Margaret Marks (2006年11月18日).
• "Insanely cool for math/language dorks" by Bynner (2006年11月15日).
• Fulminating Gold by JWB in Pepys' Diary (2006年11月12日).
• Signs, symbols and icons by Dave Gray (before 2006年11月12日).
• Harmonic series in Physics 121 by Corbin Covault (2006年11月10日).
• Tons galore... in Indonesian Wikipedia (before 2006年10月31日).
• Ideal gas law at www.squinch.org (before 2006年10月30日).
• Everything you ever wanted... by Marco Plumley (bef. 2006年10月25日).
• Subtractive principle in Roman numerals by akaSpider (2006年10月23日).
• Pound of pennies by JLam (2006年10月22日).
• U2 Puzzle (Microsoft employement test) by Steveo1983 (2006年10月22日).
• Infinity symbol (Lemniscus) by Winhill at UKC Forums (2006年10月19日).
• Dirichlet Beta Function by Gerson W. Barbosa (2006年10月16日).
• 7-11 puzzle as a homework assignment by CharlieLuciano (2006年10月15日).
• Weighing 12 coins (in Lithuanian) by Hoho (2006年10月10日).
• Omega constant (in Swedish) on Wikipedia (bef. 2006年10月08日).
• Feet, meters or leagues by Jacquelyn, a.k.a. polar griff (2006年10月04日).
• Tycho Brahe's Shakespearean relatives by Banubula (2006年10月01日).
• Tycho Brahe's Shakespearean relatives [ 1 ] by John Baez (2006年10月01日).
• Closing speed and Sagnac effect by Pupamancur (2006年09月28日).
• Speed of light in a moving liquid (Fizeau, 1851) by Montec (2006年09月14日).
• Coat-of-arms of Wernher von Braun on Wikipedia (bef. 2006年09月07日).
• Combinatorial Game Theory by Richard J. Nowakowski (bef. 2006年08月31日).
• Lanczos approximation on Wikipedia (bef. 2006年08月30日).
• Metrologist? by Dr. Russell J. Rowlett (2006年08月10日).
• Math Course by Chris Fenwick, UCL (before 2006年06月18日). *
• World Science edited by Jack Lucentini (2006年05月03日).
• Continued Fractions (SMART) by Fred Schaal (before 2004年08月31日).
• Ramanujan and Ellipses by Ken Takusagawa (2004年08月22日).