Wikipedia : Dice | Isohedra | Knucklebones
Answer : 16. (It's 240 if opposite faces needn't add up to 7.)
My local bargain store sells two types of dices from China, for 99¢ a pack.
Besides the obvious difference in sizes and colors, I noticed that the dices in the two packs were not alike because of the arrangement of the spots.
In this section, a "configuration" is understood to be merely a labeling of the faces of the die, regardless of the orientation of the labels... To each such configuration correspond 8 different orientations of traditional markings with spots. Thus, the 30 configurations enumerated below correspond to 240 ways to manufacture a die.
If the three faces 2-3-6 share one vertex (which they always do in a traditional die) then they can be arranged in two configurations (left-handed and right-handed). For either of those, there are 6 ways to arrange the "hidden" faces 1,4,5 (in only one of those is the sum of opposing faces always equal to 7). That accounts for a total of 12 possible configurations (so far).
If the 2-3-6 faces do not meet at a vertex, then two of them must be opposing each other (with the third sharing an edge with the other two). There are just three possible such configuration (think of it as choosing which of the 2-3-6 is the middle one). With respect to any such base, the other three faces can be arranged in 6 different configurations. That's a total of 18 possible configurations. Adding that to the 12 configurations described in the previous paragraph, we obtain a grand total of 30 possible configurations.
For either of the two ways the 2-3-6 faces can meet at a vertex, we have to put either 4 or 5 opposite to 6, then we have no further leeway. That accounts for 4 configurations (so far).
For any of the three ways two of the 2-3-6 faces are opposing each other, we have two choices for the face opposite to the "middle" face and two further choices for setting the remaining two faces (that are opposite to each other). That's 12 configurations to add to the 4 enumerated in the previous paragraph, for a grand total of 16.
All told, in the 30 ways to number a dice, there are two ways (right-handed and left-handed) in which opposing sides always add up to seven and 16 ways in which they never do.
A nicer and more elaborate solution :
Three-sided dice
by Don Simpson.
Dice alternatives and
substitutes
Monostatic polytopes
are "dice" with a single stable facet...
Here's a nontransitive set of three 3-sided dice (or 6-sided dice with the same spots on opposite sides) due to Dr. Nathaniel Hellerstein, CCSF :
Red = {3,5,7}. Yellow = {2,4,9}. Blue = {1,6,8}. With probability 5/9 in every case, red beats yellow, yellow beats blue and blue beats red:
Red beats Yellows
Transitivity is a fundamental property of ordering relations which formally states that if A ranks B and B ranks C, then A must rank C.
We're so used to ranking things transitively that not being able to do so is disturbing. None of the above three dice is preferable to both of the others. There are many other entertaining examples with dice.
A far more serious nontransitive case is democratic majority voting. Indeed, it's possible that voters who would prefer A to B and B to C would actually prefer C to A. That's Condorcet's paradox. An unfortunate fact of life.
Scams based on nontransitivity will fool people who don't know what to look for. One nice example is the so-called Penny Ante game which Walter Francis Penney (1913-2000) introduced in October 1969 (in 10 lines). Martin Gardner (1914-2010) discussed it in the Mathematical Games column of the October 1974 issue of Scientific American :
Each player predicts that a sequence of three heads (1) or tails (0) will occur in a sequence of flips of a fair coin before what the other has predicted. That game would be fair if both predictions were picked at random (as if inscribed on two balls drawn from an urn with 8 balls). However, the opportunity to pick a prediction knowing the choice of the first player always gives the second player an opportunity for a 2:1 advantage or better. As is often the case, an informed choice is better than a random one.
To justify the above optimal strategy, we have to compute the probability of a win for all possible responses of B to all choices of A. This is best done using an algorithm due to John Conway (1937-2020) presented in the paper of Humble and Nishiyama quoted in the footnotes below. Conway's algorithm applies to the generalized Penney game, where the two predictions need not be of length 3 (they don't even have to be of the same length).
With predictions of the same length k≥4 János Csirik (1946-) found in 1992 that Player A's best choice to limit the advantage of B is a sequence whose bits are all indentical except 3 (1 at the beginning and 2 at the end). This leaves a option with winning odds (1+2k-1):(1+2k-2).
"Optimal strategy for the first player in the Penney ante game" by J.A. Csirik,
Combinatorics, Probability and Computing, Volume 1, Issue 4 (1992), pp 311-321.
Steve Humble, MBE and Yutaka Nishiyama (1948-) prefer to play Penney Ante with a deck of cards instead (26 black cards and 26 red ones). Be that as it may, the exact probabilities involved in the "Humble-Nishiyama randomnsess game" differ slightly from those of the original Penney game and they're tougher to work out, unless a computer is used. In certain cases, there's even a nonzero probability of a tie (undecided game) which can't happen in Penney Ante with an unlimited number of coin flips. For example, when A bets "001" and B wisely replies with "100" the game is tied if "00" never occurs, which happens if and only if red and black always keep alternating, as they do with a probability of:
2 / C(52,26) = 4.03292... 10-15 which is minute but nonzero.
Intransitive dice
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Plus Magazine :
Curious Dice
by James Grime.
For sale at "Grand Illusions" :
4 Dice
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3 Dice
Intransitive_game
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Penney's game
Winning Odds
by Yutaka Nishiyama and Steve Humble.
This is not 50/50 (10:00)
by Kevin Lieber (Vsauce2, 2019年10月15日).
One die has pips 1,2,2,3,3,4 and the other is marked 1,3,4,5,6,8.
These dice were invented by Colonel George L. Sicherman (then of Buffalo, New York) whose discovery was reported by Martin Gardner, in one of his legendary Scientific American columns (1978).
The best way to investigate this matter involves generating polynomials. Besides proving the basic claim, this approach can establish the uniqueness of Sicherman's dice among 6-sided dice with nonzero markings:
To a face with n spots, we assign the monomial xn. To the whole die correspond the sum of the polynomials associated with its faces. For example, the polynomial associated to a standard die is:
S = x + x2 + x3 + x4 + x5 + x6
The number of ways we can obtain a total of n pips when we roll several dice is the coefficient of xn in the product of their polynomials (HINT: to obtain that term, you must sum up all the ways there are to pick one term from each factor so that the exponents of x add up to n).
Therefore, with two standard dice (a red one and a green one, say) the number of ways to roll a total of n pips is the coefficient of xn in the square of the above polynomial. Namely:
S2 = x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + x12
Now, the interesting remark is that S can be factored:
S = x ( 1 + x ) ( 1 - x + x2 ) ( 1 + x + x2 )
We may regroup the factors of the square S2 in the following way:
S2 = [x ( 1 + x ) ( 1 + x + x2 )] [x ( 1 + x ) ( 1 - x + x2 )2 ( 1 + x + x2 )]
The two square brackets expand respectively as follows:
x + 2x2 + 2x3 + x4
=
x + x2 + x2 + x3 + x3 + x4
and
x + x3 + x4 + x5 + x6 + x8
Those correspond to 6-sided dice marked 1,2,2,3,3,4 and 1,3,4,5,6,8. QED
What's somewhat miraculous is that we end up with a pair of 6-sided dice. To match what's done with traditional dice, those dice should be built with opposite faces adding up to 5 for the lower die and 9 for the upper die.
Other groupings of the above factors of S2 yield proper dice only when every resulting polynomial has nonnegative coefficents. We obtain:
Wikipedia
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Grand Illusions
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Col. Sicherman's home page
(CV)
"Sicherman dice" by
Ed Pegg Jr,
Alexander Bogomolny,
Arne Ledet, ...
Plus Magazine :
Let 'em Roll by Clare Hobbs (2006).
The standard set of seven polyhedral dice made popular by Dungeons & Dragons consists of the five platonic solids and a pair of 10-sided pentagonal deltohedra. One is marked from 0 to 9 and the other from 00 to 90. Those two are known as percentile dice. When rolled together, the percentile dice give any total from 0 to 99 with equal probability (1/100). In traditional role playing games (RPG) a total of zero (0+00) is interpreted as 100.
This exception wouldn't be needed if the low die was marked 1-10 instead of 0-9 (all other standard dice do start with 1). Some decimal dice are available which allow just that; they're simply not popular.
More generally, we may consider a set of p n-sided fair dice where the j+1st face of the i+1st die is marked j.ni ). When those dice are rolled, they give any total from 0 to np-1 with equal probability. Let's generalize:
For a prescribed integer M, what are the sets fair dice marked integers that will give any total between 0 and M-1 with probability 1/M ?
Well, the polynomial approach introduced in the previous section reduces this question to the factorization of the polynomial:
(1-xM ) / (1-x) = 1 + x + x2 + x3 + ... + xM-1
The factors of those polynomials are called cyclotomic ("cycle-splitting") and they've been studied and cataloged by generations of mathematicians.
Dismissing as trivial the type of splitting described in the above introduction, the first non-trivial factorization is for M = 6:
1 + x + x2 + x3 + x4 + x6 = (1+x) (1-x+x2 ) (1+x+x2 ) = (1+x3 ) (1+x+x2 )
A factorization gives a legitimate set of dice only if all the factors are polynomials whose coefficients are nonnegative integers. In this case, only three possibilities exist:
More generally, we can devise such a set of marked dice for any ordered factorization of the integer M. If M is prime, there's only one solution (a single die with M sides).
Rôle playing games (RPG) call for a variety of dice besides the traditional 6-sided cubic dice (D6). The most popular sets have 7 dice: 7 polyhedral dice
Ancient glass icosahedron (blue, severely chipped) Icosahedra were already used in Antiquity, for divination purposes. The large (52 mm) glass die shown at left is one of the most famous extant examples (c. AD 100). It was auctioned off at Christie's for 17925ドル on December 11, 2003.
Prior to that auction, it had drawn little attention and was expected to fetch between 4000ドル and 6000ドル. It would be worth a lot more now.
Note that Pluto (top symbol on the center die) was still a planet back then.
[画像: Tetrakis Cube ] In recent years, two distinct isohedra with 24 faces have been mass-produced as dice by Louis Zocchi (hear Lou's pitch). One is the isohedral tetrakis hexahedron or tetrakis cube pictured at left. The other is a large die in the shape of a deltoidal icositetrahedron [画像: Deltoidal Icositetrahedron ] (strombic icositetrahedron or trapezoidal icositetrahedron ). It's marketed by GameScience (Zocchi's company) under the name of D-Total, featuring fancy markings that are meant to facilitate the use of the die as a substitute for dice with 2, 3, 4, 5, 6, 7, 8, 10, 12, 20, 24, 30, 40, 50, 60, 70 or 80 sides. This is jointly credited to Dr. Alexander F. Simkin, Frank Dutrain (of LD Diffusion) and Louis Zocchi (2009).
GameScience Sales Pitch, by "Colonel" Lou Zocchi : 1 | 2 (2008年08月17日)
Polyhedral dice are loosely matched with 6-sided dice of similar bulk:
Oversized dice could damage dice trays. They're best tossed on carpets.
An isohedron is a polyhedron whoses faces are all equivalent. That's to say that every face can be transformed into any other through some spatial isometry (rotation or reflection) that maps the polyhedron onto itself.
The dual of an isohedron is an isogonal polyhedron, and vice-versa (duality being understood with respect to the sphere inscribed in the isohedron or circumscribed to its dual). Sometimes, the dual is more readily understood than the primal.
The dual of a convex solid (polyhedral or not) is convex. So, every convex isogonal polyhedron is associated to an isohedron and vice-versa. In particular, every Catalan solid (i.e., the dual of one of the 13 Archimedean polyhedra) is an isohedron. So are the duals of isogonal prisms and antiprisms (respectively called amphihedra and deltohedra).
However, there's no requirement that the dual of an isohedron be equilateral. So, there are isohedra that are not duals of uniform polyhedra (as uniform means both isogonal and equilateral). [画像: Isogonal Tetrahedron ]
For example, the familiar regular tetrahedron is not the only isohedral tetrahedron... Any tetrahedron whose opposing edges have the same length (as illustrated at right) is an isohedron. Such a tetrahedron is called a disphenoid. The dual of a disphenoid is another disphenoid; disphenoids are both isogonal and isohedral (they're thus noble).
Therefore, any disphenoid would make a perfectly fair 4-sided die. A disphenoid is chiral iff the three quantities a, b and c are distinct.
The full classification of all isohedra is given elsewhere on this site.
Currently, "6-sided" spherical dice are available (the inner cavity is octahedral ).
Any isohedral die could be made this way, by carving out a cavity in the shape of its dual (the dual of an isohedron is an isogonal polyhedron ). However, the fairness of such a die would only be guaranteed if the cavity was isohedral as well. In other words, it must be noble (that word simply means both isogonal and isohedral).
Among convex polyhedra, the only noble ones are the Platonic solids and the disphenoids. The latter type would yield a great way to make 4-sided dice without any sharp corners (in fact, without any corners at all). If a scalene disphenoid is used, the artefact would be a sphere where the outer markings would seem asymmetrically distributed. Yet, it would be a perfectly fair die and, therefore, a great conversation piece...
If d is the degree of every vertex in an isogonal polyhedron, it has at most d different edge lengths and d non-congruent faces. When both maxima are achieved, the isogonal polyhedron is said to be scalene.
On the other hand, an isogonal polyhedron where at least two adjoining edges have the same length can be called isosceles. Any polyhedron where all edges have the same length is said to be equilateral. A polyhedron that's both isogonal and equilateral is said to be uniform There are 75 or 76 nonprismatic uniform polyhedra (the convex ones are the 5 Platonic solids and the 13 Archimedean solids).
Consider now the three distinct types of isogonal tetradecahedra obtained by cutting off the eight corners of a cube with increasing severity:
[画像: Cube ] [画像: Truncated Cube ] [画像: Cuboctahedron ] [画像: Truncated Octahedron ]
In the first two cases, the truncation must be the same at all corners of the cube, or else we wouldn't obtain a polyhedron with isogonal symmetry.
If all such truncations are alike, then the resulting isosceles solid has square faces and is more readily obtained by "classical" truncation of a regular octahedron, without creation of new edges (the scalene version can't be so constructed, because the 8 planes supporting the hexagonal faces don't form an octahedron). Therefore, the locution isogonal truncated octahedron can only denote the tetradecahedron with square faces (and congruent hexagonal faces) discussed next as the shape of traditional Korean dice.
By definition, an isohedron is said to be scalene if its dual is (the dual of an isohedron is an isogonal polyhedron and vice-versa). For example, the dual of the polyhedron just described is a scalene isohedron (a fair die) with 24 faces congruent to the same triangle (scalene tetrakis hexahedron).
The juryeonggu is precisely an isogonal truncated octahedron.
Because its tetragonal faces are square, it's a special case of the above solid. The shape is fully specified by the length (x) of the edges between hexagons, assuming the square faces have unit sides.
For x = 1, we would obtain the uniform truncated octahedron (pictured above at left) which features regular hexagonal faces. (Don't call it a cuboctahedron, which is another uniform tetradecahedron, as is the truncated cube.) This Archimedean solid can tile space without voids (a non-uniform juryeonggu can't). It's the basis for the near-optimal foam described by Lord Kelvin in 1887 (Kelvin's cell has the same vertices as the tetradecahedron but all its edges are curved and so are its hexagonal faces).
For any x, a juryeonggu has the following dimensions (derived below). The locution "radius to" denotes the distance from the polyhedron's center. The radius to any vertex is the radius R of the circumscribed sphere.
R can be obtained as the radius of the equator circumscribed to an isogonal octagon whose sides are either the diagonal of a square face (length Ö2) or an edge of length x. In that equatorial plane, we also find h4 and R6 :
R2 = [ 1 + (1+x)2 ] / 2 = 1 + x + ½ x2
h4 = ( 1 + x ) / Ö2
R6 = 1 + x/2
An hexagonal face is obtained by subtracting three equilateral triangles of side x from an equilateral triangle of side 1+2x, so its surface area is:
A6 = Ö3 / 4 [ (1+2x)2 - 3x2 ] = Ö3 / 4 ( 1 + 4x + x2 )
The diagonal of an isogonal hexagon with sides 1 and x is (1+x). The width between parallel sides is equal to that diagonal multiplied by the sine of 60°. The least obvious quantity is r6 which can be obtained from planar cartesian coordinates (that's what we use when all else seems to fail). We then obtain h6 from the Pythagorean theorem:
R2 = r62 + h62
R4 is the height of an isosceles triangle of base 1 with two sides equal to R.
R42 = R2 - ¼ = ( 3 + 4x + 2x2 ) / 4
We may check that R4 = R6 in the uniform case (x = 1). The solid angle subtended by a square face is obtained immediately from R4 using the formula we've established elsewhere on this site. For hexagonal faces, we just use the fact that the solid angles subtended by all faces add up to 4p.
The official blog of the city of Gyeongju states that, in a traditional juryeonggu, all faces have the same surface area. This entails a quadratic equation in x, whose positive root is:
x = Ö( 3+4/Ö3 ) - 2 = 0.304213765421624907891...
Gerard Villarreal (private communication) advocates a juryeonggu with h4 = h6 so that all faces are tangent to the same inner sphere (which may then be called the inscribed sphere, by analogy with the isohedral concept). This entails a quadratic equation whose positive solution is:
x = (Ö3 - 1) / 2 = 0.36602540378443864676372317...
In either case, it was guessed that endowing an isogonal truncated octahedron with a particular property that isohedra possess would endow them with the same fairness as isohedral dice. It ain't quite so...
Oversimplifying the conclusions of the discussion below, the latter guess turns out to correspond to the situation where the die bounces a very large number of times. At the other extreme is a die that doesn't bounce at all (think of a randomly oriented die immersed in glycerol an dropped just above a sticky surface). Such a die would simply land on any face with a probability proportional to the solid angle subtended by that face from the center of the polyhedron. All faces subtend the same solid angle (2p/7) when:
x = (Ö((1 / 2sin p/14 )-½) - 1 = 0.32173356003298450750124...
Since x can't be both 0.3660 and 0.3217 (obviously) no isogonal truncated octahedron can be unconditonally fair (like an isohedron would be). However, for any given set of physical conditions and casting style, there's one isogonal truncated octahedron that looks fair...
As dice are actually rolled in specific conditions that are somewhere between the two (contrived) extremes described above, a juryeonggu that looks fair in practice would have to correspond to a parameter x determined empirically under those given conditions.
One way to do so is to build two isogonal truncated octahedra with different parameters x1 and x2 (not too far from a guess of 0.35).
Cast both 700 times and count how many times they land on an hexagonal face (N1 and N2 respectively). By linear interpolation, we'd approach a perfect score of 400 for a die having the following value of x:
The two most interesting dice to build are the ones mentioned above:
x1 = 0.32173356003298450750124...
x2 = 0.36602540378443864676372...
If you build a third die using the above interpolation, you may roll your three dice many times and plot with good precision the curve giving the probability of an hexagon as a function of x (with just 3 known points, you may as well assume the curve is a parabola or a circle). Use that information to build a fourth die, if you must. That last die may not be quite fair (with your own particulars) but it's unlikely that anybody will ever detect that!
The precision of the above method is limited by the standard deviation on N, which is about 13 for nearly-perfect dice with 700 trials (it's proportional to the square root of the number of trials).
Carved Tetradecahedron I've carved a 23.9 mm orthohedral (x = 0.366) juryeonggu to a precision of 0.05 mm and obtained the results tabulated below. They show that an hexagonal face is about twice as likely as a square one, although an hexagon subtends only 9.4% more solid angle than a square.
Orthohedral Juryeonggu on Felt-Covered Wooden Dice Trays 2 = A.B / (A+B)
Casting
the 14-sided juryeonggu Official blog of Gyeongju, South Korea (2011年07月06日)
juryeonggu.net
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Juryeonggu images from Korean pages
"Tetrakaideccahedronnocube"
"TET" & "Bocralette",
marketed as Rolla-Strike (1985).
Justin Mitchell's (D14) Fourteen-Sided Dice.
"Colonel" Lou Zocchi first earned a solid reputation in the manufacture and sale of isohedral dice which he pioneered in 1974, riding the wave of the increasing popularity of role playing games (RPG).
For lack of symmetry among their faces, the fairness of non-isohedral dice may depend critically on the way they are cast and/or on the resilience of the landing surface. Such dice may roll true on plexiglass but not on felt (say) or the imparted spin (long roll or not) may bias them.
Nevertheless, those things can substitute for fair dice with odd numbers of sides (all isohedra have an even number of faces). A good example is the 7-sided die pictured above (16.3 mm thick, as devised by Lou Zocchi) which I found to roll true under typical conditions (two dice tossed together 392 times from a ribbed cup onto a circular felt-covered wooden tray).
For non-isohedral dice to give at least the illusion of rolling true, they must be designed by trial and error and tested extensively. Sometimes, the necessary rigor isn't exercised in commercial endeavors...
In his patent application, Zocchi stated that the device had been tested by rolling it 10163 times on plexiglass and was found to be practically fair under such conditions. I find this pretty hard to believe after the following field-test:
Using my trusted ribbed cup and felt-covered dice tray, I cast 5 such dice together 100 times (for a total of 500 individual outcomes) and saw the dice land 282 times on one of the two triangular faces and only 218 times on one of the three rectangular sides. So, the 28.2 % share of each triangular face was nearly twice the 14.5 % share of each rectangular face. Bad.
If needed in actual gaming, a fair D5 die can be nicely simulated by rolling a good D6 cube until the outcome isn't 6 ("casino" dice are machined from extruded cellulose acetate to a precision better than 0.0005"). The average number N of actual rolls required for a valid outcome is only 1.2, since :
N = 1 + N/6 yields N = 6/5
Other good alternatives exist which forego extra rolls entirely, including the use of a 5-sided spindle or a 10-sided isohedron with duplicate labels.
Louis J. Zocchi (c.1935-)
is also a part-time
magician
(excerpts: 1 |
2 )
Experimentally obtained statistics
of dice rolls (6th Experimental Chaos Conference, Potsdam 2001).
Fair dice do not create randomness or uniform probability distributions; they merely conserve it. When thrown from a truly random orientation, a fair die is equally likely to land on any face.
A sufficient condition for an homogeneous convex polyhedron to be a fair is to be isohedral, for the above is then satisfied by reason of symmetry.
An isohedron is a face-transitive polyhedron. This is to say that every face is the image of any other in at least one isometric transformation of the entire polyhedron (i.e., a rotation or a mirror reflection mapping the polyhedron onto itself).
An isohedron is thus a polyhedron where all faces are congruent to each other. The converse need not be true. For example, the convex deltahedra with 12, 14 or 16 faces aren't isohedral.
Isohedral symmetry is precisely what guarantees that all faces are strictly equivalent. In any type of damped motion (including, but not limited to, inelastic shocks with fixed objects of any shape) if all initial orientations of an homogeneous rigid isohedral die are equiprobable, then it will certainly come to rest on any on its faces equiprobably.
This conclusion certainly holds for the Newtonian mechanics of rigid bodies, which will be our only concern in the rest of the discussion. It would also hold for other mechanical laws, including special and general relativity, that can deal with the fiction of homogeneous and isotropic matter and are insensitive to chirality (this last restriction comes from the fact that chirality-changing transformations are allowed as isohedral transformations). The symmetry argument is otherwise general enough to deal with fragile isohedra made from (amorphous) gelatin or rubber. Whenever the final integrity of such an isohedral die is sufficient to identify it as resting on some face, it will be any face with equal probability!
Revisit now the argument we used for isogonal truncated octahedra to prove that one value of the single parameter describing those shapes must correspond to a fair die which isn't isohedral (no truncated octahedron can possibly be isohedral, since squares and hexagons can't be congruent).
That theoretical argument (and/or the practical recipe we gave to determine something close to the correct shape using linear interpolation) was based on the implicit assumption that dice are always cast the same way (on an horizontal table covered with a particular shock-absorbing material, say).
Video : Statistical Mechanics: Lecture 1 (121 min) by Leonard Susskind (2009年03月30日)
The thesis [master's thesis | pdf] filed in 1997 by Ed Pegg, Jr. for his Master's degree at UCCS was entitled A Complete List of Fair Dice. In it, the famous recreational mathematician actually classified isohedra. He was fully aware of the remote possibility that some fair dice might exist besides isohedra (which are fair by symmetry) but he clearly estimated (rightly so) that the less technical term was more suitable for a title. Elsewhere, he answered the question "Can a non-isohedral fair die exist?" by using the example of a square pyramid with isosceles lateral faces, thrown as a die under some set of standard conditions. The four triangular faces have the probability by symmetry. The probability of the square face is above that if the pyramid is tall and below that if the pyramid is short. Therefore, an intermediate height must exist for which all faces have the same probability. He adds:
However, once the conditions changed, the die would no longer be fair. (I have a strong argument for this, but no proof.)
What follows can serve as the proof that Ed Pegg, Jr. is calling for. The main difficulty was that the lack of fairness of something like the aforementioned square pyramid can only be established if we analyze theoretically at least two sets of physical conditions. The trick is to consider two limiting cases rather than any realistic motion. A fair die could theoretically be thrown in any possible and couldn't show a difference in probabilities between those two limits, which turn out to be simple enough to analyze.
In 1981, David Singmaster (b. 1939) discussed the proposal that an homogeneous die would land on a face with a probability proportional to the solid angle subtended by that face (as seen from the center of gravity).
At first sight, the idea looks silly... For one thing, this would assign nonzero probabilities to unstable faces (whenever the orthogonal projection of the center of gravity on the plane of the face isn't inside the face, the die cannot rest on that face at all). Also, it would seem to overestimate the probabilities of lateral faces in flat prisms (our physical intuition is that a coin can stand on its edge but will never land on it).
Nevertheless, we can describe a contrived quasistatic regime that leads to that conclusion. Admittedly, dice are never cast this way but it's an idealized limit of a physical situation and fair dice ought to be fair under any conditions, including those (like isohedra are). Therefore, fair dice must be equispherical (all faces must subtend the same solid angle).
First, we assume that the horizontal plane on which dice land is infinitely sticky, so that a die can pivot about a vertex or an edge but will never roll. Thus, it can come to rest on a surface that couldn't serve as a stable resting place without such stickiness (a spindle could land on its pyramidal extremities).
Second, we assume dynamical (inertial) effects are negligible. This would happen if the die was moving at low speed in a very viscous fluid. We may also assume that this fluid is only slightly less dense than the homogeneous stuff the die is made from. The die is fully immersed in the fluid, at rest in a random orientation. Upon release, it will essentially fall at constant speed (its terminal velocity) without rotating.
discussed below is an idealized way dice could actually be cast. What's crucial is the fact that this describes an actual physical situation (or, at least, the idealized limit of such situations). The conclusions derived from this contrived model would thus be valid for any die which is intrinsically fair (i.e., unconditionally fair) in the same way convex isohedra are: A fair die lands with equal probability on any of its faces regardless on the surrounding conditions, provided it starts at rest in a random orientation.
Under such contrived conditions, the die will simply land on whichever face is directly below its center of gravity.
The assumption that the die is initially randomly oriented means that the downward vertical through the center of gravity crosses a face with a probability proportional to the solid angle it subtends, as seen from the center of gravity. That same probability is also the probability that the die will land on the prescribed face.
For example, an isogonal truncated octahedron makes a fair die under such quasistatic consitions (dead-cat bounce) when the ratio of an edge between hexagons to the side of a square face is:
x = 0.32173356003298450750124...
When placed on an agitated horizontal plate mimicking thermal motion, dice would naturally tend to orient themselves in the most energetically favorable way. That's achieved by lowering the center of gravity as much as possible.
By reducing the amplitude of the agitation gradually until motion freezes, we effectively cast the dice in a way that definitely favors, for the bottom position, the faces which are closest to the center of gravity.
An equispherical die that's fair under the previously described quasistatic conditions cannot be fair with the thermal tossing method as well, unless all its faces are also equally distant from the center of gravity.
Thus, all the faces of a fair die should be equally distant from the center of gravity and also subtend the same solid angle. Isohedra clearly meet both conditions by symmetry. Below are examples of non-isohedral dice that satisfy this restriction (the solids that don't cannot possibly be fair dice). Let's establish the vocabulary:
An orthohedron is defined as a polyhedron with an inscribed sphere; the center of that sphere is equally distant from all the faces.
A balanced isohedron with spherical inertia (i.e., its three principal moments of inertia are equal) is a fair die under all tossing conditions, by reasons of symmetry. So is a balanced isohedron with an axis of symmetry and mere cylindrical intertia about that axis (the moment of inertia about the axis of symmetry can differ from the moments about perpendicular axes). This later case applies to bipolar dice (amphihedra or deltohedra). Whichever relevant condition is automatically satisfied for isohedral solids of uniform mass density (ordinary unloaded dice).
By contrast, the fairness of a balanced mesohedron is only guaranteed for the two extreme casting methods described above (quasistatic or thermal regimes) which we may loosely think of as dead cat bounce and high resilience, respectively. For intermediate conditions, no such guarantee exists.
Wikipedia : Bisection
Pair of 10-sided rubber dice The so-called fitness dice depicted at left are a pair of 10-sided latex rubber dice meant to be rolled together to suggest a type of physical exercise and a number of repetitions to perform. The dice are large enough (7" height) to be tossed on a gym floor. A pair retails for about 30ドル.
Those seem to be shaped roughly like 10-sided mesohedra. Let's describe what a perfect 10-sided mesohedron would be:
A die with that general appearance will be equispherical if and only if both square faces subtends a solid angle of 2p/5 each.
Indeed, by symmetry, the rest of the spat (4p) is shared equally among the eight other faces, so each of those also subtends a solid angle of 2p/5.
On the other hand, a polyhedron is orthohedral iff one point (the center) belongs to all bisectors of its face angles (i.e., the dihedral angles formed when two faces meet at an edge).
Technically, a bisector is a set of two orthogonal planes consisting of all points equally distant from two intersecting planes. In the case of a convex polyhedron, we may focus our attention to only a quarter of that figure (a half-plane bordered by an edge of the polyhedron and featuring a nonempty intersection with its interior).
A mesohedron, is an equispherical orthohedron. Putting both of the above conditions together yields the following cross-section (in a plane perpendicular to half of the horizontal edges). The angle q is determined by the aforementioned equisphericity condition:
[画像: Hexahedral cross-section of 10-sided mesohedral die ]
q =
Arctg 1/5¼ = 33.7722424...°
The above area of a trapezoidal face is equal to its height f/2 multiplied into the half-sum of its bases (1+f)/2. It could also be obtained with Brahmagupta's formula (since an isosceles trapezoid is indeed a cyclic quadrilateral ). The total volume is twice the volume of a conical frustum.
Note one similarity with the geometry of a sphere: The surface area of this polyhedron happens to be four times the area of its equatorial cross-section.
Wikipedia : Bisection
Consider a tetragonal pyramid with two vertical planes of symmetry. Its horizontal base is a rhombus (i.e., an equilateral quadrilateral) and it has an inscribed sphere by reasons of symmetry, since the bisectors at the four horizontal edges do intersect at a point on the vertical axis.
Such a shape depends on three parameters; the vertical height h and the half-diagonals of the horizontal rhombus, x and y. Alternately, we could consider as parameters the three different side lengths, a, b and c :
a 2 = x 2 + h 2 b 2 = y 2 + h 2 c 2 = x 2 + y 2
One parameter determines the size. We may adjust the other two to make a uniform-density solid both equispherical and balanced. Here it goes:
The tangent of the dihedral angle based on an horizontal edge is hc/xy. (HINT: xy/c is the altitude of the right triangle of sides x and y.) The inclination of the bissecting plane (with respect to the horizontal) is half the angle corresponding to that... Now, in a straight pyramid of uniform mass density, the center of gravity is at a height z = h/4. Thus, the tangent of the bissector's inclination must be ¼ which makes the original dihedral angle's tangent 2t/(1-t2) = 8/15. All told, the solid is balanced when:
hc / xy = 8 / 15 or z (x2+y2 )½ / xy = 2 / 15
The solid is equispherical with respect to the center at altitude z if and only if the base subtends a solid angle of 4p/5. Indeed, since the other 4 faces share equally the rest of the spat (4p) by symmetry, this make them subtend the same solid angle of 4p/5. Using our expression for the solid angle subtended by a rhombus, this condition translates into:
To obtain an heptahedron with a vertical ternary axis of symmetry, we may truncate off horizontally one pole of a bipolar polyhedron with a vertical ternary axis. That's to say, either a trigonal dipyramid or a trigonal deltohedron.
In either case, the final solid can't be an orthohedron unless the untruncated side is flatter than the side of the truncated pole.
[画像: Come back later, we're still working on this one... ]
"Chestahedron" (2000) by scultptor Frank Chester (b. 1939).
A cylindrical die is an homogeneous die allowed to bounce repeatedly on an horizontal plane, without loss of energy.
[画像: Come back later, we're still working on this one... ]
"Physics
of Dice" by Antonio Recuenco-Munoz (May 2006).
"Unlikely
Landings: Dice, Coins and the Mars Pathfinder" by Gary White.
"Dice
Landing Probabilities" by Gary White & al. (Society of Physics Students)
"Predicting a Die Throw
(Science Daily, 2012年09月12日) from an AIP press release for:
"The three-dimensional dynamics of the die throw" in
Chaos, 22, 8 pages (December 2012)
by Marcin Kapitaniak, Jaroslaw Strzalko, Juliusz Grabski &
Tomasz Kapitaniak
(Lodz, Poland)
Let's consider a polygonal wheel bouncing on an horizontal track.
