The magic hexagon pictured above is (up to symmetries) the only nontrivial one which features the first integers. Larger magic hexagons exist which involve consecutive integers not starting with 1.
Its discovery (1887) is attributed to Ernst von Haselberg (1827-1905) but it was rediscovered independently many times.
Wikipedia | Louis Hoelbling | Torsten Sillke | Magic Hexagon (MathWorld)
The way in which some of our physical units where designed means that some numbers have no reason whatsoever to be "round numbers".
For example, the fact that the speed of light (Einstein's constant) is very nearly 300 000 km/s is a pure coincidence (it's now equated de jure to 299792458 m/s to define the SI meter in term of the atomic second of time).
Another pure coincidence is the fact that the diameter of the Earth is very nearly half a billion inches: The polar diameter is 500531678 inches, the equatorial diameter is 502215511 inches.
Rydberg's constant expressed as a frequency is very nearly p2/3 1015 Hz. The accuracy of this pure coincidence is better than 8 ppm:
3.289841960364(17) 1015 Hz = c R¥
[ CODATA 2010 ]
3.28986813369645287294483...
= p2/3
On the other hand, some "coincidences" were engineered or enhanced by metrologists... Consider some "nearly correct" conversion factors:
In both cases, the accuracy of the rounded inverse conversion factor keeps alive a competing unit based on that. The relevant numerical relations are:
254 . 3937 =わ 999998 13837 . 7227 = 99999999
Pairs of decimal numbers which have roughly the same number of digits and are very nearly reciprocals of each other can be designed around the factorizations into primes of numbers close to powers of ten. For example:
The second red entry is from the Iranian gold trade : 100 g = 217 MsNot all such pairs of factors are interesting, because...
[画像: Come back later, we're still working on this one... ]
At a time when authors of arithmetic textbook wanted to reward their students with interesting results to elementary problems, they would often build some of their exercises on nontrivial factorizations of 10n-1 (Cunningham numbers). The most valued prizes were products of two factors with the same number of digits:
194841 x 513239 =わ 99999999999
248399691515827 x 402576989487237
=わ 99999999999999999999999999999
The latter was known to 19th-century mathematicians but was apparently too complicated for use in elementary education (computers make it trivial).
Leonhard Euler (1707-1783) The polynomial function P(n) = n2 + n + 41 has a prime value for any integer n from 0 to 39 (it's divisible by 41 for n=40). This was first observed in 1772 by Leonhard Euler (1707-1783)
Since P(n-1) = P(-n) = n2 - n + 41 (Legendre, 1798) the above prime values of P(n) are duplicated when n goes down from -1 to -40. So, there are 80 consecutive values of n (from -40 to +39) which make P(n) prime (each such prime number being obtained twice).
Thus, the polynomial n2 - (2q-1) n + (41+q2-q) = (n-q)2 + (n-q) + 41 yields prime values for all integers from 0 to 39+q, provided that q is between 0 and 40. In particular (for q=40) the polynomial n2 - 79n + 1601 yields only prime values as n goes from 0 to 79 (namely, 40 prime values appearing twice each) as observed by Hardy and Wright in 1979.
Yes, Virginia, there is a Santa Claus.
Francis P. Church
(New York Sun,
Sept. 21, 1897)
This was first worked out by Abraham de Moivre (1667-1754) in 1730.
The challenge is to compute the integral I = ò e-x2 dx which represents the area under some Gaussian curve. The trick is to consider the square of this integral, which can be interpreted as a 2-dimensional integral which begs to be worked out in polar coordinates... The result involves the constant p.
I 2 =
ò e-x 2 dx
ò e-y 2 dy
=
òò
e-( x 2 +y 2 ) dx dy
=
ò0
2pr
e-r 2 dr
ò0
2pr
e-r 2 dr
=
p
ò0
e-u du
=
p
Therefore, the mystery integral I = ò e-t 2 dt is simply equal to Öp
Changing the variable of integration from t to x with t = Öp x yields:
Thus, the function e - p x 2 is a probability distribution whose variance is:
[ HINT: ( - x / 2p ) ( - 2p x e - p x 2 dx ) is easily integrated by parts. ]
Properly scaling the above gives the general expression of a normal Gaussian probability distribution of standard deviation s (and zero mean) :
They were discovered in 1887 by Wilhelm Killing (1847-1922). The completeness of Killing's classification of simple Lie groups was rigorously confirmed by Elie Cartan (1869-1951) in his doctoral dissertation (1894).
The most complicated of those is E8 (which may pertain to some aspects of physical reality). It describes 248 ways to rotate a 57-dimensional object.
The so-called Atlas Project culminated in an optimized computation about the representations of E8 which took 77 hours of supercomputer time to complete, on January 8, 2007. The output was a square matrix of order 453060, having entries in a set of 1181642979 distinct polynomials totalizing 13721641221 integer coefficients with values up to 11808808... all packed in 60 GB of data.
The structure of E6, identified with
SL(3,O)
by Aaron Wangberg (Ph.D. dissertation, 2007).
E7½
(2004/2005)
|
Joseph M. Landsberg &
Laurent Manivel (1965-)
The
Scientific Promise of Perfect Symmetry: A New-York Times article about E8,
by Kenneth Chang
News about
E8 by John Baez (2007年03月19日)
Jean
de Siebenthal by François de Siebenthal (2008年03月05日)
Universal Optimality of E8 and
Leech Lattice [
1
2
3
4
5
6
] Maryna Viazovska (IHES, 2019).
The Fischer-Griess Monster Group is also known as Fischer's Monster, or simply the Monster Group. It's (by far) the largest of the sporadic groups.
It was predicted independently by Bernd Fischer (1936-) and Robert L. Griess (1945-) in 1973. Calling it the Friendly Giant, Griess constructed it explicitly in 1981, as the automorphism group of a 196883-dimensional commutative nonassociative algebra over the rational numbers.
196883 = 47 . 59 . 71
In 1978 (before that proof was completed) John McKay (1939-) spotted the appearance of the number 196884 in an expansion of the modular j-function (A000521) and subsequently wondered about some unexpected relation with the Monster, in a letter to John Thompson (himself famous for the 1963 Feit-Thompson Theorem, which paved the road for a 20-year effort resulting in the final classification of finite groups).
Week 173
by John Baez (2001年11月25日)
|
Moonshine theory
|
Moonshine module (1988)
The Monster Group (8:29)
by John H. Conway (1937-2020).
