Miraculous Mathematics On the Beauty of Exceptions
and Coincidences
There are only two ways to live your life.
One is as though nothing is a miracle.
The other is as though everything is a miracle.
Albert Einstein^{ } (1979-1955)
(2004-04-02) _{ } The only magic hexagon
There's essentially only one way to arrange the first integers
in an hexagonal pattern so that all straight lines have the same sum (namely 38).
The magic hexagon pictured above is (up to symmetries) the only one
which features the first integers (although larger magic hexagons exist which
involve consecutive integers not starting with 1).
It's attributed to Ernst von Haselberg (1887)
but was rediscovered many times.
(2007-05-25) Numerical Coincidences in Man-Made Numbers
The law of small numbers applied to physical units.
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
p^{2}/3 10^{15} Hz.
The accuracy
of this pure coincidence is better than 8 ppm:
On the other hand, some "coincidences" were engineered or
enhanced by metrologists...
Consider some "nearly correct" conversion factors:
A typographer's point is 0.013837".
That's about 72.2700007227 points to the inch._{ }
An inch is 0.0254 m. That's about 39.37007874015748... inches to the meter.
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 = 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 entry is from the Iranian
gold trade :
100 g = 217 Ms
Not all such pairs of factors are interesting, because...
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
10^{n}-1
(Cunningham numbers).
The most valued prizes were
products of two factors with the same number of digits:
194841 x 513239 = 99999999999
248399691515827 x 402576989487237
= 99999999999999999999999999999
The latter was known to 19th-century mathematicians but was
apparently too complicated for use in elementary education (computers make it trivial).
(2008-01-14) Quadratic formulas giving long sequences of primes.
Quadratic polynomials giving long sequences of prime numbers.
The polynomial function P(n) =
n^{2} + 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) = n^{2} - 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 n^{2} - (2q-1) n + (41+q^{2}-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
n^{2 }- 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.
(2004-04-02) The Area under a Gaussian Curve (Gaussian integral)
A definite integral whose exact value is obtained with a unique method.
Yes, Virginia, there is a Santa Claus. ^{ }Francis P. Church (New York Sun,
Sept. 21, 1897)
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
=
ò_{0}
2pr
e^{-r 2 }dr
ò_{0}
2pr
e^{-r 2 }dr
=
p
ò_{0}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:
ò
+¥
_{ }e^{ - p x 2}
dx = 1
-¥
Thus, the function
e^{ - p x 2}
is a probability distribution whose variance is:
s^{ 2 }
= _{ }
ò
+¥
_{ } x^{ 2}e^{ - p x 2}
dx = ^{1}/_{2p}
-¥
[ 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) :
f (x) =
^{ }1^{ }
_{ } exp (
- x^{ 2}
_{ })
s
Ö
2p
2s^{ 2 }
(2007-04-18) Exceptional simple Lie groups :
E_{6} E_{7} E_{8}
F_{4} and G_{2} 5 additions to the 4 families of
simple Lie groups :
A_{n} B_{n}
C_{n} and D_{n}
Those 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 is E_{8 } (which may describe fundamental 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.
(2007-05-07) Monstrous Moonshine
(Simon Norton & John Conway)
A 1978 remark about 196884, made by John McKay to John Thompson.
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 and Robert L. Griess in 1973.
Calling it the Friendly Giant, Griess constructed it explicitely in
1981,
as the automorphism group of a 196883-dimensional commutative nonassociative
algebra over the rational numbers.
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).