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# Mathematical Miracles

There aren't enough small numbers
to meet the many demands made of them
.
Richard K. Guy  (1916-)
The Strong Law of Small Numbers (1988)

## Miraculous MathematicsOn 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.

Wikipedia   |   Louis Hoelbling   |   Torsten Sillke   |   Magic Hexagon (MathWorld)

(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  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:

• 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
NumberFactorizationPairs of Factors
999998
2 ppm
2 . 31 . 1272 2 . 499999
31 . 32258
62 . 16129
127 . 7874
254 . 3937
10000011
1.1 ppm
3 . 7 . 31 . 15361 3 . 3333337
7 . 1428573
21 . 476191
31 . 322581
93 . 107527
217 . 46083
651 . 15361
999999
1 ppm
33 . 7 . 11 . 13 . 37 3 . 333333
7 . 142857
9 . 111111
11 . 90909
13 . 76923
21 . 47619
27 . 37037
33 . 30303
37 . 27027
39 . 25641
63 . 15873
77 . 12987
91 . 10989
99 . 10101
111 . 9009
117 . 8547
143 . 6993
189 . 5291
231 . 4329
259 . 3861
273 . 3663
297 . 3367
333 . 3003
351 . 2849
407 . 2457
429 . 2331
481 . 2079
693 . 1443
777 . 1287
819 . 1221
999 . 1001
1000001101 . 9901101 . 9901
10000022 . 3 . 166667 2 . 5000013 . 3333346 . 166667
9999999
0.1 ppm
32 . 239 . 4649 3 . 3333333
9 . 1111111
239 . 41841
717 . 13947
2151 . 4649
99999999
0.01 ppm
32 . 11 . 73 .101 .137 3 . 33333333
9 . 11111111
11 . 9090909
33 . 3030303
73 . 1369863
99 . 1010101
101 . 990099
137 . 729927
219 . 456621
303 . 330033
411 . 243309
657 . 152207
803 . 124533
909 . 110011
1111 . 90009
1233 . 81103
1507 . 66357
2409 . 41511
3333 . 30003
4521 . 22119
7227 . 13837
7373 . 13563
9999 . 10001

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  10n-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)  =  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   n- 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.

Prime-Generating Polynomials  by Eric W. Weisstein

(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    =   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:

 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 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 :   E6  E7  E8  F4  and  G2
5 additions to the 4 families of simple Lie groups :   An  Bn  Cn  and  Dn

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 (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.

The Scientific Promise of Perfect Symmetry:  A New-York Times article about E8, by Kenneth Chang
News about E8   by  John Baez  (2007-03-19)
Jean de Siebenthal   by  François de Siebenthal  (2008-03-05)

(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).

Week 173  by  John Baez  (2001-11-25)