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From: MCKAY john on 1 Apr 2010 13:37


From Chapter 26 of Conway-Sloane: Sphere Packings,
Lattices & Groups -- we find that the only square
based pyramids are either the trivial (m,n) = (1,1)
or (m,n} = (24,70), this corresponding to the identity

(0^2)+1^2+3^2+...+24^2 = 70^2.

This provides the isotropic Weyl vector in the lattice

II_{25,1} - see Chapter 27 for more details.

This yields the monster group M of order about 10^54=

2^46*3^20*5^9*7^6*11^2*13^3*17*19*23*29*31*41*47*59*71

defining the 15 monstrous prime divisors

{2,3,5,7,11,13,17,19,23,29,31,41,47,59,71}.

[These appear unexplained in Erdenberger MathSCiNet
MR 2092323 too.]

There seems the possibility that an evolving universe
based on the non-commutative geometry of Connes - Marcolli
may be where to find M as the temperature gets close to
absolute zero.

An indication of the significance of M in such physics can
be checked by the reader computing:

sum (c[k]^2, k=1..24)-c[70^2] mod 70

where c[k] is the coefficient of the Fourier coefficient
q^k in the elliptic modular function j(q) whose coefficents
are the moonshine degrees of M.

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