The numbers 12 and 24 play a central role in mathematics
thanks to a series of "coincidences" that is just beginning to be
understood. One of the first hints of this fact was Euler's bizarre
"proof" that
1 + 2 + 3 + 4 + … = 1/12
which he obtained before Abel declared that "divergent series are the
invention of the devil".
Euler's formula can now be understood
rigorously in terms of the Riemann zeta function, and in physics
it explains why bosonic strings work best in 26=24+2 dimensions.
The fact that
1^{2} + 2^{2} + 3^{2} + … + 24^{2}
is a perfect square then sets up a curious link between string theory,
the Leech lattice (the densest known way of packing spheres in
24 dimensions) and a group called the Monster. A betterknown
but closely related fact is the period12 phenomenon in the theory of
"modular forms". We shall do our best to demystify some of these
deep mysteries.
