### 24

#### Posted by John Baez

My final talk in Glasgow will be about the number 24.

This has long been my favorite number, since it shows up throughout math and physics in mysterious, shocking and even *monstrous* ways… which seem to fit together in a gargantuan conspiracy, as if pure mathematics itself were the work of an insane god — or at least one with a devilish sense of humor.

You can see the transparencies for this talk by clicking on the title below. If you see mistakes in it, please let me know!

Abstract: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$1 + 2 + 3 + 4 + \cdots = -1/12$

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

$1^2 + 2^2 + 3^2 + \cdots 24^2$

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 better-known but closely related fact is the period-12 phenomenon in the theory of “modular forms”. We shall do our best to demystify some of these deep mysteries.

I’ve been writing about these things for years in *This Week’s Finds*. The Glasgow Mathematical Trust promised to help me write up my talks as papers, and I hope to use this opportunity to assemble my scattered thoughts into something nice and coherent.

You may wonder why my very *first* and *simplest* talk, on the number 5, is the last one to show up here. That’s because I think more people will come to that one, including some schoolkids of ages 16 and 17. So, I want it to be really fun.

Note added later: you can now see a streaming video of this talk.

## Re: 24

To comment on only the most important aspect: I see you’ve switched from the nickel to the brass half of Home Depot’s excellent SoftCurve 4 inch numeral range. The talk posters stick to nickel.