1+1 = 0
Math gets simpler in a world where 1+1=0, but it doesn't become self-contradictory and explode into nothing. We call this number system the field with 2 elements
About a year ago, Greg Egan and I were studying a lattice in 8 dimensions called E8 lattice, and a lattice in 24 dimensions called the Leech lattice.
In the E8 lattice each point has 240 nearest neighbors. Let's call these the first shell
. It also has 2160 second-nearest neighbors. Let's call these the second shell
We noticed some cool things. For starters, you can take the first shell, rotate it, and expand it so that the resulting 240 points form a subset of the second shell!
In fact, there are 270 different subsets of this type. And if you pick two of them that happen to be disjoint, you can use them to create a copy of the Leech lattice inside E8⊕E8⊕E8 — that is, the direct sum of three copies of the E8 lattice! Egan showed that there are exactly 17,280 ways to do this.
Tim Silverman, a friend of mine in London, has been thinking about this ever since. And he found a nice way to understand it using the field with 2 elements.
As he explains:“Everything is simpler mod p.” That is is the philosophy of the Mod People; and of all p, the simplest is 2. Washed in a bath of mod 2, that exotic object, the E8 lattice, dissolves into a modest orthogonal space, its Weyl group into an orthogonal group, its “large” E8 sublattices into some particularly nice subspaces, and the very Leech lattice itself shrinks into a few arrangements of points and lines that would not disgrace the pages of Euclid’s Elements. And when we have sufficiently examined these few bones that have fallen out of their matrix, we can lift them back up to Euclidean space in the most naive manner imaginable, and the full Leech springs out in all its glory like instant mashed potato.
Read the rest here:https://golem.ph.utexas.edu/category/2016/01/integral_octonions_part_12.html