M Theory Lesson 123

A while back we saw how the E8 lattice in eight dimensions arises from the 7 bit Hamming code with a check bit. Lattices have theta functions, such as the j invariant, which exhibits an interesting triality. The quantum E8 Dynkin diagram of Coquereaux also exhibits a kind of triality on its 24 vertices, which we think of as three copies of a cube's 8 vertices. Three cubes make up the off diagonal elements of the $3 \times 3$ octonionic exceptional Jordan algebra. The three real diagonal elements bring the dimension up to 27. As Garrett says, maybe triality has something to do with the generation structure of standard model particles and why they have different masses.