## Sunday, January 1, 2012

### Higgsy Codes II

One more pertinent fact about the number $28 = 27 + 1$: it counts the number of bitangents to a quartic curve, or pairs of lines from the $56$ lines on a (degree $2$) del-Pezzo surface. Thus it gives the number of (odd) theta characteristics. For us, $56 = 2 \times 28$ will always be the dimension of the FTS for the $3 \times 3$ octonion Jordan algebra, or the $56$ triangles of the genus $3$ Klein quartic, which has an automorphism group of order $168 = 3 \times 56$ $= 7 \times 24$, as there are $7$ sides on the heptagon tiles.

The Mathieu group $M_{24}$ can also be constructed using the Klein quartic symmetries, along with an extra permutation associated to the (Leech lattice's) small cubicuboctahedron, as mentioned recently by kneemo. If we can have this much fun with classical codes, just imagine how much fun we can have with braids!

1. Steven Cullinane has a whole website based on the number $759$. His blog links to this paper by R. T. Curtis. As usual, I do not have access to it.