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!

6 years ago

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.

ReplyDeleteKea -- Apparently access to my weblog is very difficult. Comments require moderation, and I never, these days, see any comments to moderate... which is how I like it. Only two commenters, apart from occasional spam, have managed somehow to access Log24. For their comments, see my m759.net/wordpress/?p=**** posts numbered 4736 ("The Curve of Beauty." Nov. 21, 2009)) and 11075 ("Reflection," Oct. 5, 2010). (I am having trouble putting HTML for links in

ReplyDeletethiscomment). At any rate, you have not beendeliberatelyblocked from commenting.Hi, m759. Yes, comment moderation is a pain, but necessary. As it happens, I never tried to comment on your blog anyway. I prefer to 'keep track' of theory related things here, on this blog.

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