Comments on: what does the monster see?

By: Hurwitz’s Theorem on Automorphisms « Rigorous Trivialities

Hurwitz’s Theorem on Automorphisms « Rigorous Trivialities — Fri, 01 Aug 2008 15:04:00 +0000

[...] details of this have been done better than I could do them elsewhere, so I’ll just link to Lieven le Bruyn and Marni Dee [...]

By: Daniel Moskovich

Daniel Moskovich — Mon, 21 Jul 2008 11:28:28 +0000

Whoa!!! This is really cool!
There’s got to be an amazing story behind this… it’s got to be tremendously fun to do this kind of research!!

By: lieven

lieven — Sun, 20 Jul 2008 06:15:47 +0000

@mark : yes, i borrowed the monster-picture from Verrill's old monster-page and acknowledged this four years ago. You're right, the page seems to have disappeared (at least the links in that post no longer work...)

By: mark a. thomas

mark a. thomas — Sun, 20 Jul 2008 04:27:46 +0000

Noticed the reference. I wonder what ever happened to Helena Verrill’s wonderful Monster page?

By: javier

javier — Thu, 17 Jul 2008 08:25:49 +0000

That was really a fine piece of math… Thanks for sharing it!

By: lieven

lieven — Thu, 17 Jul 2008 07:33:27 +0000

@Jonathan,

in the paper by Wilson linked to in the post there is this table of info on the sporadic groups

For the 12 cases with (2,3,7)-generators one can play the same game as above, that is, there is a tiling of the curve with #G/7 heptagons, meeting in triples, and its genus g satisfies 84(g-1)=#G.

The dessin is just the Cayley graph for the group G wrt. the order 2 and 3 generators. For most other sporadics there are generators of order 2 and 3 so they also have an associated dessin and curve but it is a lot harder to say something about the genus of those...

By: Kea

Kea — Thu, 17 Jul 2008 01:14:37 +0000

Such a cute monster! Oh, now we can play around with a group algebra version of a PSL(2,Z) trinity, in the monster!

By: Jonathan Vos Post

Jonathan Vos Post — Wed, 16 Jul 2008 23:08:45 +0000

This is wonderful!

The order of the Monster has no reason but convention to be written in base 10. In base 7, for example, it is:

4431342225625260560256532232030236515450064652421143534121000000

In base 11 it is: 6291662405214765809949462152A63928802964363717906600

and in base 13: 2986240A341C4104C6A3418BB961286CCB150892CAA682000

and in hex: 86FA3F510644E13FDC4C5673C27C78C31400000000000

Since 2 and 13 divide it, we could get a pure alphabetic representation modulo 26. Or, since 2 and 3 divide it with sufficiently high powers, we could have a modulo 36 representation with all arabic digits and where A=10, B=11, ..., Z=26. Sadly, neither of these spells "MONSTER" or "FAFNIR" or "HURWITZ" any more than the digits of pi draw a circle, as Carl Sagan suggested in his one novel.

Can you summarize what we know of the Reimann surfaces from the sporadic groups in between the Klein quartic and the empire of the monster?