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Tag: symmetry

megaminx

In a few
weeks I will give a _geometry 101_ course! It was decided that in
this course I should try to explain what rotations in $\mathbb{R}^3’$
are, so the classification of all finite rotation groups seemed like a
fun topic. Along the way I’ll have to introduce groups so bringing in a
little bit of GAP
may be a good idea. Clearly, the real power of GAP is lost on the
symmetry groups of the Platonic solids so I’ll do the traditional
computation of the transformation group of the Rubik’s cube. But
then I discovered that there is also a version of it on the dodecahedron
which is called megaminx so I couldn’t resist trying to work out the order of its
transformation group. Fortunately Coreyanne Rickwalt did already the
hard work giving a presentation as
a permutation group. So giving the generators to GAP


f1:=(1,3,5,7,9)(2,4,6,8,10)(20,31,42,53,64)(19,30,41,52,63)(18,29,40,51,62);
f2:=(12,14,16,18,20)(13,15,17,19,21)(1,60,73,84,31)(3,62,75,86,23)(2,61,74,85,32);
f3:=(23,25,27,29,31)(24,26,28,30,32)(82,95,42,3,16)(83,96,43,4,17)(84,97,34,5,18);
f4:=(34,36,38,40,42)(35,37,39,41,43)(27,93,106,53,5)(28,94,107,54,6)(29,95,108,45,7);
f5:=(45,47,49,51,53)(46,48,50,52,54)(38,104,117,64,7)(39,105,118,65,8),(40,106,119,56,9);
f6:=(56,58,60,62,64)(57,59,61,63,65)(49,115,75,20,9)(50,116,76,21,10),(51,117,67,12,1);
f7:=(67,69,71,73,75)(68,70,72,74,76)(58,113,126,86,12)(59,114,127,7,13),(60,115,128,78,14);
f8:=(78,80,82,84,86)(79,81,83,85,87)(71,124,97,23,14)(72,125,98,24,15),(73,126,89,25,16);
f9:=(89,91,93,95,97)(90,92,94,96,98)(80,122,108,34,25)(81,123,109,35,26),(82,124,100,36,27);
f10:=(100,102,104,106,108)(101,103,105,107,109)(91,130,119,45,36),(92,131,120,46,37)(93,122,111,47,38);
f11:=(111,113,115,117,119)(112,114,116,118,120)(102,128,67,56,47),(103,129,68,57,48)(104,130,69,58,49);
f12:=(122,124,126,128,130)(123,125,127,129,131)(100,89,78,69,111),(101,90,79,70,112)(102,91,80,71,113);

and defining the
megaminx group by


megaminx:=Group(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12); Size(megaminx);

and asking for its order I was a bit surprised to get
after a couple of minutes the following awkward number


33447514567245635287940590270451862933763731665690149051478356761508167786224814946834370826
35992490654078818946607045276267204294704060929949240557194825002982480260628480000000000000
000000000000000

or if you prefer it is
$2^{115} 3^{58} 5^{28} 7^{19} 11^{10} 13^9 17^7 19^6 23^5 29^4 31^3
37^3 41^2 43^2 47^2 53^2 59^2 61 .67 .71. 73. 79 .83 .89 .97. 101 .103.
107 .109 .113$

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Borcherdsโ€™ monster papers


Yesterday morning I thought that I could use some discussions I had a
week before with Markus Reineke to begin to make sense of one
sentence in Kontsevich’ Arbeitstagung talk Non-commutative smooth
spaces :

It seems plausible that Borcherds’ infinite rank
algebras with Monstrous symmetry can be realized inside Hall-Ringel
algebras for some small smooth noncommutative
spaces

However, as I’m running on a 68K RAM-memory, I
didn’t recall the fine details of all connections between the monster,
moonshine, vertex algebras and the like. Fortunately, there is the vast
amount of knowledge buried in the arXiv and a quick search on Borcherds gave me a
list of 17 papers. Among
these there are some delightful short (3 to 8 pages) expository papers
that gave me a quick recap on things I once must have read but forgot.
Moreover, Richard Borcherds has the gift of writing at the same time
readable and informative papers. If you want to get to the essence of
things in 15 minutes I can recommend What
is a vertex algebra?
(“The answer to the question in the title is
that a vertex algebra is really a sort of commutative ring.”), What
is moonshine?
(“At the time he discovered these relations, several
people thought it so unlikely that there could be a relation between the
monster and the elliptic modular function that they politely told McKay
that he was talking nonsense.”) and What
is the monster?
(“3. It is the automorphism group of the monster
vertex algebra. (This is probably the best answer.)”). Borcherds
maintains also his homepage on which I found a few more (longer)
expository papers : Problems in moonshine and Automorphic forms and Lie algebras. After these
preliminaries it was time for the real goodies such as The
fake monster formal group
, Quantum vertex algebras and the like.
After a day of enjoyable reading I think I’m again ‘a point’
wrt. vertex algebras. Unfortunately, I completely forgot what all this
could have to do with Kontsevich’ remark…

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