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Refurio Anachro (Feb 06 2021 at 13:31):H.S.M. Coxeter found the 57-cell in 1982, a few years after Branko Grünbaum had found the similar 11-cell, or hendekachoron, in 1976. But Coxeter didn't notice, so he independently rediscovered the 11-cell in 1984.
Both are abstract, locally projective polytopes, that is, their faces and vertex figures are tessellations of projective planes. They are similar to the hemi-platonic polyhedra one can obtain by identifying opposite features of platonic polyhedra.
The 57-cell is a regular 4-polytope, meaning its largest-dimensional toroidal cells have 4 dimensions. It consists of 57 hemi-dodecahedra, the same number of vertices, making up 171 pentagons and again, the same number of edges!
It is self-dual, its symmetry is L_2(19), also known as PSL(2,F_19), and of order 57•60=3420.
The 57-cell's skeleton is the Perkel graph, which is the unique distance-regular graph with intersection array {6, 5, 2; 1, 1, 3}. It has the Petersen graph as subgraph, 57 times, forming a single orbit!
Refurio Anachro (Feb 06 2021 at 13:32):The image by Roccini shows some of its embeddings with 19-fold symmetry. Source:
https://en.m.wikipedia.org/wiki/File:Perkel_graph_embeddings.svg
Refurio Anachro (Feb 06 2021 at 13:32):1024px-Perkel_graph_embeddings.svg.png
Refurio Anachro (Feb 06 2021 at 13:34):This graph is also distance-transitive, which means that whenever we have two pairs of vertices with the same distances between their vertices, then there's an automorphism between the two pairs.
This is true for any skeleton of a platonic graph, but in fact there are only 12 3-valent distance-transitive graphs, which implies we can't keep building ever higher-dimensional 3-valent vertex-transitive polytopes where this holds.
The definition of distance-transitive graphs was given only in 1971 by Norman Linstead Biggs and D. H. Smith, preceding the discovery of the 57-cell by over a decade.
That there are graphs that are distance-regular but not distance-transitive was discovered just two years earlier in 1969 by a group lead by Georgy Adelson-Velsky. Two nice examples are the 3-valent Tutte 12-cage and the 6-valent Shrikhande graph.
Anyways, Manley Perkel was looking for bounds on the valence of polygonal graphs with odd girth when he came up with his example in 1979.
To find a realization of the 57-cell one has to go up to dimension 56! But even that is not faithful: its 3-cells are skew, which means they aren't contained in flat R³.
It seems, Coxeter hadn't heard about any of this. He certainly wouldn't have needed to. Typically, he would systematically classify all the things, before widening his perspective slightly so he could continue his quest: cartographing as much of geometrical cosmos as he could!
https://en.wikipedia.org/wiki/Abstract_polytope
https://en.wikipedia.org/wiki/11-cell
https://en.wikipedia.org/wiki/57-cell
https://en.wikipedia.org/wiki/Distance-transitive_graph
https://en.wikipedia.org/wiki/Perkel_graph
For some more details about the Perkel graph see here:
https://www.win.tue.nl/~aeb/graphs/Perkel.html
Carlo H. Séquin and James F. Hamlin attempted to visualize this wildly sel-intersecting and warped 57-cell for SIGGRAPH '07, giving a nice construction and insights about its embeddings in R³:
http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
Note that they did a similar thing for the 11-cell, see their paper for a reference.
For full detail look at Coxeters paper "Ten toroids and fifty-seven hemidodecahedra"
doi:10.1007/BF00149428
John Baez (Feb 06 2021 at 16:16):When I hear "57" I think of the 57-dimensional space that's the lowest-dimensional manifold on which E8 acts. I also think about ketchup.
But I doubt either of these are connected to the 57-cell.
John Baez (Feb 06 2021 at 16:22):The easiest way to get a handle on the 57 here might be to think about PSL(2,F_19). The number of 2x2 matrices with entries in F_19 is . Requiring that the determinant is 1 knocks this down to , I think. I'm not remembering how the count changes when you mod out by the center and get PSL(2,F_19).
John Baez (Feb 06 2021 at 16:22):But for some reason it's 57 60 = 3420, if you're right about this.
John Baez (Feb 06 2021 at 16:24):The fact that 57 = 19 3 is clearly gonna be relevant here.
Todd Trimble (Feb 06 2021 at 17:04):John Baez said:
The easiest way to get a handle on the 57 here might be to think about PSL(2,F_19). The number of 2x2 matrices with entries in F_19 is . Requiring that the determinant is 1 knocks this down to , I think. I'm not remembering how the count changes when you mod out by the center and get PSL(2,F_19).
Well, GL(2, F_19) has (19^2 - 1)(19^2 - 19) elements, which is 20 x 18 x 19 x 18, and so SL(2, F_19) has 20 x 19 x 18 elements, and then divide by 2 to get PSL(2, F_19). That's 10 x 18 x 19 = 10 x 6 x 3 x 19 = 60 x 57.
Todd Trimble (Feb 06 2021 at 17:06):John Baez said:
When I hear "57" I think of the 57-dimensional space that's the lowest-dimensional manifold on which E8 acts. I also think about ketchup.
But I doubt either of these are connected to the 57-cell.
Could you clarify? E8 acts on a point, for instance. Do you mean acts faithfully, or something?
John Baez (Feb 06 2021 at 17:35):Well, GL(2, F_19) has (19^2 - 1)(19^2 - 19) elements, which is 20 x 18 x 19 x 18, and so SL(2, F_19) has 20 x 19 x 18 elements, and then divide by 2 to get PSL(2, F_19). That's 10 x 18 x 19 = 10 x 6 x 3 x 19 = 60 x 57.
Thanks. I was not awake enough, just waking up early to go grocery shopping. As I was driving along I started trying to do the calculation right. When you have a square matrix over a finite field it's not true that all determinants are equally likely: that was my downfall. If it were true, SL(2,F_19) would have 19 x 19 x 19 elements, which is 19 more than the actual answer, 20 x 19 x 18. But once you know the matrix is invertible, all determinants are equally likely.
John Baez (Feb 06 2021 at 17:40):(Here "likely" is with respect to counting measure: I could have said "numerous" but it doesn't have the same ring.)
John Baez (Feb 06 2021 at 17:49):Todd Trimble said:
John Baez said:
When I hear "57" I think of the 57-dimensional space that's the lowest-dimensional manifold on which E8 acts. I also think about ketchup.
Could you clarify? E8 acts on a point, for instance. Do you mean acts faithfully, or something?
I meant nontrivially. Using some technology I don't understand you can work out the maximal Lie subgroups of a simple Lie group, where "maximal" means "maximal proper", and then look for the one of highest dimension. This gives the lowest-dim manifold on which the group acts nontrivially.
John Baez (Feb 06 2021 at 18:34):Hmm, it clearly gives the lowest-dim manifold on which the group acts nontrivially and transitively. Are the orbits of a Lie group action on a manifold always submanifolds?
John Baez (Feb 06 2021 at 18:35):That should be the same dimension, but right now I'm not seeing why...
John Baez (Feb 06 2021 at 18:36):Oh, and another thing: all this stuff is for the complex form of E8, so my dimensions are complex dimensions!
Refurio Anachro (Feb 06 2021 at 23:30):Yes, the numerology is strong with this one. I could barely resist mentioning that 57 isn't a prime =)
I thought you might like it. Yes, octonions, they have most likely nothing to do with this.
Coxeter and his relentless churning through the landscape, such an inspiring character! I bet there are many more layers of polysomethings to be uncovered by careful generalization of geometry. Much of what he did seems so unavoidable, and not at all arbitrary.
I wish I had a much better view on these things. I want to learn more about algebraic geometry, even tough that seems pretty un-Coxeterian to me. And mighty difficult =)
John Baez (Feb 06 2021 at 23:41):I got a bit sidetracked, but I was going to say: maybe there's a nice 60-element subgroup of PSL(2,F_19), say H, such that PSL(2,F_19)/H can naturally be identified with the 57 cells in the 57-cell.
Any 60-element H will give |PSL(2,F_19)/H| = 57, but we want to make sure this group gives the right structure.
Now, the obvious guess for H is the rotational symmetry group of the dodecahedron, also known as the alternating group A_5, since:
1) this H has 60 elements,
2) you said the cells of the 57-cell are "hemidodecahedra" and we probably want H to be the stabilizer of one of these,
3) this H is isomorphic to PSL(2,F_5).
John Baez (Feb 06 2021 at 23:42):But to get this to work, we need a monomorphism
PSL(2,F_5) PSL(2,F_19)
John Baez (Feb 07 2021 at 00:12):It would be fun to think about this without "cheating" and looking it up.
John Baez (Feb 07 2021 at 07:52):Refurio Anachro said:
I want to learn more about algebraic geometry, even though that seems pretty un-Coxeterian to me.
I suggest going back to the algebraic geometry of the 1800's and early 1900's. It seems more Coxeterian, because it's more visual and full of nice examples... which later blossomed into the general theory. For example, try these two related topics:
or this:
which explores a beautiful relation between a star polytope that Coxeter liked and some algebraic geometry.
To get really deep into some beautiful old algebraic geometry, try Klein's 1884 Lectures on the Icosahedron - or, for a modernized account of the same material, Jerry Shurman's Lectures on the Quintic. Klein used the icosahedron to solve the quintic equation!
John Baez (Feb 07 2021 at 07:53):John Baez said:
Todd Trimble said:
John Baez said:
When I hear "57" I think of the 57-dimensional space that's the lowest-dimensional manifold on which E8 acts. I also think about ketchup.
Could you clarify? E8 acts on a point, for instance. Do you mean acts faithfully, or something?
I meant nontrivially. Using some technology I don't understand you can work out the maximal Lie subgroups of a simple Lie group, where "maximal" means "maximal proper", and then look for the one of highest dimension. This gives the lowest-dim manifold on which the group acts nontrivially.
John Baez (Feb 07 2021 at 08:12):Let me clarify this a bit. I just found what I was looking for in Varadajan's book Lie Groups, Lie Algebras and Their Representations: he shows that if a Lie group G acts on a manifold X, every orbit is a manifold in its own right (injectively immersed in X, though not necessarily embedded).
So, suppose the lowest-dimensional manifold on which G acts nontrivially has dimension d. Then G acts nontrivially on some orbit, which must also be a manifold, which must therefore also have dimension d. So, G has a nontrivial transitive action on a manifold of dimension d. So, the lowest-dimensional manifold on which G acts nontrivially and transitively also has dimension d. (Clearly it cannot be lower; we've just shown it's not more.)
Another little fact: every transitive action of G on a manifold is isomorphic to its action on G/H for some Lie subgroup H, and dim(G/H) = dim(G) - dim(H), so
iff
Putting these facts together:
Theorem. Given a Lie group G, the dimension of the lowest-dimensional manifold on which G acts nontrivially is dim(G) - dim(H), where H is a proper Lie subgroup of G of maximal dimension.
John Baez (Feb 07 2021 at 08:16):Then someone showed that the maximum possible dimension of a proper Lie subgroup of the complex real form of E8 is 248-57. I describe that subgroup in my review of Conway's book on octonions.
David Michael Roberts (Feb 07 2021 at 08:25):Presumably one works with the Lie algebra instead of the Lie group, to find a maximal proper Lie subgroup, since then it reduces to linear algebra.
John Baez (Feb 07 2021 at 17:38):Right! The Lie algebra has a very nice 5-grading - a -grading where only grades -2, -1, 0, 1, 2 are occupied. The subspace consisting of grades -2, -1, 0 form a Lie subalgebra, clearly, and it turns out to be a maximal Lie subalgebra. When we use this trick to get a nontrivial homogeneous space of of minimum dimension, its tangent space at the basepoint can be identified with the subspaces consisting of grades 1 and 2.
The dimensions of the grades go like this:
1, 56, 134, 56, 1.
So the Lie subalgebra has dimension 1+56+134, and the homgeneous space has dimension 56+1.
John Baez (Feb 07 2021 at 17:39):This general type of trick works for lots of simple Lie algebras.
John Baez (Feb 07 2021 at 17:39):A lot of them have 3-gradings, but is more complicated.
