Re: An error in Coxeter's regular polytopes?


On 20-Jan-2006, "*** T. Winter" <***.Winter@xxxxxx>
wrote in message <ItFBrJ.9tB@xxxxxx>:

> I think I found a serious error in Coxeter's book Regular Polytopes, of
> which I use the original edition.
>
> First a definition. Given a regular convex polytope, we can look at
> the graph consisting of the edges connecting all vertices to each
> other. From this we can select subgraphs where all edges have the
> same length. These are the "sections". Sections are numbered from
> 1 onwards with increasing length of the edges.
>
> Next some background. I have always been interested in regular
> polytopes, especially in higher dimensions, and I have been trying
> to visualise it using "numerical" information. During that process I
> became interested in the regularities that occur in the various
> sections in the regular convex polytopes, and there are far more
> regularities than can be found in the book. But that is not an
> error, it is just a different focus of interest (I am looking at
> all regular objects that can be found, so triangles in 4-D are
> also of interest to me, although they do not form higher dimensional
> objects).
>
> The problem comes at section 18 of {5, 3, 3} (the 120-cell). According
> to table VI(iv) in the table section you can find a compound consisting
> of {5, 3, 3}[120{3, 3, 3}]{3, 3, 5}. However, during my calculations
> I found 840 {3, 3, 3}'s, and my initial thinking was that the compound
> was actually: 7{5, 3, 3}[840{3, 3, 3}]7{3, 3, 5}. But the factor 7
> is very strange in this context. What would the symmetry group of this
> polytope (of order 14400) do with those objects?
>
> Further investigation lead me to the following conclusion, the
> symmetry group of a convex regular polytope is transitive on the
> vertices (trivial) and almost always on the edges in all of its
> sections. That it can not be true in general shows section 15
> of {5, 3, 3}. There are 16200 edges but the order of the symmetry
> group is 14400. Now I did the calculations and the exceptions
> occurr only with {5, 3, 3} in the sections 8, 12, 15, 18 and 22,
> for all other polytopes and sections the symmetry group is
> transitive on the edges.
>
> Section 12 of {5, 3, 3} is instructive. It has 8400 edges.
> Transitivity splits it in two sets, one of 1200 edges and one
> of 7200 edges. In the second set you will find triangles and
> squares, and even 400 odd configurations of 9 vertices, 18
> edges, 6 triangles and 9 squares that cover everything. In
> the first set I have been unable to find any regularity at all.
>
> This brings me back to section 18. When I consider the group
> operations, I find that the 840 {3, 3, 3}'s split in a group
> of 120 {3, 3, 3}'s and another group of 720 such polytopes.
> So the (in my opinion correct) entry would be:
> 7{5, 3, 3}[840{3, 3, 3}]7{3, 3, 5}
> which can be split in two sub-compounds, one as shown in the
> table and one with 7 replaced by 6 and 840 replaced by 120.
> Two quite distinctive compounds. (I do not yet know whether
> the last compound can be split in sub-compounds, but even if
> that is possible, those sub-compounds in no way resemble the
> compound given by Coxeter, there are at least two essentially
> different compounds.)
>
> Has anything been written about this before? I think it is
> quite intriguing. (Did Coxeter miss the non-transitivity?)
>
> I am still looking at the other sections that are non-transitive.

The last time I looked at Coxeter's _Regular Polytopes_, quite a
few years ago now, I found it completely impenetrable. :-( Maybe
it's time I gave it another try. :-/

So I'm not familiar with all your terminology and notation above;
I'm not even sure I fully grasp your definition of sections. I do
know that the longest minimal graph distance from one vertex of A =
{5,3,3} to another, along edges joining adjacent vertices, is from
a vertex to its opposite and is 15 steps. How does that map into
your sections?

Anyway, you're right that the vertices of A are also those of 840
interlocking {3,3,3}'s, which split into 2 orbits of sizes 120 and
720 under the action of G = Aut(A).

If I understand correctly, one of your questions is whether the 2nd
orbit, the 720 one, is in fact an orbit; that is, whether it
doesn't further split into smaller orbits. The answer is no. To see
this:

Each vertex V of A belongs to 6 {3,3,3}'s in the 720 group, and any
2 of these 6 {3,3,3}'s share only V in common. The other (5-1)*6 =
24 vertices of the 6 {3,3,3}'s belong to a single orbit of the
stabilizer S in G of V, so S acts transitively on the 6 {3,3,3}'s.
And of course G acts transitively on the 600 vertices of A, so it
therefore also acts transitively on the set of 720 {3,3,3}'s, by
composing an element of G that takes any given vertex to V with an
appropriate element of S.

--
Jim Heckman
.

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