An error in Coxeter's regular polytopes?

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.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

Quantcast