Re: An error in Coxeter's regular polytopes?

In article <11tccrndegi94f3@xxxxxxxxxxxxxxxxxx> "Jim Heckman" <wnzrfeurpxzna@xxxxxxxxxxxxxxxxx> writes:
> On 23-Jan-2006, "*** T. Winter" <***.Winter@xxxxxx>
> wrote in message <ItKvMn.K0C@xxxxxx>:
....
> > Are you sure it is 15 and not 30? But I just checked, and indeed it is.
> > What I am also consideringA There is no mapping between the graph
> > distance and the Euclidean distance. I was considering the Euclidean
> > distance (otherwise use of the symmetry group makes nearly no sense).
>
> OK, I see now: The vertices of an embedded {3,3,3} of {5,3,3} are
> at the 18th Euclidean distance from each other, so that must be
> what your "sections" are.

Yes, it conforms to what Coxeter calls sections.

> But the symmetry group _does_ make sense for the graph distances as
> well as the Euclidean; in both cases it must map vertices at equal
> distances among themselves. Even better, it must, by definition,
> map the vertices in a vertex-stabilizer orbit among themselves.
> {5,3,3} has 45 such orbits (including the 2 singletons of the
> vertex and its opposite), a finer division of the 600 vertices than
> either the graph or Euclidean distances (16 and 31, respectively).

I have already looked at such, but it did not give me very much.

> > Yes, and I just found another thing, prompted by this reply. The
> > {3,3,3}'s in the orbit of size 120 have graph-distance between the
> > vertices of 10, while the other 720 have graph-distance of 9, but
> > the Euclidean distance puts them all in section 18.
>
> Right. Did you notice though that the graph distance of the
> opposites of both sets of vertices (section 12) is 7?

Yes, I noticed. There are three Eclidean sections that split over
two different graph sections: #8, #18 and #22. However, the symmetry
group is also not transitive over the edges in sections #12 and #15.
Now I think I think that the non-transitiveness in section #12 is
induced by that in section #18, but I do not yet see what the
essential difference is between the edges in the two orbits of
section #15.

> > This is a section where graph-distances do not mix.
>
> Yes. As mentioned above, graph distances can never mix.

In some sense they can. In sections #12, #18 and #22 all regular
triangles and squares have edges from one orbit, and the group
is transitive over the subsets, this remains true if you go to
higher dimensional regular objects. However, in section #8 there
are three kinds of regular triangles, two kinds have three similar
edges, but one kind of triangle has edges of two difference kinds.

> > No, the question was more, can the 720 {3,3,3}'s (that together cover
> > each vertex 6 times) be put in subsets that cover each vertex fewer
> > times. For instance, in the dodecahedron you can find 10
> > tetrahedrons. That set of tetrahedrons covers each vertex of the
> > dodecahedron twice. You can split it in two subsets of 5
> > tetrahedrons, where the subsets cover each vertex only once
....
> > (but
> > clearly not all associated edges are covered by these subsets).
>
> What do you mean by "associated edges"?

The set of edges of the appropriate length. Each group of 5
tetrahedrons has 30 edges, but there are in total 60 edges of
this length in the dodecahedron.

> > I know already that such grouping is not always possible. I have
> > already seen some strange counter-examples.
>
> Sounds interesting. What are they?

In Eclidean section #2 of the 3,3,5. You can find there (amongst
others) 25 {3,4,3}, that can be split in 5 groups of 5 {3,4,3}
(actually you can to this grouping in two ways), where each group
covers the vertices exactly once. In that same section you can
also find 120 {3,5} (icosahedrons) that do not form a 4-dimensional
object. Together they cover the vertices 12 times. You will not
find a subset of 10 {3,5}'s that covers the vertices only once, but
you can find subsets of 20 {3,5}'s that cover the vertices twice.
And you can indeed group them in 6 such subsets (in twelve ways!).

> > But there is again enough to do more thinking. I am trying to put all
> > the information I find on a web-page, but it is becoming huge.
>
> Is it somewhere under http://www.cwi.nl/~***/ ? I'll try to take a
> look at it.

Not yet, it will still take a few days before it is web accessible
(at least, what I already have).
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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