Re: An error in Coxeter's regular polytopes?


On 23-Jan-2006, "*** T. Winter" <***.Winter@xxxxxx>
wrote in message <ItKvMn.K0C@xxxxxx>:

> In article <11t93h9sgvi1668@xxxxxxxxxxxxxxxxxx> "Jim Heckman"
> <wnzrfeurpxzna@xxxxxxxxxxxxxxxxx> writes:

[...]

> > 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?
>
> 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.

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).

> > 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).
>
> 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?

> This is a section where graph-distances do not mix.

Yes. As mentioned above, graph distances can never mix.

The graph distances are in bijection with the Euclidean distances
(and stabilizer orbits) for all regular convex polytopes in all
dimensions apart from the exceptional {3,4,3}, {3,3,5} and {5,3,3}
in 4-d. All of these have cases of more than one Euclidean distance
for some of their graph distances, and {5,3,3} also has cases of
more than one graph distance for some of its Euclidean
distances--as you saw above. (This occurs in sections 8, 18 and 22.
Each of these involves a stabilizer orbit of size 4--the smallest
orbits other than the 2 singletons--which I like to think of as
'holes'.)

> I think I will have to look a bit more into the graph-distance.
>
> > 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:
>
> 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

Right. The two subsets of 5 are even 'natural' in the sense that
they are separate orbits of the rotation subgroup ~= A_5 of the
dodecahedral symmetry group ~= A_5 x C_2. I'll have to think about
whether there's any similar natural grouping for the relevant 720
{3,3,3}'s of {5,3,3}.

> (but
> clearly not all associated edges are covered by these subsets).

What do you mean by "associated edges"?

> I know already that such grouping is not always possible. I have
> already seen some strange counter-examples.

Sounds interesting. What are they?

> 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.

--
Jim Heckman
.