From: Douglas Zare <zare@math.columbia.edu>
Subject: Re: Edge-transitive POLYHEDRA - only a finite number?
Date: Wed, 20 Oct 1999 21:20:28 -0400
Newsgroups: sci.math

Bill Taylor wrote:

> Looking here at edge-transitive polyhedra:- that is, polyhedra for which,
> given any two edges there is an automorphism that takes one to the other.
>
> The question can be asked for graphs in general; but then there are
> infinitely many cases, and indeed a great many different families
> of cases.  The complete graphs, the 1-regulars, the cycles, the clusters
> of equal length cycles, a torus gridded with a rectangular net, or
> hexagonal net, various duals, all their complements...  phew!
>

... the Cayley graphs with generator set an orbit of an automorphism of the
group... I don't know that there will be a low-dimensional space such that the
graph can be embedded so that the symmetries can be recognized.

> [BTW - I often want a name for a graph which is a collection of cycles;
>  a collection of trees is called a "forest", so maybe a collection
>    of cycles should be called a "bike shed"?]
>
> ANYWAY.
>
> Getting back to edge-transitive polyhedra - convex polyhedra even.
> Are there only a finite number of these?  The edge-transitive ones include
> all the regulars, two of the semi-regulars, (cuboctahedron and icosidodec),
> and their duals - 9 in all.
>
> Are there any others?

These come in dual pairs (plus the tetrahedron). One can pay attention to the
degrees of the at most 2 types of vertices and the number of sides of the at
most 2 types of faces.

Tetrahedron (3,3)
Cube (3,4)
Octahedron (4,3)
Dodecahedron (3,5)
Icosahedron (5,3)
Cuboctahedron (4,{3,4})
Rhombic Dodecahedron ({3,4},4)
Icosidodecahedron (4, {3,5})
Rhombic 30-hedron ({3,5},4)

If there are two types of faces, they must alternate around the vertices since
each edge at that vertex is between two different faces. So, the degree must
be 4. Also, the angles must add up to less than 360 degrees, so the above are
the only possible sets of information.

I think it should not be too hard to show that the information determines the
combinatorial type. Perhaps it would be easier to show that the symmetries act
transitively on either the vertices or the faces, hence the polyhedron is
Archimedean or its dual is. There certainly cannot be more than two orbits on
the set of vertices or faces. If this is valid on more general surfaces, then
the classification of graphs with edge-transitive symmetries is not more
complicated than the classification of Archimedean tilings.

Douglas Zare