Edge-Transitive Polyhedra
Posted: Sun Aug 28, 2022 2:20 am
One can find a lot of things written about face- and vertex-transitive polyhedra, especially regular-faced ones. There is not much on edge-transitive (isotoxal) ones, and all of it is incomplete. Frequently one encounters the assertion that any edge-transitive polyhedron must also be face- or vertex-transitive. It turns out, though, that there are 11 isotoxal polyhedra that are neither.
I found them back in 1998, along with the rest of the isotoxal polyhedra, empirically using VRML models. As far as I know, no one else has peeked into that polyhedral corner. I recently got around to writing up a proof of completeness and creating a 3D application to view all the isotoxal polyhedra. You can find them at https://isotoxals.github.io. The app does a few things that Stella doesn't, but does not support transformations.
To facilitate exploring these in Stella, I have created OFF files for the 22 vertex-intransitive isotoxal polyhedra. I found it straightforward to add them to Great Stella's Library. There's a zip archive of them attached to this post. Showing edges and vertices and exploding the models help greatly to comprehend the more complex ones. Leonardo-style models would be most understandable. I have some of those that I can post if there is interest.
My count of 11 uses the common intuitive definition of "polyhedron" that results in 75 uniform ones (other than prisms and antiprisms). If one accepts the more exotic things that Branko Gruenbaum wrote so much about, one should be able to come up with many more. I myself have gone back and forth on a couple of objects that I ended up rejecting. I know that there are members here who would count more, just as they would count more uniform polyhedra than are usually listed. It is not my intent to argue details or change people's minds, but just to make my own results known. I think those 11 polyhedra are quite beautiful.
I found them back in 1998, along with the rest of the isotoxal polyhedra, empirically using VRML models. As far as I know, no one else has peeked into that polyhedral corner. I recently got around to writing up a proof of completeness and creating a 3D application to view all the isotoxal polyhedra. You can find them at https://isotoxals.github.io. The app does a few things that Stella doesn't, but does not support transformations.
To facilitate exploring these in Stella, I have created OFF files for the 22 vertex-intransitive isotoxal polyhedra. I found it straightforward to add them to Great Stella's Library. There's a zip archive of them attached to this post. Showing edges and vertices and exploding the models help greatly to comprehend the more complex ones. Leonardo-style models would be most understandable. I have some of those that I can post if there is interest.
My count of 11 uses the common intuitive definition of "polyhedron" that results in 75 uniform ones (other than prisms and antiprisms). If one accepts the more exotic things that Branko Gruenbaum wrote so much about, one should be able to come up with many more. I myself have gone back and forth on a couple of objects that I ended up rejecting. I know that there are members here who would count more, just as they would count more uniform polyhedra than are usually listed. It is not my intent to argue details or change people's minds, but just to make my own results known. I think those 11 polyhedra are quite beautiful.