Edge-Transitive Polyhedra

For general discussion of polyhedra, not necessarily Stella-specific.
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Gordon
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Edge-Transitive Polyhedra

Post by Gordon » 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.
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Ulrich
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Re: Edge-Transitive Polyhedra

Post by Ulrich » Fri Sep 02, 2022 1:55 pm

This is great! Thanks for sharing.

Ulrich

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Re: Edge-Transitive Polyhedra

Post by Ulrich » Sat Sep 03, 2022 10:43 am

I‘d love to have Leonardo-style shapes of them because the understanding is much better, so please post off-files of them too.

Don Romano from Denver asked me to thank you on his behalf for the great work and inspiration. He immediately started to build a paper model of the first one in your list.

Ulrich

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Gordon
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Re: Edge-Transitive Polyhedra

Post by Gordon » Sun Sep 04, 2022 2:07 am

Thanks, Ulrich! Please send my regards to Don Romano.

Attached is a zip archive of the OFF files for Leonardo-style models of the 11 vertex-intransitive face-intransitive isotoxals. It is more complicated to generate such models for the vertex-transitive ones, but those are much easier to understand anyway and most are familiar. These models are best displayed without showing vertices and edges, as Stella does not distinguish between the "real" ones and the ones that are only there to delimit the struts.

Unfortunately, these models generate some warning messages which must be clicked through each time. Say "no" to blending faces, "yes" to building the model, and you probably want to cancel stellation, as Stella estimates that it would take a long time. I don't know why these are assessed as so much more complicated than the builtin Leonardo-style models.

If you select a gray face and tell Stella to hide all of that color you will get a model that is like the ones that the interactive application on my website shows when you select all faces to be shown as strips (the default). There are other things as well that that application will do to aid understanding.
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Re: Edge-Transitive Polyhedra

Post by Ulrich » Thu Sep 29, 2022 2:53 pm

Here is my first try of a paper version of one of your models: O32b_2. I made it because it is very pretty and because there are only 12 rhombs crossing each other. In my physical models I want to display the inner structure. Here I had to construct pyramidal building blocs with their top vertices pointing towards the middle. Thus 240 triangles are meeting in the center. In the case of your shapes with 30 rhombs the corners of 1800 triangles coincide in the center, so I think I won‘t try to make models of them. :wink:

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Gordon
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Re: Edge-Transitive Polyhedra

Post by Gordon » Fri Sep 30, 2022 5:51 am

It looks great, Ulrich! I was hoping that you'd make some of these. I like that you made separate pieces for the higher-density regions of the faces. It's a shame that you are stopped by the large number of central facelets that the I52 and I32 polyhedra require. Perhaps 3-D printing is the only way to get physical models of those. I don't think that one can get paper-thin membranes with that technology - yet.

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Re: Edge-Transitive Polyhedra

Post by Ulrich » Sun Oct 02, 2022 5:11 am

You just need a device that can print it with an edge length of 1 m or so and with 40 colours and an algorithm that can display the patterns in the faces.

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