EdgeTransitive Polyhedra
EdgeTransitive Polyhedra
One can find a lot of things written about face and vertextransitive polyhedra, especially regularfaced ones. There is not much on edgetransitive (isotoxal) ones, and all of it is incomplete. Frequently one encounters the assertion that any edgetransitive polyhedron must also be face or vertextransitive. 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 vertexintransitive 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. Leonardostyle 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 vertexintransitive 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. Leonardostyle 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.
Re: EdgeTransitive Polyhedra
This is great! Thanks for sharing.
Ulrich
Ulrich
Re: EdgeTransitive Polyhedra
I‘d love to have Leonardostyle shapes of them because the understanding is much better, so please post offfiles 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
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
Re: EdgeTransitive Polyhedra
Thanks, Ulrich! Please send my regards to Don Romano.
Attached is a zip archive of the OFF files for Leonardostyle models of the 11 vertexintransitive faceintransitive isotoxals. It is more complicated to generate such models for the vertextransitive 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 Leonardostyle 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.
Attached is a zip archive of the OFF files for Leonardostyle models of the 11 vertexintransitive faceintransitive isotoxals. It is more complicated to generate such models for the vertextransitive 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 Leonardostyle 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.
Re: EdgeTransitive Polyhedra
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.
Re: EdgeTransitive Polyhedra
It looks great, Ulrich! I was hoping that you'd make some of these. I like that you made separate pieces for the higherdensity 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 3D printing is the only way to get physical models of those. I don't think that one can get paperthin membranes with that technology  yet.
Re: EdgeTransitive Polyhedra
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.
Re: EdgeTransitive Polyhedra
Recently I got a chance to revisit the OFF files that I posted here a while ago, in particular the Leonardostyle ones.
I tried to address the issues that Great Stella had with them. One of these had to do with having more than two faces at an edge. I normally would agree that this is not valid, but the ones for which this occurred are specialpurpose objects. Nevertheless, I found a way to define the "strips" around the edges of the faces in a way that avoids having more than 2 faces at an edge while preserving symmetry. (These really should be annular faces, but I'm not aware of any 3D application or toolkit that can handle such things.)
These facelets are in the "Faces As Edge Strips" and Leonardo versions. For the "as Strips" ones, as before, one can select one of the inner gold facelets and tell the software to hide faces of that color. That gives the best view of the interior structure. Of course, the result is not a closed surface; the Leonardostyle models provide those.
The coordinates in the new files, posted below, are accurate to 20 decimal places. Great Stella now recognizes the full symmetry of the polyhedra and will calculate the stellations instantly without warnings  even the ones with over 10,000 cells.
One still must decline the offer to blend coplanar faces. There does not appear to be an option to turn that off globally, but it only matters for the files that don't represent the full polyhedra. It looks like one can bypass this question by saving a polyhedron as a stel file, but I don't want to supply those as I don't know what else is in them.
For the Leonardostyle versions, I tried out ones with struts having rhombic cross sections instead of triangular ones, more in da Vinci's style. This usually results in fewer stellation cells, on account of the parallel face planes, but creates much more overlap and blocks the view of the interior more, undermining the purpose. One must use thinner struts, which sometimes look a bit spindly and I'm not sure work as well for paper models. I have included those anyway in the attached collection. da Vinci's approach certainly works best with convex prototypes.
For quite a few of these there was a tradeoff between thinness of struts, visibility into the interior, and comprehension of the overall structure. The latter is better when showing edge cylinders, but only on the real edges, those between the red and blue facelet strips. In Great Stella, only the "full" files look good with edge cylinders and vertex spheres.
Another consideration was that I ran into a problem with Great Stella (I'm running version 5.4) frequently hanging or crashing when trying to draw nets, especially ones with tabs. Whether or not it does so depends oddly on the precise dimensions of the facelets. All the files in the archive posted below should work OK as long as you have tabs turned off. I haven't tried them all with tabs on. Apparently I'm pushing the boundaries of what this excellent program is for.
For I52c_2, a couple of the Leonardo nets are a bit weird. While technically made up of several narrow coplanar trapezoids, Great Stella fills in the holes between them and draws the holes as separate kiteshaped quadrilaterals off to the side. The program handles holes just fine in other polyhedra. Using a straightedge you can draw them where they should be on printed nets.
I hope that people find these new files useful.
I tried to address the issues that Great Stella had with them. One of these had to do with having more than two faces at an edge. I normally would agree that this is not valid, but the ones for which this occurred are specialpurpose objects. Nevertheless, I found a way to define the "strips" around the edges of the faces in a way that avoids having more than 2 faces at an edge while preserving symmetry. (These really should be annular faces, but I'm not aware of any 3D application or toolkit that can handle such things.)
These facelets are in the "Faces As Edge Strips" and Leonardo versions. For the "as Strips" ones, as before, one can select one of the inner gold facelets and tell the software to hide faces of that color. That gives the best view of the interior structure. Of course, the result is not a closed surface; the Leonardostyle models provide those.
The coordinates in the new files, posted below, are accurate to 20 decimal places. Great Stella now recognizes the full symmetry of the polyhedra and will calculate the stellations instantly without warnings  even the ones with over 10,000 cells.
One still must decline the offer to blend coplanar faces. There does not appear to be an option to turn that off globally, but it only matters for the files that don't represent the full polyhedra. It looks like one can bypass this question by saving a polyhedron as a stel file, but I don't want to supply those as I don't know what else is in them.
For the Leonardostyle versions, I tried out ones with struts having rhombic cross sections instead of triangular ones, more in da Vinci's style. This usually results in fewer stellation cells, on account of the parallel face planes, but creates much more overlap and blocks the view of the interior more, undermining the purpose. One must use thinner struts, which sometimes look a bit spindly and I'm not sure work as well for paper models. I have included those anyway in the attached collection. da Vinci's approach certainly works best with convex prototypes.
For quite a few of these there was a tradeoff between thinness of struts, visibility into the interior, and comprehension of the overall structure. The latter is better when showing edge cylinders, but only on the real edges, those between the red and blue facelet strips. In Great Stella, only the "full" files look good with edge cylinders and vertex spheres.
Another consideration was that I ran into a problem with Great Stella (I'm running version 5.4) frequently hanging or crashing when trying to draw nets, especially ones with tabs. Whether or not it does so depends oddly on the precise dimensions of the facelets. All the files in the archive posted below should work OK as long as you have tabs turned off. I haven't tried them all with tabs on. Apparently I'm pushing the boundaries of what this excellent program is for.
For I52c_2, a couple of the Leonardo nets are a bit weird. While technically made up of several narrow coplanar trapezoids, Great Stella fills in the holes between them and draws the holes as separate kiteshaped quadrilaterals off to the side. The program handles holes just fine in other polyhedra. Using a straightedge you can draw them where they should be on printed nets.
I hope that people find these new files useful.
 Attachments

 Isotoxals_24.05.03.zip
 (162.92 KiB) Downloaded 198 times
Re: EdgeTransitive Polyhedra
The original zip archives have been quite popular, but the files in the new one are so much of an improvement that I have deleted the old ones. So they no longer show as attachments above, which may make the earlier posts a bit confusing. I would love to copy and rename the new one and have it replace the old ones, but that doesn't appear possible.
Re: EdgeTransitive Polyhedra
These are fascinating to read about, and to examine the one thumbnail image which has been posted. They sound beautiful. But there is no polyhedron software which installs cleanly on my system and can view .off files equally cleanly. And Ulrich's image server is either broken or hates my system.
The one shown by Ulrich is clearly neither a stellation nor a facetting of any convex polyhedron. It has faces through the centroid, so its standard dual is an infinite "Wenninger dual". One could jiggle the dual to make it finite, but then that would not be isotoxal. Are they all like that, or are there any finite dual pairs?
Is there any chance of more plain, boring inanimate 2D images?
The one shown by Ulrich is clearly neither a stellation nor a facetting of any convex polyhedron. It has faces through the centroid, so its standard dual is an infinite "Wenninger dual". One could jiggle the dual to make it finite, but then that would not be isotoxal. Are they all like that, or are there any finite dual pairs?
Is there any chance of more plain, boring inanimate 2D images?
Cheers,
Guy. Guy's polyhedra pages
Guy. Guy's polyhedra pages
Re: EdgeTransitive Polyhedra
Thanks for your interest in these polyhedra. For more on them, check out https://isotoxals.github.io and the pages linked from https://polytope.miraheze.org/wiki/Isotoxal_polytope. The latter have images with good density shading made using Great Stella. Click on the images to enlarge them.
All of the isotoxals that are both vertex and faceintransitive have central faces and none can be derived from convex polyhedra by any of the wellknown operations. None has an uncontroversial dual either  as you note, one has to accept infinite tubes. Four of the vertexintransitive, facetransitive ones also do not have ordinary duals, since adjacent faces share pairs of collinear edges. The rest of the isotoxals are uniform polyhedra and duals thereof. There are several dual pairs among them.
All of the isotoxals that are both vertex and faceintransitive have central faces and none can be derived from convex polyhedra by any of the wellknown operations. None has an uncontroversial dual either  as you note, one has to accept infinite tubes. Four of the vertexintransitive, facetransitive ones also do not have ordinary duals, since adjacent faces share pairs of collinear edges. The rest of the isotoxals are uniform polyhedra and duals thereof. There are several dual pairs among them.