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.
EdgeTransitive Polyhedra
EdgeTransitive Polyhedra
 Attachments

 VI_Isotoxals_OFF.zip
 (15.4 KiB) Downloaded 24 times
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.
 Attachments

 VI_Isotoxals_OFF_Leonardo.zip
 (36.85 KiB) Downloaded 24 times
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.