To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting.
After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation:
This is the 18th stellation:
The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron.
This one is the 22nd stellation:
This is the 30th stellation:
The next really interesting stellation I found was the 69th:
At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals.
This is the third stellation of this dual — and another distorted Platonic dodecahedron.
This is the dual’s 7th stellation:
And this one is the dual’s 18th stellation:
At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared:
Here is this convex hull’s dual:
Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result:
While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this:
I also found an interesting polyhedron as the 19th stellation of the dual which is three images above:
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