Shown below are the snub dodecahedron and its dual, the pentagonal hexecontahedron.
Seeking a way to make a “new” polyhedron (one never seen before), I augmented each face of the orange dual, above, with prisms. These prisms have a height equal to twice the average edge length of their bases.
Next, I used the software I use to manipulate polyhedra (Stella 4d, available here) to create the convex hull of this augmented pentagonal hexecontahedron.
Finally, I used Stella’s “try to make faces regular” function, and obtained this result, which I liked enough to stop here. There’s no way for me to know with certainty that this polyhedron has never been seen before, of course, but that didn’t stop me from having fun making it.
Viewers will be the judges of how successful this attempt to blend these polyhedra actually is. I made it using Stella 4d, software you can try right here.
This was made by the augmentation of an icosahedron, using snub dodecahedra on each of its twenty faces. I used software available at http://www.software3d.com/stella.php.
It can sometimes be difficult to distinguish between kites and rhombi. In this case, the edges between rhombi which meet five at a vertex are slightly shorter than the other types of edges here, making the yellow faces here, of which there are sixty, kites. The blue equilateral triangles are twenty in number.
Software credit: visit http://www.software3d.com/stella.php for more information on the program used to make this rotating image. A free trial download is available.