This polyhedron has, as faces, a dozen regular pentagons, thirty rhombi, and sixty irregular heptagons. I made this using Stella 4d, which is available as a free trial download at http://www.software3d.com/Stella.php.
The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.
The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.
Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.
Next, here is the compound of the last two polyhedra shown.
Continuing this process, here is the convex hull of the compound shown immediately above.
This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.
This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.
All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at http://www.software3d.com/Stella.php.
In a kite-rhombus solid, or KRS, all faces are either kites or rhombi, and there are at least some of both of these quadrilateral-types as faces. I have found eight such polyhedra, all of which are formed by creating the convex hull of different Archimedean-Catalan base-dual compounds. Not all Archimedean-Catalan compounds produce kite-rhombus solids, but one of the eight that does is derived from the truncated dodecahedron, as explained below.
The next step is to create the compound of this solid and its dual, the triakis icosahedron. In the image below, this dual is the blue polyhedron.
The convex hull of this compound, below, I’m simply calling “the KRS derived from the truncated dodecahedron,” until and unless someone invents a better name for it.
The next KRS shown is derived, in the same manner, from the truncated tetrahedron.
Here is the KRS derived from the truncated cube.
The truncated icosahedron is the “seed” from which the next KRS shown is derived. This KRS is a “stretched” form of a zonohedron called the rhombic enneacontahedron.
Another of these kite-rhombus solids, shown below, is based on the truncated octahedron.
The next KRS shown is based on the rhombcuboctahedron.
Two of the Archimedeans are chiral, and they both produce chiral kite-rhombus solids. This one is derived from the snub cube.
Finally, to complete this set of eight, here is the KRS based on the snub dodecahedron.
You may be wondering what happens when this same process is applied to the other five Archimedean solids. The answer is that all-kite polyhedra are produced; they have no rhombic faces. Two are “stretched” forms of Catalan solids, and are derived from the cuboctahedron and the icosidodecahedron:
If this procedure is applied to the rhombicosidodecahedron, the result is an all-kite polyhedron with two different face-types, as seen below.
The two remaining Archimedean solids are the great rhombcuboctahedron and the great rhombicosidodecahedron, each of which produces a polyhedron with three different types of kites as faces.
The polyhedron-manipulation and image-production for this post was performed using Stella 4d: Polyhedron Navigator, which may be purchased or tried for free at http://www.software3d.com/Stella.php.
If you stellate the snub dodecahedron 43 times, this is the result. The yellow faces are kites, not rhombi.
Like the snub dodecahedron itself, this polyhedron is chiral. Here is the mirror-image of the polyhedron shown above.
Any chiral polyhedron may be combined with its own mirror-image to create a compound.
This is the dual of the snub dodecahedron’s 43rd stellation.
This dual is also chiral. Here is its reflection.
Finally, here is the compound of both duals.
I used Stella 4d: Polyhedron Navigator to create these images. You may try this program for yourself at http://www.software3d.com/Stella.php.
The polyhedron above had 422 faces and 360 vertices. In dual polyhedra, these numbers are reversed, so the next polyhedra (the dual of the first one) has 360 faces and 422 vertices. Both were created using Stella 4d, available here.
The blue figure below is the rhombic triacontahedron. It has thirty identical faces, and is one of the Catalan solids, also known as Archimedean duals. This particular Catalan solid’s dual is the icosidodecahedron.
I use a program called Stella 4d (available here) to transform polyhedra, and the next step here was to augment each face of this polyhedron with a prism, keeping all edge lengths the same.
After that, I created the convex hull of this prism-augmented rhombic triacontahedron, which is the smallest convex figure which can enclose a given polyhedron.
Another ability of Stella is the “try to make faces regular” function. Throwing this function at this four-color polyhedron above produced the altered version below, in which edge lengths are brought as close together as possible. It isn’t possible to do this perfectly, though, and that is most easily seen in the yellow faces. While close to being squares, they are actually trapedoids.
For the next transformation, I looked at the dual of this polyhedron. If I had to name it, I would call it the trikaipentakis icosidodecahedron. It has two face types: sixty of the larger kites, and sixty of the smaller ones, also.
Next, I used prisms, again, to augment each face. The height used for these prisms is the length of the edges where orange kites meet purple kites.
Lastly, I made the convex hull of the polyhedron above. This convex hull appears below.
The dual of the snub dodecahedron (above) is called the pentagonal hexacontahedron (below, left). The compound of the two is shown below, at right. (Any of the smaller images here may be enlarged with a click.)
Like all chiral polyhedra, both these polyhedra can form compounds with their own mirror-images, as seen below.
Finally, all four polyhedra — two snub dodecahedra, and two pentagonal hexacontahedra — can be combined into a single compound.
This polyhedral manipulation and .gif-making was performed using Stella 4d, a program you can find here.