To make this polyhedron, I first changed the symmetry-type of a dodecahedron from icosahedral to tetrahedral, then stellated it twice. This was done using Stella 4d, a program you may try for free at http://www.software3d.com/Stella.php.
This expanded version of the rhombicosidodecahedron has, as faces, 30 rhombi, 60 almost-square trapezoids, twelve regular pentagons, and twenty equilateral triangles, for a total of 122 faces. I made it using Stella 4d, software you may try for free at http://www.software3d.com/Stella.php.
I call this an “expansion of the rhombicosidodecahedron” because it is similar in appearance to that Archimedean solid. However, it is formed by augmenting the thirty faces of a rhombic triacontahedron with prisms, making the convex hull of the result, and then using Stella‘s “try to make faces regular” function on that convex hull.
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 rhombicuboctahedron.
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 rhombicuboctahedron 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.
This is the rhombicosidodecahedron. It is considered by many people, including me, to be the most attractive Archimedean solid.
To create a faceted polyhedron, the first step is to get rid of all the faces and edges, leaving only the vertices, as shown below.
In the case of this polyhedron, there are sixty vertices. To create a faceted version of this polyhedron, these vertices are connected by edges in ways which are different than in the original polyhedron. The new positioning of edges defines new faces, often in the interior of the original polyhedron. Here is one such faceting, with the red hexagonal faces in the interior of the now-removed original polyhedron.
The rhombicosidodecahedron can be faceted in many different ways. I don’t know how many possible facetings this polyhedron has, but it is a finite number much larger than the ten shown in this post. Here’s another one.
In faceted polyhedra, many faces intersect other faces, as is the case with the red and yellow faces above. The next faceting demonstrates that faceted polyhedra are sometimes incredibly complex.
Faceted polyhedra can even contain holes that go all the way through the solid, as seen in the next image.
Sometimes, a faceting of a non-chiral polyhedron can be chiral, as seen below. Chiral polyhedra are those which exist in “left-handed” and “right-handed” reflections of each other.
Any chiral polyhedron may be fused with its mirror image to form a compound, and that’s exactly what was done to produce the next image. In addition to being a polyhedral compound, it is also, itself, another faceted version of the rhombicosidodecahedron.
All these polyhedral manipulations and gif-creations were performed using a program called Stella 4d: Polyhedron Navigator. If you’d like to try Stella for yourself, please visit http://www.software3d.com/Stella.php, where a free trial download is available.
The rest of the rhombicosidodecahedron-facetings needed to round out this set of ten are shown below, without further comment.
I made this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.
Programs used to make this included Geometer’s Sketchpad, MS-Paint, and Stella 4d: Polyhedron Navigator, which was used to assemble everything else into what you see here. You may try Stella for free at http://www.software3d.com/Stella.php.
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