# Eight Kite-Rhombus Solids, Plus Five All-Kite Polyhedra — the Convex Hulls of the Thirteen Archimedean-Catalan Compounds

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.

# Ten Different Facetings of the Rhombicosidodecahedron

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.

# Special PCSSD Board Meeting, 3:00 pm, Saturday, April 28 — Please Attend, and Spread the Word!

Dr. Janice Warren is the interim superintendent of the Pulaski County Special School District. It has become clear in recent weeks that she is not being treated fairly by the PCSSD’s Board of Education — even though she is, in my opinion, the best superintendent I’ve ever seen (and I have seen many).

Dr. Warren now needs our help, at a special meeting of the PCSSD school board, at 3:00 pm tomorrow. I’m asking teachers, parents, and other members of our community to come to this meeting, to show our support for Dr. Warren.

Please come to this important meeting if you can — and even if you cannot be there yourself, please help spread the word to others. We need to pack the boardroom tomorrow afternoon!

# Purify Your Water First, Michigan!

The “Pure Michigan” ad campaign should wait until they’ve replaced ALL of the lead pipes in Flint. Until then, the whole thing is just an exercise in hypocrisy.