Uniform polyhedra

In the regular polyhedron page we saw a detailed description of the Platonic solids, and in the Archimedean polytope page we presented the Archimedean solids.

However, as we've seen for the regular polyhedra, when we drop the condition of convexity, additional forms are allowed, the regular star polyhedra. This also allows many additional uniform polyhedra: among the star polyhedra, we have already seen the rectifications of the regular star polyhedra (Fig. 6a and b) and the three partially regular polyhedra (Fig. 7). But how many such forms are there in total?

It was proved by Skilling (1975) that there are 75 uniform non-prismatic polyhedra. A careful description of these was made earlier by Coxeter, Longuet-Higgins and Miller (1954).

In the 1990's, I tried to build paper models of the sub-set that is mirror-symmetric (i.e., not including the objects without mirror symmetry, the snub polyhedra). I used mostly the nets in Magnus Wenninger's "Polyhedron Models" (Cambridge). And I almost finished the whole set! However, some of the more complex models proved to be too tedious, though, and I had other things to worry about. 3-D printing might be the way to do eventually finish this collection.

Fig. 1: I made these 60 uniform polyhedra in the later 90's. These are in my parents' country house in Portugal. This set includes the models of the regular and quasi-regular polyhedra displayed in my office, but on a smaller scale.

Another way of representing polyhedra is with edge models. One possibility is doing this with Zometool, another one, presented in the next figure, is 3-D printing.

Fig. 2: Here is a 3-D printed edge model of two complex polyhedra and two polyhedron compounds.

Model printed by Shapeways, design by Mathematical Creations.

This model represents two of the most complex uniform star polyhedra, which I have not built as paper models. The first is the Great dirhombicosidodecahedron, also known as "Miller's Monster". This was the last entry in the 1954 list of uniform polyhedra compiled by Coxeter, Longuet-Higgins and Miller (1954).

There is another remarkable polyhedron that shares this set of edges. In the 1975 paper where he proved that the 1954 list of uniform polyhedra is complete, John Skilling generalized the concept of Uniform polyhedron: instead of two faces per edge, there can be any even number (or, equivalently, edges can be multiple). This relaxation yields only one additional polyhedron, the Great dis-nub dirhombidodecahedron.

Finally, the edges also belong to two different uniform polyhedron compounds: The Compound of twenty octahedra and the Compound of twenty tetrahemihexahedra.

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The following video and pictures show how the paper models used to hang from the ceiling of the attic.

In these pictures, all triangles are red, all squares are green, all pentagons are dark blue, all hexagons are yellow, all octagons are pink, all decagons are light blue, all pentagrams are gold, all octagrams are a lighter shade of pink and all decagrams are brown. These are the colors used in Roman Maeder's website. If a uniform polyhedron has a single type of face (i.e., a single color in this collection) then by definition it is regular.

Some of the complex, non-convex polyhedra with icosahedral symmetry.

The image above shows mainly Archimedean polyhedra with icosahedral symmetry. Among these are two regular polyhedra: the icosahedron (all in red) and the dodecahedron (all in dark blue).

The non-prismatic convex uniform polyhedra: the five regular Platonic solids and the 13 Uniform Archimedean solids.

A severe polyhedral storm!

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