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 in turn allows additional non-convex
uniform polyhedra, like their rectifications (Figs. 3.6a and b) and the three partially
regular polyhedra (Fig. 3.7). But how many such polyhedra are there in total?
The first enumeration of all 75 uniform
non-prismatic uniform polyhedra was made by Coxeter, Longuet-Higgins and Miller
(1954), by a systematic evaluation of all possible Wythoff constructions of the full list
of Schwarz triangles. 21 years later, Skilling (1975) proved that there are no additional
uniform polyhedra.
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 Wenninger (1974). And I almost finished the whole set! However, some of the more
complex models proved to be too tedious. 3-D printing might be the way to do eventually
finish this collection.
Fig. B.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. Edge models for all these
uniform polyhedra were also published by Coxeter, Longuet-Higgins and Miller (1954). One
possibility is doing this with the Zometool, another one,
presented in the next figure, is, again, 3-D printing.
Fig. B.2: Here is a 3-D printed edge model of two complex polyhedra and two polyhedron compounds.
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); it is a "monster"
because it was the only uniform polyhedron that is not strictly derivable via the Wythoff
construction from a Schwarz triangle, but instead from a spherical quadrilateral.
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
generalised the concept of Uniform polyhedron: instead of two faces per edge, there can be
any even number (or, equivalently, edges can be superposed). This relaxation yields only
one additional polyhedron, the Great dis-nub
dirhombidodecahedron.
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.
Fig. B.3a
Fig. B.3b
Fig. B.3c: Some of the complex, non-convex polyhedra with icosahedral symmetry.
Fig. B3d: Mainly Archimedean polyhedra with icosahedral symmetry. Among these
are two regular polyhedra: the icosahedron (all in red) and the dodecahedron (all in dark
blue).
Fig. B3e: The non-prismatic convex uniform polyhedra: the five regular Platonic solids and the 13 Uniform Archimedean solids.
