- Convex Uniform Polyhedra:
- Platonic solids (these 5 are regular)
- Archimedean solids (there are 13 of these)
- Convex prisms and antiprisms (two infinite families)
- Nonconvex Uniform Polyhedra:
- Kepler-Poinsot polyhedra (these 4 are regular)
- Nonconvex uniform polyhedra (there are 53 of these)
- Nonconvex prisms, antiprisms, and crossed antiprisms (three infinite families)
- compound of snub disicosidodecahedron and dual
- compound of small triambic icosidodecahedron and dual
- compound of small icosic icosidodecahedron and dual
- compound of great dodecicosidodecahedron and dual
- compound of great stellated truncated dodecahedron and dual
- compound of great vertisnub icosidodecahedron and dual
- prism:
- antiprism:
- pentagrammatic antiprism
- pentagrammatic trapezohedron (face)
- compound of pentagrammatic antiprism and dual
- crossed antiprism:

There are 75 uniform polyhedra, plus an infinite number of prisms and antiprisms. Uniform polyhedra can be organized in the following taxonomy. Nine of these are regular and the remainder are semi-regular.

The snub solid just derived is chiral, like the two Archimedean snubs. Among the uniform polyhedra there are also reflexible snubs, for example the snub disicosidodecahedron. The essence of "snubness" here is that some of the faces (all the triangles) have the property that there is no rotation of the whole polyhedron which brings the whole back on to itself and also brings a given triangle back to itself. So the triangles are snub faces, but they come in pairs in such a way that the entire solid is reflexible. A similar property holds for the retrosnub disicosidodecahedron.

Nine of the 53 are "hemihedra" --- the infix "-hemi-" in their name
indicates that some of the faces span a complete hemisphere (if you project
their edges on to a surrounding sphere). This means that these faces pass
through the center of the polyhedron. For example, the small
icosihemidodecahedron consists of 20 triangles and six decagons whose
centers coincide at the center of the polyhedron. A most interesting example
in this class is the tetrahemihexahedron,
also called the *one-sided heptahedron*, which is the only nonprismic
uniform polyhedron with an odd number of sides: four triangles plus three
squares. The hemihedra have a number of special properties because they
fall within the class of quasi-regular
polyhedra.

One particularly interesting uniform polyhedron is the great snub disicosidisdodecahedron. It has eight faces per vertex (4, 5/2, 4, 3, 4, 5/2, 4, 3) and is the only one with more than six faces per vertex. All three types of faces come in coplanar pairs. The squares pass through the center, so it has no finite dual. The squares are snub faces because there is no rotation which brings the whole back to itself and some square back to itself.

The duals to the uniform polyhedra are *facially
regular*, meaning that they are composed of a single type of face, every
face in the same relation to the whole, and their vertex figures are regular
polygons. Because each dual has a different type of nonregular face, and
the faces partially obscure each other, it can be unclear at first that
a visible facet is not a complete face. In order to know what to look for,
I have included a picture of the complete dual face for each of these polyhedra.
Looking at a face alone lets one know what to look for in the whole and
so lets one understand more clearly how the dual is constructed. As an
example, the small rhombicube (8,
4, 8, 4) has the small rhombihexacron
as its dual. It is easier to understand the latter after seeing that the
face
is a butterfly-shaped crossed quadrilateral, which appears as two triangles.
However, the point at which the two triangles join is just an edge-crossing,
not a vertex of the polyhedron and so does not correspond to a face of
the underlying primal.

In a compound of a nonconvex uniform solid and its dual, the dual often hides most or all of the uniform solid, so I have rendered these compounds with the dual polyhedron partially transparent. This lets you look through it to see otherwise hidden structure. (However not all VRML viewers render partially transparent surfaces correctly, so you may see them as solid.) In their compound, partial transparency lets you see how the dual completely surrounds the inner primal.

Here are a few attractive, never-before-seen compounds which you might like. Each shows a nonconvex polyhedron with its dual:

At left is an antiprism based on the {25/12} polygon. Although prism symmetry is usually considered a "lower" symmetry than other polyhedral symmetry, examples of this sort show that at high order they can be very attractive. (Be sure and fly through this and the following compound, along their axes.)

There is also a kind of crossed antiprism, e.g., the pentagrammatic
crossed antiprism also (5/2, 3, 3, 3), but the vertex figure is a butterfly-shaped
(crossed) quadrilateral and the triangular faces cross the n-fold axis.
The first type of antiprism can be constructed on any base *n/m*,
but the crossed antiprism can only be constructed on a base *n/m*
where *2< n/m < 3*.

To view any of an infinite number of such prisms, antiprisms, and crossed
antiprisms, use the **prism generator**
page which allows you to type in any desired *n* and *m* and
it will create a never-before-seen virtual object for you on the fly.

Dual to prisms and antiprisms are dipyramids and trapezohedra on nonconvex bases. In the case of crossed antiprisms, the dual trapezohedron is composed of concave quadrilaterals. At right is the compound of a {20/7} crossed antiprism and its dual concave trapezohedron. The following triples illustrate some simpler examples, where the second line in each case is the dual.

References: The first complete list is in the 1953 paper by Coxeter et al. but the list wasn't proven to be complete until Skilling's 1975 computer search. Wenninger's 1971 book shows paper models and templates for them all, and his 1983 book does the same for their duals. As far as I know, the compounds and the high order prisms and antiprisms have not been illustrated in any previous publication.

The software I used to generate these is an adaptation of Maeder's adaptation
of Har'El's Kaleido program. The names
for these polyhedra and their duals are largely the creations of Norman
Johnson whose forthcoming book on uniform polyhedra contains a great deal
more information. The names used here are based on his latest suggestions
and differ in some places from his earlier terminology, used in the above
works of Wenninger.