#
Uniform Polyhedra and their Duals

Uniform polyhedra have regular faces, and identical vertex figures, meaning
that each vertex is surrounded by faces is the same way. The faces needn't be
the same though, and the polyhedra need not be convex. The set includes
the Platonic solids,
the Kepler-Poinsot solids,
the Archimedean solids,
and prisms and antiprisms,
which are shown on their own pages.
What remains are the nonconvex equivalents of the Archimedean solids, and this
is what appears below. These are just the models I have made, but not the
complete set.
A uniform polyhedron may be completely specified by its *vertex
description*, which describes the sequence of faces around each vertex. For
example, 8.8.3 indicates a truncated cube, because each
vertex is surrounded by two regular octagons and an equilateral triangle.
Some uniform polyhedra also make use of nonconvex regular polygons, such as
pentagrams. A regular pentagram (5-pointed star) is described as 5/2, meaning
that its 5 sides go around the polygon's centre twice before repeating.
Uniform polyhedra make use of pentagrams (5/2), octagrams (8/3) and decagrams
(10/3) in addition to other convex regular polygons. One such example is
8/3.3.8/3.4 which specifies the great
cubicuboctahedron, having a cycle of faces around each vertex as follows: a
regular octagram, equilateral triangle, another octagram, and a square.

These models were made using nets generated by
**Great Stella**
and can also be made using
**Stella4D**.
When you load a uniform polyhedron in Stella, it is generated on-the-fly just
from the vertex description. That is, the final geometry is NOT already stored
anywhere in the program.

Click on the images below to see a bigger picture and find more photos and
information about them.

### Uniform Polyhedra

### Duals of Uniform Polyhedra

Copyright © 2001-2015,
Robert Webb.