# The Uniform Polyhedra

## Contents

*Uniform polyhedra*
consist of regular faces and congruent vertices.
Allowing for non-convex faces and vertex figures, there are 75 such polyhedra, as well as 2 infinite families of prisms and
antiprisms.
A recently discovered uniform way of computing their vertex coordinates
[Harel93]
is the basis for a program to display all of these solids,
among which are many beautiful and stunning shapes.
This text is an excerpt of Chapter 9 of
R. Maeder's book The Mathematica Programmer II.
An expanded version of these Web pages can also be found in
the *Mathematica* notebook `Polyhedra.ma`

from the
Illustrated Mathematics CD-ROM.

The metric properties and graphics data were computed with a
*Mathematica*
program,
developed by R. Maeder, based on a C program by Zvi Har'El.
The ray-traced images on these pages were rendered with
POV-ray,
from data computed with the program mentioned above,
using a conversion program
from *Mathematica* graphics format to
POV-ray input.

### Programs and Images are Available!

The *Mathematica* programs to compute and render the polyhedra
are included on the CD-ROM that comes with
The Mathematica Programmer II.
The book contains also high-resolution color images of all uniform polyhedra.
Follow the link to the book's home page
for more information and
direct ordering in association with amazon.com.

### Graphic Resources

There is one page for each polyhedron with a high-resolution image
and geometrical information. The pages can be accessed in these ways:
- A visual index (sensitive map) of all 80 polyhedra
- List and thumbnail pictures of all Uniform Polyhedra
(Warning: contains 80 thumbnail pictures)
- A
list sorted by Wythoff symbol
(Warning: contains 80 thumbnail pictures)
- A guided tour of all 80 polyhedra starts here

#### Animations

See the polyhedra spin about a symmetry axis for better visualization.
The animations are linked through the high-resolution images on the
individual polyhedra pages.
The animations use a rather high number of frames for smoother motion
and are, therefore, quite large.
This listing indicates which type of symmetry the polyhedron has:
dihedral, tetrahedral, octahedral, and icosahedral.
The listing does not indicate whether the polyhedron possesses full
(reflective) symmetry, or only rotational symmetry (most snub polyhedra).
The vertex configuration is the sequence of faces arranged around a vertex.
Since vertices are congruent, this sequence is the same for all vertices.
A regular *n*-sided polygon (an *n*-gon) is described by *n*.
Star polygons are described by *n*/*d*, where *n* is the number of vertices, connected *d* apart.
For example, 5/2 is the pentagram.
Some polyhedra contain retrograde faces.
For example, 4/3 is a square, traversed in the opposite direction.
A regular 2-gon is degenerate and can be left out if it occurs in the formulae.

You may also want to have
a look at the collection of all stellated icosahedra.