The Uniform Polyhedra

Introduction
About the Images on These Pages
Background Information
The Wythoff Symbol
Geometric Properties

Introduction

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.

About the Images on These Pages

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
A list sorted by Wythoff symbol
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.

Background Information

The Wythoff Symbol

Geometric Properties

Symmetry Group

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).

Vertex Configuration

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.

Programs and high-resolution images for uniform polyhedra are available in the book The Mathematica Programmer II by R. Maeder.
All 75 uniform polyhedra, with background information, a clickable map, and animations.

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