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The Uniform Polyhedra

Introduction

Uniform polyhedraconsist 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 andantiprisms.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 ofR. Maeder's bookThe Mathematica Programmer II.An expanded version of these Web pages can also be found intheMathematica notebookPolyhedra.ma from theIllustrated Mathematics CD-ROM.

About the Images on These Pages

The metric properties and graphics data were computed with aMathematicaprogram,developed by R. Maeder, based on aC program by Zvi Har'El.

The ray-traced images on these pages were rendered withPOV-ray,from data computed with the program mentioned above,using a conversion programfromMathematica graphics format toPOV-ray input.

Programs and Images are Available!

TheMathematica programs to compute and render the polyhedraare included on the CD-ROM that comes withThe Mathematica Programmer II.The book contains also high-resolution color images of all uniform polyhedra.Follow the link to thebook's home pagefor more information anddirectordering in association with amazon.com.

Graphic Resources

There is one page for each polyhedron with a high-resolution imageand geometrical information. The pages can be accessed in these ways:

Avisual index (sensitive map) of all 80 polyhedra
List and thumbnail pictures of all Uniform Polyhedra
Alist sorted by Wythoff symbol
Aguided 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 theindividual 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 regularn-sided polygon (ann-gon) is described byn.Star polygons are described byn/d, wheren is the number of vertices, connectedd 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 havea look at the collection of allstellated icosahedra.

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

A service provided byMathConsult AG,http://www.mathconsult.ch/.

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