dodecahedron The Uniform Polyhedra



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 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

Volume 1 / Plate 6
Volume 2 / Plate 5
Volume 2 / Plate 6


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
GIF-animations of all polyhedra! See them 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.

© 1998 MathConsult Dr. R. Mäder  
Comments to; Last update: 31.07.1998