Chapter 9
Uniform Polyhedra

Overview

Uniform polyhedra consist of regular faces and congruent vertices. Allowing for nonconvex faces and vertex figures, there are 75 such polyhedra, as well as two infinite families of prisms and antiprisms. A recently discovered uniform way of computing their vertex coordinates is discussed in Section 2. It is the basis for our program to display all of these solids (Sections 3 and 4), among which are many beautiful and stunning shapes. The program requires a flexible way to store properties of objects.

We shall also develop some utilities for drawing spherical triangles in Section 5 and for storing properties of objects in a flexible way.

Programs

UniformPolyhedra.m

Computation of all uniform polyhedra.

PolyhedraExamples.m

Standard list of all 75 uniform polyhedra plus 5 prisms and antiprisms.

SphericalTriangles.m

Illustrations of spherical triangles.

Notebooks

UniformPolyhedra.nb

Examples in the chapter.

Color Plates

Plate 5

Plate 8

Resources on the Internet

Zvi Har'El's original program kaleido.c and paper

Har'El, Zvi. Uniform solution for uniform polyhedra. Geometriae Dedicata, 47, 1993.

The Uniform Polyhedra

Images of all uniform polyhedra and part of the text from this chapter.