TABLE OF CONTENTS Polyhedra, in their infinite variety, have been objects of study for more than 2500 years. In ancient Greece, contemplation of the regular polyhedra and a few semiregular derivative shapes led to the establishment of the classical world's most famous school, Plato's Academy. We find polyhedral forms expressed concretely in the civil and monumental architecture of antiquity. Their usefulness to us, and our interest in them, continues undiminished today. We now use polyhedra in the mathematics of geometric modeling to represent more complex shapes. We use assemblies of them, like building blocks, to create computer graphics displays. Alone and in Boolean combinations, their properties are relatively easy to compute. And, what is most amazing, all possible polyhedra are defined by only the three simplest of geometric elements points, lines, and planes.
Regular and semiregular polyhedra are the most interesting. Many are found in nature, as well as in art and architecture. Economy of structure and aesthetically-pleasing symmetries characterize these shapes, and help account for their prevalence in nature and in man's constructions. Salt crystals are cubic polyhedra. Certain copper compounds are octahedral. Buckminster Fuller's geodesic dome, in its simplest form, is an icosahedron whose faces are subdivided into equilateral triangles.
This chapter defines convex, concave, and stellar polyhedra, with particular attention to the five regular polyhedra, or Platonic solids, and the 13 semiregular, or Archimedean, polyhedra and their duals. Using Euler's formula, we will prove that only five regular polyhedra are possible in a space of three dimensions. Other topics include...
