Platonic Solids

The regular polyhedra have been known to mathematicians for over 2,000 years, and have played an important role in the development of Western philosophy and science. Drawing on the teaching of his predecessors Pythagoras (sixth century B.C.) and Empedocles (c. 490-c. 430 B.C.), and contributing many original insights, the Greek philosopher Plato (c. 427-347 B.C.) discusses the regular polyhedra, subsequently named after him, in Timaeus, his seminal cosmological work. Plato's narrator, the astronomer Timaeus of Locri, uses triangles—as fundamental figures—to create four of the five regular polyhedra (tetrahedron, hexahedron, octahedron, icosahedron). Timaeus's four polyhedra are further identified with the four basic elements-the hexahedron with earth, the tetrahedron with fire, the octahedron with air, and the icosahedron with water. Finally, in Plato's view, the regular polyhedra constitute the building-blocks not merely of the inorganic world, but of the entire physical universe, including organic and inorganic matter. Plato's ideas greatly influenced subsequent cosmological thinking: for example, Kepler's fundamental discoveries in astronomy were directly inspired by Pythagorean-Platonic ideas about the cosmic significance of geometry. Platonic geometry also features prominently in the work of the noted American inventor and philosopher R. Buckminster Fuller (1895-1983).