# Platonic Solids

## Historical significance

The term platonic solids refers to regular polyhedra. In geometry, a polyhedron, (the word is a Greek neologism meaning many seats) is a solid bounded by plane surfaces, which are called the faces; the intersection of three or more edges is called a vertex (plural: vertices). What distinguishes regular polyhedra from all others is the fact that all of their faces are congruent with one another. (In geometry, congruence means that the coincidence of two figures in space results in a one-to-one correspondence.) The five platonic solids, or regular polyhedra, are: the tetrahedron (consisting of four faces that are equilateral triangles), the hexahedron, also known as a cube (consisting of six square faces), the octahedron (consisting of eight faces that are equilateral triangles), the dodecahedron (12 pentagons), and the icosahedron (20 equilateral triangles).

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).

## Resources

### Books

Coplestone, Frederick. Greece and Rome. Vol. 1, A History of Philosophy. Garden City, NY: Doubleday, 1985.

Kline, Morris. Mathematics in Western Culture. London: Oxford University Press, 1964.

Koestler, Arthur. The Sleepwalkers. New York: Grosset & Dunlap, 1959.

Millington, T. Alaric, and William Millington. Dictionary of Mathematics. New York: Harper & Row, 1966.

Stewart, Ian, and Martin Golubitsky. Fearful Symmetry: Is God a Geometer? London: Penguin Books, 1993.

Zoran Minderovic