Works Cited

The true elements of the material world, Timaeus says, are not earth, air, fire, and water, but two sorts of right-angled triangles, the one which is half a square and the one which is half an equilateral triangle... The above two sorts of triangles, we are told, are the most beautiful forms, and therefore God used them in constructing matter. By means of these two triangles, it is possible to construct four of the five regular solids, and each atom of one of the four elements is a regular solid. Atoms of earth are cubes; of fire, tetrahedra; of air, octahedra; and of water, icosahedra. (I shall come to the dodecahedron presently.)...

The regular tetrahedron, octahedron, and icosahedron, have equilateral triangles for their faces; the dodecahedron has regular pentagons, and cannot therefore be constructed out of Plato's triangles. For this reason he does not use it in connection with the four elements.

As for the dodecahedron, Plato says only "there was yet a fifth combination which God used in the delineation of the universe." This is obscure, and suggests that the universe is a dodecahedron; but elsewhere it is said to be a sphere. The pentagram has always been prominent in magic... It seems that it owed is properties to the fact that the dodecahedron has pentagons for its faces, and is, in some sense, a symbol of the universe.


Bertrand Russell, The History of Western Philosophy