3.2.2 The Five Regular Polyhedra

In accordance with the four classical elements and their quintessence, there are exactly five geometric solids whose faces are regular and identical polygons meeting at equal three-dimensional angles. These five regular polyhedra are the only solid shapes with this sort of complete symmetry. Many philosophers wondered why there cannot be more, or fewer, so perfectly symmetrical shapes. They are usually called Platonic Solids, because Plato composed a complete theory of the cosmos based on their definite shapes, associating them with the four classical elements and the Quintessence.

The names of these polyhedra reflect the number of faces that each one possesses, and these same names may be also used to describe other three-dimensional solids that have the same number of faces. At least some of these solids were known by Pythagoras, or even before, but according to Euclid, the Octahedron and Icosahedron were first discussed by the Theaetetus (c. 417–369 BC). Plato was greatly impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of non-planar points in space, so he expounded a complete theory of the cosmos based explicitly on these five solids. In his dialogue Timaeus, he associated four of these polyhedrons with the four basic elements: assigning the Tetrahedron to the element Fire, because of its sharp points and edges, while the Cube corresponded to Earth because of its four-square regularity, and the Octahedron to Air since its minuscule components are so smooth that one can barely feel it, and finally the Icosahedron to Water that flows out of one’s hand when picked up, as if it is made of tiny little balls. The fifth solid, the Dodecahedron, with its twelve pentagonal faces, was assigned to the Heavens with its twelve zodiac constellations, since also it is so different from the other polyhedra, in virtue of its pentagonal faces. Timaeus contains a very detailed discussion of virtually all aspects of physical existence, including biology, cosmology, geography, chemistry, physics, psychological perceptions, all expressed in terms of these four basic elements and their transmutations from one into another by means of the constituent triangles being broken apart and re-assembled into other forms.

With these hypothetical relations, Plato developed a theory of the Universe based on the five regular polyhedra, hence they became known as the Platonic solids, though even this correspondence was defined earlier by the philosopher Empedocles. Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.

The symmetry, structural integrity, and beauty of these solids have inspired architects, artists, and artisans from ancient Egypt to the present. Euclid devoted the last book of the Elements to the regular polyhedra, where he provided the first known proof that exactly five regular polyhedra exist. It is natural to wonder why there should be exactly five regular polyhedra, and whether there might conceivably be one that simply hasn’t been discovered yet. However, it is not difficult to show that there must be five, and that there cannot be more than five. Thus Plato concluded that they must be the fundamental building blocks of nature.

The symmetries of these five regular polyhedra and relations to the four elements and their quintessence have been discussed with more details in Chapter IV of the Ultimate Symmetry.