Platonic Solids

In Plato’s Timaeus the Demiurge fashions the body of the cosmos not from formless stuff but from geometry. The four elements are given the shapes of four of the five regular solids — the only solids whose faces, edges, and angles are all equal — and these are in turn built up from two kinds of right triangle. Matter, at bottom, is mathematical.

The five and their assignments

Fire is the tetrahedron (sharp, mobile, piercing); earth the cube (the most stable and immovable); air the octahedron; water the icosahedron. The fifth, the dodecahedron, the god used “for the whole” — for the heavens themselves. Because three of the four element-solids share the same basic triangle, they can transform into one another, which is Plato’s geometric account of elemental change.

Afterlife of the idea

Rooted in Pythagorean geometry (Philolaus, Archytas) and carried forward through Euclid’s Elements, Kepler’s cosmology, and modern mathematics, the “Platonic solids” became one of the most enduring images of a mathematically ordered nature.