/pləˈtɒn.ɪk ˈsɒl.ɪdz/

Platonic solids

noun · sacred geometry

What it is

A Platonic solid satisfies three conditions at once: every face is the same regular polygon, the same number of faces meet at every vertex, and the whole figure is convex. Euclid proved in the Elements that exactly five forms can meet all three — four triangular forms (tetrahedron, octahedron, icosahedron), one square (cube), and one pentagonal (dodecahedron). No sixth is geometrically possible, which is why the set is treated as complete and closed.

The solids come in dual pairs: the cube and octahedron, the dodecahedron and icosahedron, with the tetrahedron its own dual. Each can be inscribed in a single sphere with all vertices touching, which is why sacred-geometry traditions nest them inside one another and inside the larger circular figures.

Etymology

The forms are named for Plato, who in the Timaeus (c. 360 BC) assigned four of them to the classical elements — tetrahedron to fire, cube to earth, octahedron to air, icosahedron to water — and the dodecahedron to the cosmos as a whole. The forms were studied earlier by the Pythagoreans, and the proofs were systematised by Theaetetus, but Plato's cosmology fixed his name to them. They are also called the cosmic or perfect solids.

Examples in use

In sacred geometry, all five Platonic solids can be derived from the thirteen circles of Metatron's Cube by joining its centres — the construction that links the solids to the Flower of Life lineage. Johannes Kepler used the nested solids in his 1596 Mysterium Cosmographicum to model planetary spacing.

Robert Lawlor's Sacred Geometry presents the solids as the three-dimensional resolution of the plane figures: the hexagonal and pentagonal symmetries of the flat diagrams stand up into the octahedron and the dodecahedron.