What are the 5 Platonic solids?

Every corner of a solid needs at least three faces, and their angles must add up to less than 360°. If they make exactly 360° the corner lies flat, like floor tiles. If they make more, the faces overlap. Only room for a gap lets the corner fold up into 3D.

Triangles have 60° corners, so 3, 4 or 5 of them fit (180°, 240°, 300°): the tetrahedron, octahedron and icosahedron. Six triangles make 360°, flat. Squares have 90° corners, so only 3 fit: the cube. Pentagons have 108°, so only 3 fit: the dodecahedron. Hexagons have 120°, and three already make 360°. Nothing with more sides can work at all. That is five, and only five.

This is the last proposition of Euclid’s Elements, Book XIII, written around 300 BCE:

“No other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another.”Euclid, Elements XIII.18 · translated by T. L. Heath

Why Platonic

Plato’s fire, earth, air and water

In his dialogue Timaeus, around 360 BCE, Plato built the four elements out of these shapes. Fire was the sharp, light pyramid: “the pyramid is the solid which is the original element and seed of fire.” Earth was the cube: “To earth, then, let us assign the cubical form; for earth is the most immoveable of the four.” Air got the octahedron and water the icosahedron.

The fifth shape he left almost unnamed. Plato writes only that “there was yet a fifth combination which God used in the delineation of the universe.” Readers have taken it to be the dodecahedron ever since; Benjamin Jowett’s famous translation says so in its introduction, but the dialogue itself never uses the word.

Plato did not discover them. The mathematician Theaetetus was the first to study the octahedron and icosahedron, and Book XIII of Euclid is believed to rest on his work. An old note quoted by the historian Thomas Heath credits the cube, pyramid and dodecahedron to the Pythagoreans.

After Plato

Kepler’s nested universe and the stone balls

In 1596 Johannes Kepler tried to explain the solar system with them. In Mysterium Cosmographicum he nested the five solids between the spheres of the six known planets: a cube between Saturn and Jupiter, a tetrahedron before Mars, a dodecahedron between Mars and Earth, an icosahedron between Earth and Venus, and an octahedron between Venus and Mercury. The fit was off by less than 10%, which impressed him and was still wrong. He later found the real laws of planetary motion.

The carved stone balls of Scotland are a story that grew. More than 400 of these Neolithic balls are known, carved around 3000 to 2500 BCE, and some books say they show all five Platonic solids a thousand years before Plato. The photo behind that claim traces back to a ball at the Ashmolean Museum that, when counted, had 14 knobs, not the icosahedron’s 12. The mathematician John Baez and the historian D. R. Lloyd went through the evidence and found no support for it.

Out in the world

Where you meet them

Dice. A fair die needs identical faces, so the Platonic solids are the natural shapes. Twenty-sided icosahedron dice survive from Greco-Roman Egypt.

Viruses. Many viruses wrap their genes in a protein shell with icosahedral symmetry. The simplest kind uses 60 copies of one protein; many small plant and animal viruses use 180. Caspar gave the first evidence of this symmetry in 1956.

Crystals. Pyrite grows as cubes and octahedra, and fluorite as cubes and octahedra too. Pyrite’s famous twelve-sided crystal, the pyritohedron, looks like a dodecahedron but its pentagons are not regular.

Sea plankton. In 1904 the biologist Ernst Haeckel opened his book Art Forms in Nature with plate 1, a microscopic radiolarian he named Circogonia icosahedra.

Sacred geometry. Four of the five can be traced as flat drawings inside Metatron’s Cube, and the dodecahedron and icosahedron are built on the golden ratio.