The Platonic solids

The teaching

As the Greek geometry of the solids from Plato's Timaeus to Euclid teaches it.

The five and their counts

A regular solid has faces that are all one regular figure, with the same number meeting at every corner. There are five. The tetrahedron has 4 triangles, 6 edges and 4 corners, three faces meeting at each. The cube has 6 squares, 12 edges and 8 corners, three at each. The octahedron has 8 triangles, 12 edges and 6 corners, four at each. The dodecahedron has 12 pentagons, 30 edges and 20 corners, three at each. The icosahedron has 20 triangles, 30 edges and 12 corners, five at each.

Why there are only five

Euclid closes the Elements with the proof. At any corner of a solid at least three faces must meet, and their angles there must add to less than a full turn of 360 degrees, or the corner would lie flat. The angle of a triangle is 60 degrees, so three, four or five triangles can meet, making 180, 240 or 300; six make 360 and lie flat. The angle of a square is 90, so three can meet, making 270; four lie flat.

The angle of a pentagon is 108, so three can meet, making 324; four would make 432. The angle of a hexagon is 120, and three already make 360. Every figure with more sides has larger angles still. So there are five ways for regular faces to meet at a corner, and five solids.

Pairs and a rule of two

Join the centres of the faces of a cube and an octahedron appears inside it; join the centres of the octahedron's faces and a cube appears. The dodecahedron and the icosahedron are paired in the same way, and the tetrahedron is paired with itself. Each pair swaps its numbers: the cube's six faces and eight corners are the octahedron's six corners and eight faces, and both have twelve edges. For every one of the five, corners less edges plus faces makes two: for the cube, 8 less 12 plus 6. Euler found the rule, and it holds for any solid without holes.

Beyond the five

Loosen the rule and more solids appear. Allow two or more kinds of regular face, with the same arrangement at every corner, and there are thirteen more, the Archimedean solids, such as the cuboctahedron of squares and triangles. Let the faces pass through one another as stars, and four more stand up, found by Kepler and by Poinsot. The five alone keep the strict rule: one face, one corner, everywhere the same.

Plato's elements

In the Timaeus the maker of the cosmos builds the four elements from two kinds of right-angled triangle: the half square, with two equal sides, and the half of an equilateral triangle, which Plato calls the most beautiful of triangles. From the half-equilateral triangles come the faces of the tetrahedron, the octahedron and the icosahedron; from the half squares come the square faces of the cube.

Each element takes the solid that fits its nature. Fire is the tetrahedron, with the fewest faces and the sharpest points, the lightest and quickest, which cuts and burns. Earth is the cube, the most stable, set firm on its broad faces. Air is the octahedron, and water the icosahedron, with the most faces, the roundest, which rolls and flows.

How the elements change into one another

Because fire, air and water share one triangle, they can break apart and join again as each other. A particle of water, twenty faces, cut up by fire, may become one particle of fire, four faces, and two of air, eight faces each; a particle of air may become two of fire. Earth is made of the other triangle and never becomes any of the three: broken up, its pieces drift until they meet and join again as earth.

The fifth figure

One solid is left over. Plato says the maker used the fifth for the whole, to decorate it with figures, and later readers took its twelve faces for the twelve signs of the zodiac. When Aristotle added a fifth element, the aether of the heavens, the dodecahedron was given to it as the quintessence.

Kepler's nest of solids

Kepler set the five one inside another between the paths of the six planets then known: the cube between Saturn and Jupiter, the tetrahedron between Jupiter and Mars, the dodecahedron between Mars and the Earth, the icosahedron between the Earth and Venus, and the octahedron between Venus and Mercury. In his Harmony of the World he kept Plato's matching of solids to elements and gave reasons for each, the sharp tetrahedron to fire and the steady cube to earth.