The Five Platonic Solids, Their 13 Archimedean Cousins, and Why That's All There Is

Exactly five, and no more

A Platonic solid is a convex polyhedron built from one single type of regular polygon, with the same number of faces meeting at every vertex. There are exactly five of them — tetrahedron, cube, octahedron, dodecahedron, icosahedron — and that count is not an accident of taste; it falls straight out of a simple angle-sum argument. At any vertex, the interior angles of the faces meeting there must sum to strictly less than 360°, or the surface can't fold up into a solid at all (exactly 360° gives a flat tiling, not a 3D shape). Since a regular polygon's interior angle grows with its number of sides, only triangles (60° each), squares (90°) and pentagons (108°) leave enough angular room to combine — hexagons and beyond hit or exceed 360° with as few as three meeting, and that closes off every other possibility.

3 triangles/vertex (180°)  → tetrahedron   (4 faces)
4 triangles/vertex (240°)  → octahedron    (8 faces)
5 triangles/vertex (300°)  → icosahedron   (20 faces)
3 squares/vertex   (270°)  → cube          (6 faces)
3 pentagons/vertex (324°)  → dodecahedron  (12 faces)

Euclid proved this closure in the final book of the Elements around 300 BCE, and it remains one of the cleanest finiteness results in all of mathematics: a purely local angular constraint at a single vertex forces a hard global cap on how many such solids can exist.

Euler's formula: the invariant that survives distortion

Every one of these five, and every convex polyhedron in general, obeys Euler's polyhedron formula, published by Leonhard Euler in 1758: vertices minus edges plus faces always equals 2.

V - E + F = 2

tetrahedron:   4  -  6 + 4  = 2
cube:          8  - 12 + 6  = 2
octahedron:    6  - 12 + 8  = 2
dodecahedron: 20  - 30 + 12 = 2
icosahedron:  12  - 30 + 20 = 2

This invariant is topological, not geometric — it holds equally for a lopsided, irregular convex polyhedron as for a perfect Platonic solid, because it is really a statement about the sphere's topology (V - E + F = 2 is the Euler characteristic of any surface with the topology of a sphere) rather than about angles or edge lengths. It fails, predictably, for shapes with holes: a torus-shaped polyhedron gives V - E + F = 0, and the formula generalizes to V - E + F = 2 - 2g for a genus-g surface with g holes.

Duality: cube and octahedron are the same idea inside out

Every Platonic solid has a dual: place a vertex at the centre of each face and connect vertices whose original faces shared an edge, and you get another Platonic solid with vertex and face counts swapped. The cube (6 faces, 8 vertices) is dual to the octahedron (8 faces, 6 vertices); the dodecahedron (12 faces, 20 vertices) is dual to the icosahedron (20 faces, 12 vertices); the tetrahedron is self-dual (4 faces, 4 vertices — dual to itself). Duality preserves the edge count exactly (all five have E unchanged under duality) and swaps V and F, which is exactly why Euler's formula, being symmetric in that swap plus a constant, holds for both members of every dual pair automatically.

Archimedean solids: relaxing 'one polygon type'

Drop the constraint that every face is the same regular polygon, but keep every vertex looking identical to every other vertex (vertex-transitivity) and every face a regular polygon, and you get the 13 Archimedean solids — attributed to Archimedes, though his original text is lost and the list survives through later reconstructions, notably Kepler's in 1619. Many arise by truncation: slicing the corners off a Platonic solid. Truncating a cube's eight corners turns each square face into an octagon and each removed corner into a new triangle, producing the truncated cube (8 triangles + 6 octagons). Truncating every corner of an icosahedron produces the truncated icosahedron — 12 pentagons and 20 hexagons — the shape of a classic football and, geometrically, of the buckminsterfullerene C60 molecule.

Symmetry groups and why 3D rotation needs a different tool

Each Platonic solid's rotational symmetries form a finite group — the tetrahedron's 12 rotations, the cube/octahedron pair's 24 (they share a symmetry group because they're dual), the dodecahedron/icosahedron pair's 60. Rendering a rotatable 3D solid on a flat canvas means projecting rotated 3D vertex coordinates down to 2D, and doing that rotation with plain (x,y,z) Euler angles runs into gimbal lock — a configuration where two of the three rotation axes align and a degree of freedom is lost — which is why smoother interactive 3D tools use rotation matrices or quaternions internally even for a task as visually simple as 'drag to spin the shape.'

Frequently asked questions

Try it live

Everything above runs in your browser — open Polyhedra Explorer and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.