Five cubes hide inside a dodecahedron

2026-05-13

A regular dodecahedron has twenty corners. They look haphazard - twelve pentagonal faces, each sharing corners with three others, fitted together like a stitched leather ball. But the corners aren't haphazard. They're holding a secret.

Pick the right eight of the twenty. They form a perfect cube. Pick a different eight. Another perfect cube. There are exactly five ways to do it. Five cubes, all the same size, sharing the dodecahedron's vertices in groups of two.

Drag the figure to spin it. The faint grey outline is the dodecahedron itself - twenty corners, thirty edges, twelve pentagonal faces. Inside it: five cubes in five different colours, each using a different selection of eight dodecahedron vertices. Every vertex belongs to exactly two cubes. Every cube shares some vertices with every other cube.

All five at once is a beautiful tangle but a hard read. Use the five colour swatches under the diagram to hide the cubes one at a time. Leave one on and you can see a single perfect cube inscribed in the dodecahedron, with its eight corners poking out at eight of the twenty vertices. Turn another on and you can see how a pair of cubes shares some vertices, twists against each other, and avoids the rest.

Where the cubes come from

Start with the dodecahedron oriented so two opposite pentagonal faces are horizontal - one at the top, one at the bottom. The dodecahedron has a five-fold rotational symmetry around the axis through those two faces: rotate 72° and the dodecahedron maps to itself.

Now pick the eight vertices of one cube. Rotate by 72° - those eight vertices map to eight different vertices, which form a different cube of the same size. Rotate again. And again. Five cubes, related by 72° rotations. After five turns you're back where you started.

The reason there are exactly five and not three or seven is hiding in the order of the icosahedral symmetry group: 60 rotations, which factors as 5 × 12. Each cube has twelve rotational symmetries of its own; the five cubes are the cosets of the cube's group inside the dodecahedron's.

The edges as face diagonals

A dodecahedron face is a pentagon. A pentagon has five diagonals - the chords inside it that aren't edges. Each cube's edges, drawn against the dodecahedron, run along these diagonals. One diagonal per pentagonal face per cube. Twelve faces × five cubes = sixty cube edges in total, which is exactly five diagonals per pentagonal face. Every diagonal of every face is part of exactly one of the five cubes.

That is the cleanest statement of what the five cubes are: a partition of the dodecahedron's face-diagonals into five groups of twelve, each group forming a cube.

A puzzle

The five cubes have a handedness. Look closely as the figure rotates: the cubes are all twisted the same way. There is also a second compound of five cubes, twisted the opposite way, which exists in the mirror-image dodecahedron. Combine both compounds and you have ten cubes filling a dodecahedron - a compound first described by Edmund Hess in 1876. We are only drawing five of them. The other five would double the lines and lose the dance.

Try this: as the figure rotates, fix your eye on one corner. Watch the two cubes that pass through it. They emerge, twist, withdraw. Watch a second corner. Two cubes again, but a different pair. Every corner has two cubes. Every cube has eight corners. The arithmetic works out: 5 × 8 = 40 = 2 × 20.

Five cubes interlocked inside a dodecahedron, sharing every vertex, related by five-fold rotation. The dodecahedron has been quietly containing them since Euclid.