Two tetrahedra weave a cube and an octahedron

2026-05-13

A cube has eight corners. Look at them and the eye sees a box. Look harder and the eye sees something else: the corners come in two flavours. Four of them form one regular tetrahedron. The other four form another. Put both tetrahedra back in the cube together and a small star appears - eight outer points, twelve edges crossing through each other, six tidy points in the middle.

Kepler drew the figure in 1609 and called it the stella octangula - the eight-pointed star. He noticed what the figure was hiding: not two solids, but three.

Drag to spin. The blue and red tetrahedra share no corners. Each tetrahedron has four corners and six edges; together that is eight corners and twelve edges. Around them is the faint dashed cube whose eight corners are exactly the eight corners of the compound. Inside them, sitting at the points where the tetrahedra's edges cross, is a small octahedron. Toggle any layer off with the swatches under the figure.

Why the corners split in two

Label the eight corners of the cube by their three signs: (+,+,+), (+,+,-), and so on, up to (-,-,-). Now sort them by how many of the three signs are negative. Four have an even count of minuses - zero or two. Four have an odd count - one or three. That is the entire trick: even-parity corners form one tetrahedron, odd-parity corners form the other.

Why does that work? Two cube-corners are diagonally opposite on a face exactly when they share one sign and disagree on the other two - which flips parity by zero. So same-parity corners are connected by face-diagonals, not by cube edges. There are six pairs of same-parity corners in each group of four, and each pair is at the same distance: the face-diagonal length, 2√2 for our unit cube. Six edges, all the same length - that is a regular tetrahedron.

The cube itself owns no edges in the compound. The cube's edges run between corners of different parity - a blue corner and a red one. The compound's edges are all the face-diagonals; the cube only frames them.

Where the octahedron comes from

Look at one face of the cube. It is a 2×2 square. The blue tetrahedron contributes one diagonal of that square; the red one contributes the other. Two diagonals of a square cross at the centre of the square. So on each of the cube's six faces, one blue edge and one red edge cross at the face's centre.

Six faces, six crossings, six points - one at each face centre. Connect every pair of them whose connecting segment is a face of the inner solid and out falls a regular octahedron, edge length √2, sitting inside the cube. Three Platonic solids in one figure: two tetrahedra, the cube they fit into, and the octahedron they fight over.

A small symmetry

The compound is left alone by the same rotations as the cube, with one extra trick: half the cube's symmetries swap the two tetrahedra. The compound has the cube's 24 rotations as a symmetry group, but only 12 of them fix each tetrahedron in place. The other 12 flip blue and red. The figure is a mirror of itself across that swap.

This is what gives the stella octangula its quiet magic. Spin the figure until you cannot tell which tetrahedron is which. You are not lost: you are inside the symmetry, in a place where blue and red are the same shape and only their colour is keeping them apart.

Two tetrahedra share a centre, a cube, and an octahedron. The cube around them is older than they are, but they were always hiding inside it.