Start from a square cut by its diagonals, with the incentre of each triangle marked. This page lifts that drawing into a cube.

Cut from the centre, the cube splits into six pyramids, one on each face. Each dot is the incentre of its pyramid: the centre of the largest ball that fits inside, equally far from all five faces. Red lines run from it to the pyramid’s five corners. Drag to turn it.

Shape

Pull apart Tinted faces Touching spheres Slow spin Flat projection (no perspective)

Look along

The 24 tetrahedra

Highlight them Split each into mirror halves (48)

Use the “Look along” buttons to close the cube and view it down each of its three kinds of symmetry axis.

Facts hiding in it

1. 24 pieces, 24 turns. Each dot plus one triangle face of its pyramid makes a small tetrahedron. There are 6 × 4 = 24, all identical, and the cube can be turned exactly 24 ways onto itself. Turning the cube carries any one tetrahedron onto any other. This 24-to-24 match is special to 3D (see the long read).
2. 48 with mirrors. Each tetrahedron has a mirror plane through it. Split along it and you get 48 halves, one for each of the cube’s 48 symmetries when reflections count. The flat drawing does the same with 8 triangles for the square’s 8 symmetries.
3. The number is tan 22.5°. With a cube of side 2, each dot is √2 − 1 ≈ 0.414 from every face of its pyramid. That is tan 22.5°, half of 45°, and 1 / (1 + √2), the reciprocal of the “silver ratio” that governs the regular octagon.
4. The six spheres kiss. Neighbouring dots are 2(√2 − 1) apart, exactly two radii. So each ball touches its four neighbours, on the shared wall between them, and touches the centre of its own cube face. The four circles in the flat drawing touch the same way, on the diagonals.
5. The dots make an octahedron. The six dots sit on the three axes, at the corners of a regular octahedron, the cube’s dual shape. The 12 kissing points are the octahedron’s edge midpoints, one for each of the cube’s 12 edges.
6. A small hole in the middle. The six balls leave a gap at the centre. Its radius is 3 − 2√2 = (√2 − 1)² ≈ 0.17, the same number squared.

Atoms, tilings and dimensions: the illustrated long read →