Shave a cube's corners and you get an octahedron

2026-06-09

A cube and an octahedron are duals: swap the cube's faces for corners and you get the octahedron. That makes them feel like opposites. But they are something closer than opposites - they are the two ends of one continuous slide, and you travel from one to the other by doing nothing fancier than slicing off corners.

Drag the slider, or tap a named stop to jump straight to it. Each notch shaves a little more off the cube's eight corners. A green face opens wherever a corner used to be, while the six blue faces - the originals from the cube - shrink to make room. The colours track the two families of face, not their shape, so each face keeps its colour the whole way even as it gains and loses sides. Watch the counts: the cube's 8 corners and 6 faces drift toward the octahedron's 6 corners and 8 faces, trading places as you slide.

Two stops along the way are famous in their own right. Just past the cube, the blue faces become regular octagons and the green corners are neat triangles: the truncated cube, one of the Archimedean solids. Three-quarters of the way, the green faces have grown into hexagons against the blue squares: the truncated octahedron. Both are tap-stops on the slider.

The shape in the middle

Halfway along, something exact happens. The cuts reach the midpoint of every edge, so the growing triangles and the shrinking squares meet corner to corner, and every edge in the figure is suddenly the same length. This is the cuboctahedron: 8 triangles and 6 squares, 12 vertices, 24 edges. It is the one shape that is exactly as much cube as octahedron - the balance point of the slide, and a beautiful solid in its own right.

Past the midpoint the squares keep shrinking. The cuts, which started at the corners, have eaten all the way to the centre of each original face. When the last square closes to a point, the six faces are gone and only the eight triangles remain, three to a corner: the octahedron. The corners you started shaving have become the faces you end with.

Why the dual was always reachable

Duality told us the octahedron was hiding inside the cube, its corners sitting on the cube's face-centres. Truncation shows that hiding place is not a trick of nesting - it is where you arrive if you simply keep cutting. The eight triangles of the final octahedron are the eight corners of the cube, opened out and grown until they are all that is left.

The same slide runs between the other dual pair too, passing through a shape called the icosidodecahedron at its midpoint - and the tetrahedron, slid against itself, passes through the octahedron on the way. Every dual pair is two stops on one road.

The cube and the octahedron aren't two solids that happen to be related. They are one shape, caught at two ends of a slice - and the cuboctahedron is the moment exactly between them.