Every Platonic solid is hiding its opposite inside

2026-06-09

A cube has 6 faces and 8 corners. An octahedron has 8 faces and 6 corners. Look at those numbers again: they are the same two numbers, with the roles of face and corner swapped. That is not a coincidence. The two solids are the same object seen two ways - and you can build one out of the other with a single move.

Mark the centre of every face. Then join two marks whenever their faces share an edge. The marks and the joins are the corners and edges of a brand-new solid, sitting inside the first. It is called the dual.

Start with the dodecahedron (the default above). It has 12 pentagon faces. Drop a dot on each and join the neighbours: 12 dots become the 12 corners of an icosahedron. The icosahedron's 20 faces match the dodecahedron's 20 corners. Every count swaps; the edge count, 30, stays put. Try each solid with the buttons and watch the readout swap its first and last numbers every time.

Why faces and corners trade places

The move turns every face into a corner: one dot per face, so the dual gets exactly as many corners as the original had faces. It also turns every corner back into a face. Look at a single corner of the original - say a corner where three squares of a cube meet. Those three faces give three dots, and the three dots, joined up, ring that corner with a little triangle. The corner has become a triangular face of the dual. A corner where k faces meet becomes a k-sided face.

And edges simply pair off. Every edge of the original is shared by exactly two faces - so it corresponds to exactly one join between their two dots. One edge in, one edge out. That is why the middle number never changes: the cube and octahedron both have 12 edges, the dodecahedron and icosahedron both have 30.

Written together, the swap is Euler's little bookkeeping rule seen in a mirror. Both solids obey V − E + F = 2; the dual just reads it right-to-left.

The three pairings, and the loner

There are only five Platonic solids, and this move sorts them into pairs. The cube and octahedron are duals. The dodecahedron and icosahedron are duals. That leaves the tetrahedron, which has 4 faces and 4 corners - so its dual is another tetrahedron. It is its own opposite. (Two of them, sharing a centre, make the star you may have met as the stella octangula.)

Do the move twice - dual of the dual - and the dots-of-dots land back on a shrunken copy of the solid you began with. Duality is a round trip. Nothing is added and nothing is lost; the cube and the octahedron were always two readings of one shape, and the icosahedron was always a dodecahedron turned inside out.

Put a dot on every face. Connect the neighbours. The solid you get is the one you started with, holding up a mirror - faces for corners, corners for faces, and the same edges between them.