Slice a tetrahedron's corners and an octahedron appears

2026-06-09

The tetrahedron is the odd one out among the Platonic solids: it is its own dual. It has 4 faces and 4 corners, so swapping faces for corners gives back a tetrahedron. You might expect, then, that slicing its corners off - the move that carries a cube to its octahedron - would just lead to a smaller tetrahedron. It leads somewhere more surprising.

Drag the slider. Each corner you shave opens a small green triangle, while the four original blue faces shrink. A little way in is the truncated tetrahedron - 4 triangles and 4 hexagons, an Archimedean solid in its own right. But keep going to the exact midpoint.

The midpoint is an octahedron

When the cuts reach the midpoints of the edges, the four shrunken blue faces and the four grown green corners become the same size. Now there are eight identical triangles, four of each colour, meeting four at every corner. That is a regular octahedron. The colours give the secret away: the four blue faces were the tetrahedron's faces, the four green ones were its corners - and an octahedron is exactly those two sets of four, made equal.

This is the rectified tetrahedron, and "rectify a tetrahedron and you get an octahedron" is the kind of fact that sounds like a mistake until you watch it happen. Slide past the middle and the octahedron's faces drift apart again into another truncated tetrahedron, then close down to the dual tetrahedron - the same shape you started with, turned inside out, pointing the other way.

Two tetrahedra, one octahedron

The two end tetrahedra - the one you start at and the one you finish at - are the very pair that interpenetrate to make the stella octangula, and the octahedron in the middle here is the same octahedron that sits at their intersection there. Truncation walks you smoothly from one tetrahedron, through their shared octahedron, to the other - the static compound's three solids, set in motion.

A self-dual solid, halfway through losing its corners, is not a smaller copy of itself. It is an octahedron - four faces and four corners, made equal at last.