09 Mar

Icosahedron in Cube

As mentioned and illustrated in the post on the Anticos, it’s possible to inscribe an icosahedron in a cube. (In this case, that technically means that given a cube, you can choose two points on each face of the cube such that the convex hull of the resulting set of twelve points is a regular icosahedron.)

But why should this be so? To see this, it’s easiest to start with a regular dodecahedron, say with unit edge length. Notice the interesting pattern of the blue face diagonals in this diagram:


Notice that during this transformation, each of the edges remains in the plane perpendicular to that axis. Therefore, twelve of the vertices of an icosahedron lie on the surface of a cube, two on each face. Since an icosahedron only has twelve vertices, it is inscribed in a cube.

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