Icosahedron and inscribed cube We can inscribe a cube in dodecahedron (see this), where $12$ faces of dodecahedron give the $12$ edges of the cube. !enter image description here Can we inscribe cube in icosahedron?

Dodecahedron and icosahedron are dual to one another. So there would be a way where each vertex of the icosahedron corresponds to an edge of the cube. So you'd have corners of the cube in the centers of _some_ of the faces.

!Cube in Icosahedron

!Cube in Dodecahedron in Icosahedron

Pictures created using a Cinderella Application created by J. Richter-Gebert.