Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

The Icosahedron in the Cube

A nice model not in the book is to take a regular icosahedron, and build a yellow pyramid on each of eight faces chosen so that their eight apexes are the vertices of a cube.  (By duality, if you built all 20 pyramids, you would get the vertices of a dodecahedron, but here you are choosing just eight.)  Any height pyramid can be used to locate a cube's vertices, (including negative or zero height) but with the yellow pyramid, you can add blue struts to build the cube and see that six of the icosahedron's edges lie in its face planes.

Here is a 15th century image of this
construction, by Piero della Francesca.