Pyritohedron

When you fold a dodecahedron into a cube, inside the cube there is an empty space.

A Cube can be inscribed in a Dodecahedron. A Dodecahedron can be seen as a cube with six 'roofs'. You can fold a dodecahedron into a cube.

We can fill this empty space with a polyhedron that is a kind of pyritohedron, that is to say, this polyhedron is an irregular dodecahedron composed of identical irregular pentagons. In this case is a concave pyritohedron an is called concave pyritohedral dodecahedron.

We encourage you to build your own pyritohedron:

We can build this pyritohedron using Zome:

Eight vertices are the vertices of a cube:

In the interior, twelve vertices are in three golden rectangles:

The three golden rectangles in three different colors:

These twelve vertices are the vertices of an icosahedron:

To calculate the volume of this pyritohedron of side length 1 you should remember some properties of the golden ratio, and it is very similar to the calculation of the volume of a dodecahedron. The roof that was outside the cube it is now inside the cube.

The diagonal of a regular pentagon are in golden ratio to its sides and the point of intersection of two diagonals of a regular pentagon are said to divide each other in the golden ratio or 'in extreme and mean ratio'.
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

The volume of this pyritohedron is the volume of a cube minus six times the volume of a roof.

Remember the volume of the cube:

And the volume of one roof is:

Then the volume of this pyritohedron is:

REFERENCES

Zome is a wonderful tool to build polyhedra.