The Pyritohedral Golden Icosahedron

Both the Platonic icosahedron and the golden icosahedron have twenty triangular faces. In the Platonic version, these faces are all equilateral triangles. The golden icosahedron has eight such triangles, but the other twelve are golden triangles, which have a leg-to-base ratio which is the golden ratio. These golden triangles appear in pairs, and the six pairs are arranged in such a way as to make this a solid with pyritohedral symmetry: the symmetry of a standard volleyball.

A net for the golden icosahedron appears below. Both images were made using a program called Stella 4d, which you can try for free right here.

A Golden Tessellation of Quadrilaterals

In this tessellation, golden rectangles are shown in yellow. The orange darts are each made of two golden gnomons, joined at a leg — while the blue rhombi are each made of two golden triangles, sharing a base.

The Golden Rhombus, the Rhombic Triacontahedron, and the Rhombic Hexecontahedron

There’s a special rhombus which is called the “golden rhombus,” because its diagonals are in the golden ratio. To construct it with compass and straight edge, you first construct a golden rectangle (shown with blue edges and a yellow interior), and then connect the midpoints of its sides to form a rhombus (with edges shown in red).

Several polyhedra can be made which use golden rhombi as their faces. The most well-known of these polyhedra is the rhombic triacontahedron, which has 30 such faces. It is the dual of the icosidodecahedron.

If the rhombic triacontahedron is stellated 26 times, the result is the (non-convex) rhombic hexecontahedron. It has 60 golden rhombi as faces.

Both of these polyhedra can be constructed with Zometools (available at http://www.zometool.com). With white Zomeballs and red Zomestruts, these polyhedra look a lot like this:

The flat image at the top of this post was created using Geometer’s Sketchpad and MS-Paint. The four rotating polyhedral images were created using Stella 4d: Polyhedron Navigator, which you can purchase, or try for free, at http://www.software3d.com/Stella.php.