The Magnum Opus

Earlier, we looked at this dynamic compound:

If you were astute, you might have wondered what shape the rhombic dodecahedron and cuboctahedron make together. In other words, what polyhedron is the external hull that stabilises this union? Well, it’s called the triacontahedron. If you take the vertices of the Philosopher’s Stone shape and extend them slightly (specifically by the factor of $\varphi )\; then$they reach out and define the vertices of the shape below.

In formal geometry, we call this radial scaling. In the specific context of the Philosopher’s Stone, the most accurate term is golden harmonisation. Here we are taking two “jagged” or “unsynchronised” shapes (the 12 and the 20) and symmetrising them. In polyhedral math, when we take points that are “lumpy” and push them out onto the surface of a perfect globe it is called Projecting Onto the Canonical Sphere.

The resulting polyhedron has 30 rhombic faces, 60 edges and 32 vertices of two types. The dihedral angles are 144°. It is a Catalan solid and the dual polyhedron of the icosidodecahedron. It is also a zonohedron and can be seen as an elongated rhombic icosahedron. The ratio of the long diagonal to the short diagonal of each face is equal to the golden ratio. A rhombus so obtained is called a golden rhombus:

The triacontahedron’s vertices include the arrangement of four Platonic solids—ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centres of the faces contain five octahedra. It is the only solid that uses 30 identical golden diamonds to perfectly resolve the conflict between the 3-fold mineral world (the IVM) and the 5-fold biological world (DNA).

Above: the orthogonal projection of the triacontahedron

Most importantly, if we include the centre it becomes a 33-point radial system. This makes it a Christ Number—a map of a singular source radiating into a complex body. So the triacontahedron is the most accurate 3D candidate for the Philosopher’s Stone because it is the only shape that successfully wraps the union of opposites in a skin of golden proportions.