Getting Phi in the Game

:: vZome by Koski ::

Kirby Urner: in the calcs below, the assumption is always tetravolumes, meaning we've calibrated to the concentric hierarchy with the volume 20 cuboctahedron.

The icosahedron with that same edge, of volume ~18.51 is the icosahedron of volume 5 Φ² √2 mentioned below.

The pentagonal dodecahedron is its partner in the rhombic triacontahedron, inside of which one finds these unit-edge cubes (but not of unit volume in this schema).

David Koski:

Icosahedron edge = 1 in tetra volumes (blue)
= ~18.51229587
= 5 Φ² √2

Pentagonal dodecahedron edge Φ^-1 or .618034 (orange)
= ~15.35001821
= (Φ² + 1) 3√2

Rhombic triacontahedron diagonals 1 and Φ^-1 (red)
= ~21.21320244
= 15√2

Cube edge = 1 (purple)
= ~8.485281374
= 6√2