__Phi in the icosahedron and dodecahedron__

The golden ratio (phi) is defined as (sqrt(5)+1) / 2 and is approximately equal to 1.618...

Phi occurs repeatedly in the construction of the icosahedron and dodecahedron. A simple example shows how phi occurs in the co-ordinates of these polyhedra

The twelve vertices of the icosahedron can all be
defined by placing three orthogonal rectangles with side ratio phi:1 at the
origin. If we define t = phi/2, the resulting co-ordinates are then

(*±t, ±½, 0*), (*±½, 0, ±t*) and (*0, ±t, ±½*).

The vertices of the dodecahedron can be formed in
a similar fashion, in this case place three orthogonal rectangles with side
ratio phi²:1 at the origin. if we define u=phi²/2, the resulting twelve
co-ordinates are then (*±u, ±½, 0*), (*±½, 0, ±u*) and (*0,
±u, ±½*). The remaining eight vertices can then the formed by
placing a cube of side length phi at the origin with its faces parallel to the
planes of the rectangles. With t = phi/2 as before, the resulting
co-ordinates are then (*±t, ±t, ±t*).

I am indebted to Daud Saum's book 'Polyhedra' [ref] for bringing these relationships to my attention.