__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, °°, 0*), (*°°, 0, °t*) and (*0, °t, °°*).

The vertices of the dodecahedron can be formed in
a similar fashion, in this case place three orthogonal rectangles with side
ratio phi°:1 at the origin. if we define u=phi°/2, the resulting twelve
co-ordinates are then (*°u, °°, 0*), (*°°, 0, °u*) and (*0,
°u, °°*). 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.