Mathematical Experiment 5

Construct the part of the cylinder

x2 + y2 =  1

lying between the planes z= -1 and z = x.

Combine the above surface with its reflection across the place x = z to form this model.

Join four cylinders of the same size symmetrically as thus:

Join eight cylinders of the same size symmetrically as thus:

Join six cylinders of the same size symmetrically as thus:

Construct three mutually orthogonal "golden rectangles" whose 12 vertices form a regular icosahedron.

Join 12 cylinders of the same size symmetrically as thus:

Let τ denote the golden ratio

τ = .

Illustrate this interesting fact: the points

 [+τ, +τ-1, 0], [0, +τ, +τ-1],

[+τ-1, 0, +τ], [+1, +1, +1]

comprise the vertices of a regular dodecahedron.