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Adapted from Platonic & Archimedean Solids by Daud Sutton, Wooden Books 2002

Dividing a line so that the shorter section is to the longer as the longer section is to the whole line defines the golden ratio. It is an irrational number, inexpressible as a simple fraction…Its value is one plus the square root of five, divided by two – approximately 1.618 … It is innately related to five-fold symmetry; each successive pair of red lines in the pentagram below is in the golden ratio.

Golden Rectangle

A golden rectangle has sides in the golden ratio. If a square is removed from one side, the remaining rectangle is another golden rectangle. This process can continue indefinitely and establishes a golden spiral.

An icosahedron’s twelve vertices are defined by three perpendicular golden rectangles.
Twelve of the twenty vertices of the dodecahedron are defined by three perpendicular golden rectangles with long edge squared, and the remaining eight vertices are found by adding a cube of edge length 1.618