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Platonic Solids

23 22 the golden ratIo and some intriguing juxtapositions 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 below. It is an irrational number, inexpressible as a simple fraction see page 55. Its value is one plus the square root of five, divided by two approximately 1.618. It is represented by the Greek letter phi or sometimes by tau. has intimate connections with unity times itself 2 is equal to plus one 2.618..., and one divided by equals minus one 0.618.... It is innately related to fivefold symmetry, each successive pair of heavy lines in the pentagram below is in the golden ratio. 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 below right. Remarkably, an icosahedron s twelve vertices are defined by three perpendicular golden rectangles opposite top. The dodecahedron is richer still. Twelve of its twenty vertices are defined by three perpendicular 2 rectangles, and the remaining eight vertices are found by adding a cube of edge length lower opposite.
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