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Platonic Solids

25 24 Polyhedra wIthIn Polyhedra and so proceed ad infinitum The Platonic Solids fit together in remarkable and fascinating ways the appendix on page 54 shows many relationships. The upper stereogram pair opposite shows a dodecahedron with edge length one. Nested inside it is a cube, edge length , and a tetrahedron, edge length 2 times the cube s see page 55. The tetrahedron occupies one third of the cube s volume. In the lower stereogram pair opposite, the six edge midpoints of the tetrahedron define the six vertices of an octahedron. As well as halving the tetrahedron s edges this octahedron has half its surface area and half its volume, perfectly embodying the musical octave ratio of 12. Similarly the twelve edges of the octahedron correspond to the twelve vertices of a nested icosahedron. The icosahedron s vertices cut the octahedron s edges perfectly into the golden ratio see page 16 for instructions on how to make these stereogram images seem to jump into 3D. Imagine these two sets of nestings combining to give all five Platonic Solids in one elegant arrangement. Since the outer dodecahedron defines a larger icosahedron by their dual relationship, and the inner icosahedron likewise defines a smaller dodecahedron, the nestings can be continued outwards and inwards to infinity. The tetrahedron, octahedron and icosahedron, made entirely from equilateral triangles, are convex deltahedra. The five other possible convex deltahedra are shown opposite below.
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