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Platonic Solids

27 26 comPound Polyhedra a stretch of the imagination The interrelationships on the previous page generate particularly beautiful compound polyhedra. Fix the position of an icosahedron, and octahedra can be placed around it in five different ways, giving the compound of five octahedra top left. Similarly the cube within the dodecahedron, placed five different ways, generates the compound of five cubes top right. The tetrahedron can be placed in the cube two different ways to give the compound of two tetrahedra shown on page 16. Replace each of the five cubes in the dodecahedron with two tetrahedra to give the compound of ten tetrahedra middle left. Remove five of the tetrahedra from the compound of ten, to leave the compound of five tetrahedra middle right. This occurs in two versions, righthanded or dextro and lefthanded or laevo the two versions cannot be superimposed and are described as each others enantiomorphs. Polyhedra or compounds with this property of handedness are described as chiral. Returning to the cube and dodecahedron, and this time fixing the cube, there are two ways to place the dodecahedron around it. The result of both ways used simultaneously is the compound of two dodecahedra lower left. In the same way the octahedron and icosahedron pair gives the compound of two icosahedra lower right. Many other extraordinary compound polyhedra are possible, for example Bakos compound of four cubes is shown on the very first page of this book.
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