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Platonic Solids

43 42 four exPloSIonS expanding from the centre Exploding the faces of the cube or the octahedron outwards until they are separated by an edge length below defines the rhombicuboctahedron opposite top left. The same process applied to the dodecahedron or icosahedron gives the rhombicosi dodecahedron opposite top right. The octagonal faces of the truncated cube, or the hexagonal faces of the truncated octahedron, explode to give the great rhombicuboctahedron opposite lower left. The decagonal faces of the truncated dodecahedron, or the hexagonal faces of the truncated icosahedron, explode to give the great rhombicosidodecahedron opposite lower right. Kepler called the great rhombicuboctahedron a truncated cuboctahedron, and the great rhombicosidodecahedron a truncated icosidodecahedron. The two truncations he refers to, however, do not produce square faces, but 2 and rectangles. These four polyhedra have face planes in common with either the cube, octahedron and rhombic dodecahedron see page 47, or the icosahedron, dodecahedron, and rhombic triacontahedron see page 47, hence the prefix rhombi in their names.
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