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Platonic Solids

37 36 the cuboctahedron 14 faces 24 edges 12 vertices The cuboctahedron combines the six square faces of the cube with the eight triangular faces of the octahedron. It has octahedral symmetry. Joining the edge midpoints of either the cube or the octahedron traces out a cuboctahedron shown below as a stereogram pair. According to Heron of Alexandria 1075 AD, Archimedes ascribed the cuboctahedron to Plato. Quasiregular polyhedra such as the cuboctahedron are made of two types of regular polygon, each type being surrounded by polygons of the other type. The identical edges, in addition to defining the faces themselves, also define equatorial polygons. For example the cuboctahedrons edges define four regular hexagons lower centre opposite. The radial projections of quasiregular polyhedra consist entirely of complete great circles lower left opposite. Twelve spheres pack around an identical thirteenth to produce a cuboctahedron lower right opposite. Greengrocers use this system to stack oranges in offset hexagonal layers. Known to chemists as hexagonal close packing each sphere is surrounded by twelve others, their centres defining a strong lattice of tetrahedra and octahedra.
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