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In particular he identified the five elements, fire, earth, air, water and celestial matter with the five regular solids, the tetrahedron, cube, octahedron, icosahedron and the dodecahedron.
Water, because it is the most mobile and fluid, has to be an icosahedron , the regular solid that rolls most easily.
Four more Archimedeans appear in the Short book on the five regular solids: the truncated cube, the truncated octahedron, the truncated icosahedron and the truncated dodecahedron.
One of Plato's perfect solids, the icosahedron has 20 triangular faces.
Hypsicles relates that his father and Basilides studied a treatise by Apollonius on a dodecahedron and an icosahedron in the same sphere and decided that Apollonius's treatment was not satisfactory.
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