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The tetrahedron has four triangular faces,
the cube six square faces,
the octahedron eight triangular faces,
the dodecahedron twelve pentagonal faces,
the icosahedron twenty triangular faces.
Plato proposed that four of these solids built the Four Elements: sharp-pointed tetrahedra give the sting of Fire, smooth-sliding octahedra give easily-parted Air, droplety icosahedra give Water, and lumpish, packable cubes give Earth.
The dodecahedron, at last, is the shape of the Universe as a whole. Later Aristotle emended Plato’s system, suggesting that dodecahedra provide a fifth essence
—the space-filling Ether.
Plato’s ideas lent dignity and grandeur to the study of geometry, and greatly stimulated its development. The thirteen and final book of Euclid’s Elements, the grand synthesis of Greek geometry that is the founding text of axiomatic mathematics, culminates with the construction of the five Platonic solids, and a proof that they exhaust the possibilities. Scholars speculate that Euclid planned the Elements with that climax in mind from the start.