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Platonic Solids

15 14 a Short Proof are there really only five A regular polygon has equal sides and angles. A regular polyhedron has equal regular polygon faces and identical vertices. The five Platonic Solids are the only convex regular polyhedra. At least three polygons are needed to make a solid angle. Using equilateral triangles this is possible with three A, four B and five C around a point. With six the result lies flat D. Three squares make a solid angle E, but with four F a limit similar to six triangles is reached. Three regular pentagons form a solid angle G, but there is no room, even lying flat, for four or more. Three regular hexagons meeting at a point lie flat H, and higher polygons cannot meet with three around a point, so a final limit is reached. Since only five solid angles made of identical regular polygons are possible, there are at most five possible convex regular polyhedra. Incredibly, all five regular solid angles repeat to form the regular polyhedra. This proof is given by Euclid of Alexandria c. 325 BC 265 BC in Book XIII of his Elements. The angle left as a gap when a polyhedrons vertex is folded flat is its angle deficiency. Ren Descartes 15961650 discovered that the sum of a convex polyhedrons angle deficiencies always equals 720o, or two full turns. Later, in the eighteenth century, Leonhard Euler 17071783 noticed another peculiar fact In every convex polyhedron the number of faces minus the number of edges plus the number of vertices equals two.
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