Proof
Compare the side lengths: tetrahedron , octahedron
, cube , icosahedron (minor irrational),
dodecahedron (apotome). For the uniqueness clause: at each vertex of
a regular polyhedron, the sum of face-angles must be less than four
right angles (XI.21). Equilateral triangles (): 3, 4, or
5 around a vertex — tetrahedron, octahedron, icosahedron. Squares
(): only 3 around a vertex — cube. Regular pentagons
(): only 3 around a vertex — dodecahedron. Hexagons
(): three would tile flat, no vertex — impossible.
Larger polygons: even three exceed . Hence exactly five
regular polyhedra exist.
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Full neighborhood
Depends on (7)
- I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
- XI.21Proposition XI.21Any solid angle is contained by plane angles less than four right angles.
- XIII.13Proposition XIII.13To construct a pyramid (regular tetrahedron), to comprehend it in a given sphere, and to prove that the square on the…
- XIII.14Proposition XIII.14To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the…
- XIII.15Proposition XIII.15To construct a cube and comprehend it in a sphere, as in the preceding case; and to prove that the square on the…
- XIII.16Proposition XIII.16To construct an icosahedron and comprehend it in a sphere, as in the case of the aforesaid figures; and to prove that…
- XIII.17Proposition XIII.17To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of…
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