Proposition·Untested·2605.00009

Proposition XIII.17

To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the dodecahedron is the irrational straight line called apotome.

Proof

The dodecahedron has twelve regular pentagonal faces; the side is the apotome formed when the cube-edge is cut in extreme and mean ratio (XIII.6). Inscribe by placing pentagonal faces on the six square faces of the inscribed cube (XIII.15).

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