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Geometric ideas and concepts

The Dodecahedron

from Curious and Interesting Geometry by David Wells

Dodecahedra have 12 faces, and there fore include the regular dodecahedron, with 12 regular pentagonal faces, and the rhombic dodecahedron, with 12 rhombic faces.

The regular dodecahedron has 31 axes of symmetry: 10 are threefold, passing through pairs of opposite vertices; 6 are fivefold, passing through the centers of opposite faces; and 15 are twofold, passing through the mid-points of opposite sides.

The regular icosahedron has the same number of axes of symmetry, but with 'vertices' and 'faces' reversed in their description.

The relationship between the dodecahedron and the cube can be seen either by joining the mid-point of faces to form the vertices of three rectangles (whose edges are in the golden ratio) which are mutually perpendicular, or by choosing eight vertices of the dodecahedron which are also the vertices of a cube:

(figures not shown on this web site)

A perhaps surprising fact is that when a regular dodecahedron and a regular icosahedron are inscribed in the same sphere, the dodecahedron occupies a larger proportion of the spheres' volume. The icosahedron has more faces, but the faces of the dodecahedron are more nearly circular.

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