Rhombic Triacosiohedron

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congruent golden rhombi are aligned along the edges of a dodecahedron. The cluster produces a polyhedron with 280 vertices, 600 edges, and 300 congruent faces, each a rhombus the ratio of whose diagonals equals the golden ratio.

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The coordinates of the vertices of the polyhedron are expressed in terms of the acute angle of the rhombus. If you vary this angle from 0° to 90°, then 30 of the rhombi are deformed to a cube and the rest are deformed to a rectangle. (At both extremes the cubes and rectangles come from the same rhombi.)

The Euler characteristic of is . Consequently, its genus , which means that it is topologically equivalent to a doughnut with 11 holes, or, if you prefer, to a sphere with 11 handles.

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Contributed by: Lajos Szilassi and Sándor Kabai (June 2008)
Additional contributions by: Gábor Gévay
Open content licensed under CC BY-NC-SA


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