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.


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.


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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