Rhombic Triacosiohedron

angle of rhombus

20+30×2

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

P

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

P

is

χ=F-E+V=-20

. Consequently, its genus

g=(1/2)(2-χ)=11

, which means that it is topologically equivalent to a doughnut with 11 holes, or, if you prefer, to a sphere with 11 handles.