Great rhombidodecacron

Great rhombidodecacron
Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 42 (χ = −18)
Symmetry group	Ih, [5,3], *532
Index references	DU73
dual polyhedron	Great rhombidodecahedron

In geometry, the great rhombidodecacron (or Great dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great rhombidodecahedron. It is visually identical to the great deltoidal hexecontahedron. Its faces are antiparallelograms.

Each antiparallelogram has two angles of ${\displaystyle \arccos({\frac {1}{2}}+{\frac {1}{5}}{\sqrt {5}})\approx 18.699\,407\,085\,15^{\circ }}$ and two angles of ${\displaystyle \arccos(-{\frac {5}{8}}+{\frac {1}{8}}{\sqrt {5}})\approx 110.211\,801\,805\,89^{\circ }}$. The diagonals of each antiparallelogram intersect at an angle of ${\displaystyle \arccos({\frac {1}{8}}+{\frac {9{\sqrt {5}}}{40}})\approx 51.088\,791\,108\,96^{\circ }}$. The dihedral angle equals ${\displaystyle \arccos({\frac {-19+8{\sqrt {5}}}{41}})\approx 91.553\,403\,672\,16^{\circ }}$. The ratio between the lengths of the long edges and the short ones equals ${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}{\sqrt {5}}}$, which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 88

• Weisstein, Eric W. "Great pentagrammic hexecontahedron". MathWorld.