# Rhombicosacron

Rhombicosacron Type Star polyhedron
Face Elements F = 60, E = 120
V = 50 (χ = −10)
Symmetry group Ih, [5,3], *532
Index references DU56
dual polyhedron Rhombicosahedron

In geometry, the rhombicosacron (or midly dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.

## Proportions

Each face has two angles of $\arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }$  and two angles of $\arccos(-{\frac {1}{6}})\approx 99.594\,068\,226\,86^{\circ }$ . The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{8}}+{\frac {7{\sqrt {5}}}{24}})\approx 38.996\,309\,663\,87^{\circ }$ . The dihedral angle equals $\arccos(-{\frac {5}{7}})\approx 135.584\,691\,402\,81^{\circ }$ . The ratio between the lengths of the long edges and the short ones equals ${\frac {3}{2}}+{\frac {1}{2}}{\sqrt {5}}$ , which is the square of the golden ratio.