Rhombicosacron

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.

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

Proportions

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

References

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

External links

• Weisstein, Eric W. "Rhombicosacron". MathWorld.
• Uniform polyhedra and duals