Great deltoidal hexecontahedron

In geometry, the great deltoidal hexecontahedron (or great sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models.

Great deltoidal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 62 (χ = 2)
Symmetry group	Ih, [5,3], *532
Index references	DU67
dual polyhedron	Nonconvex great rhombicosidodecahedron

3D model of a great deltoidal hexecontahedron

It is also called a great strombic hexecontahedron.

Proportions

The darts have two angles of ${\displaystyle \arccos({\frac {1}{2}}+{\frac {1}{5}}{\sqrt {5}})\approx 18.699\,407\,085\,15^{\circ }}$, one of ${\displaystyle \arccos(-{\frac {1}{4}}+{\frac {1}{10}}{\sqrt {5}})\approx 91.512\,394\,720\,74^{\circ }}$ and one of ${\displaystyle 360^{\circ }-\arccos(-{\frac {1}{8}}-{\frac {9}{40}}{\sqrt {5}})\approx 231.088\,791\,108\,96^{\circ }}$. 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 and short edges is ${\displaystyle {\frac {21+3{\sqrt {5}}}{22}}\approx 1.259\,463\,815\,11}$.

References

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

External links

• Weisstein, Eric W. "Great deltoidal hexecontahedron". MathWorld.
• Uniform polyhedra and duals