Small hexagrammic hexecontahedron

Small hexagrammic hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 180 V = 112 (χ = −8)
Symmetry group	Ih, [5,3], *532
Index references	DU72
dual polyhedron	Small retrosnub icosicosidodecahedron

In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.

3D model of a small hexagrammic hexecontahedron

Geometry

Its faces are hexagonal stars with two short and four long edges. Denoting the golden ratio by $\phi$ and putting $\xi ={\frac {1}{4}}+{\frac {1}{4}}{\sqrt {1+4\phi }}\approx 0.933\,380\,199\,59$, the stars have five equal angles of $\arccos(\xi )\approx 21.031\,988\,967\,51^{\circ }$ and one of $360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 254.840\,055\,162\,43^{\circ }$. Each face has four long and two short edges. The ratio between the edge lengths is

$1/2-1/2\times {\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.428\,986\,992\,12$.

The dihedral angle equals $\arccos(\xi /(1+\xi ))\approx 61.133\,452\,273\,64^{\circ }$. Part of each face is inside the solid, hence is not visible in solid models.

References

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

External links

• Weisstein, Eric W. "Small hexagrammic hexecontahedron". MathWorld.