Small hexagrammic hexecontahedron

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.

Small hexagrammic hexecontahedron

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

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.

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