Small icosacronic hexecontahedron

In geometry, the small icosacronic hexecontahedron (or small lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform small icosicosidodecahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.

Small icosacronic hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 52 (χ = −8)
Symmetry group	Ih, [5,3], *532
Index references	DU31
dual polyhedron	Small icosicosidodecahedron

3D model of a small icosacronic hexecontahedron

Proportions

The kites have two angles of ${\displaystyle \arccos({\frac {3}{4}}-{\frac {1}{20}}{\sqrt {5}})\approx 50.342\,524\,343\,87^{\circ }}$, one of ${\displaystyle \arccos(-{\frac {1}{12}}-{\frac {19}{60}}{\sqrt {5}})\approx 142.318\,554\,460\,55^{\circ }}$ and one of ${\displaystyle \arccos(-{\frac {5}{12}}-{\frac {1}{60}}{\sqrt {5}})\approx 116.996\,396\,851\,70^{\circ }}$. The dihedral angle equals ${\displaystyle \arccos({\frac {-44-3{\sqrt {5}}}{61}})\approx 146.230\,659\,755\,53^{\circ }}$. The ratio between the lengths of the long and short edges is ${\displaystyle {\frac {31+5{\sqrt {5}}}{38}}\approx 1.110\,008\,944\,41}$.

References

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

External links

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