Small icosacronic hexecontahedron

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

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.

Proportions

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