Great hexacronic icositetrahedron

In geometry, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.

Great hexacronic icositetrahedron
Type	Star polyhedron
Face
Elements	F = 24, E = 48 V = 20 (χ = −4)
Symmetry group	Oh, [4,3], *432
Index references	DU14
dual polyhedron	Great cubicuboctahedron

Proportions

The kites have two angles of ${\displaystyle \arccos({\frac {1}{4}}-{\frac {1}{2}}{\sqrt {2}})\approx 117.200\,570\,380\,16^{\circ }}$, one of ${\displaystyle \arccos(-{\frac {1}{4}}+{\frac {1}{8}}{\sqrt {2}})\approx 94.199\,144\,429\,76^{\circ }}$ and one of ${\displaystyle \arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }}$. The dihedral angle equals ${\displaystyle \arccos({\frac {-7+4{\sqrt {2}}}{17}})\approx 94.531\,580\,798\,20^{\circ }}$. The ratio between the lengths of the long and short edges is ${\displaystyle 2+{\frac {1}{2}}{\sqrt {2}}\approx 2.70710678118655}$.

References

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

External links

• Weisstein, Eric W. "Great Hexacronic Icositetrahedron". MathWorld.