Great rhombihexacron

In geometry, the great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U₂₁).[1] It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges.[2]

Great rhombihexacron
Type	Star polyhedron
Face
Elements	F = 24, E = 48 V = 18 (χ = −6)
Symmetry group	Oh, [4,3], *432
Index references	DU21
dual polyhedron	Great rhombihexahedron

It has 12 outer vertices which have the same vertex arrangement as the cuboctahedron, and 6 inner vertices with the vertex arrangement of an octahedron.

As a surface geometry, it can be seen as visually similar to a Catalan solid, the disdyakis dodecahedron, with much taller rhombus-based pyramids joined to each face of a rhombic dodecahedron.

Proportions

Each bow-tie has two angles of ${\displaystyle \arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }}$ and two angles of ${\displaystyle \arccos(-{\frac {1}{4}}+{\frac {1}{2}}{\sqrt {2}})\approx 62.799\,429\,619\,84^{\circ }}$. The diagonals of each bow-tie intersect at an angle of ${\displaystyle \arccos({\frac {1}{4}}-{\frac {1}{8}}{\sqrt {2}})\approx 85.800\,855\,570\,24^{\circ }}$. 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 edges and the short ones equals ${\displaystyle {\sqrt {2}}}$.

Notes

1. Weisstein, Eric W. "Great rhombihexacron". MathWorld.
2. Great Rhombihexacron—Bulatov Abstract Creations

References

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