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	O_h, [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 $\arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }$ and two angles of $\arccos(-{\frac {1}{4}}+{\frac {1}{2}}{\sqrt {2}})\approx 62.799\,429\,619\,84^{\circ }$ . The diagonals of each bow-tie intersect at an angle of $\arccos({\frac {1}{4}}-{\frac {1}{8}}{\sqrt {2}})\approx 85.800\,855\,570\,24^{\circ }$ . The dihedral angle equals $\arccos({\frac {-7+4{\sqrt {2}}}{17}})\approx 94.531\,580\,798\,20^{\circ }$ . The ratio between the lengths of the long edges and the short ones equals ${\sqrt {2}}$ .

Notes

Weisstein, Eric W. "Great rhombihexacron". MathWorld.
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

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[1] Weisstein, Eric W. "Great rhombihexacron". MathWorld.

[2] Great Rhombihexacron—Bulatov Abstract Creations