Great rhombihexacron

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

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

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 $\arccos (\frac{1}{2}+\frac{1}{4}\sqrt{2})\approx 31.39971480992^{\circ }$ and two angles of $\arccos (-\frac{1}{4}+\frac{1}{2}\sqrt{2})\approx 62.79942961984^{\circ }$. The diagonals of each bow-tie intersect at an angle of $\arccos (\frac{1}{4}-\frac{1}{8}\sqrt{2})\approx 85.80085557024^{\circ }$. The dihedral angle equals $\arccos (\frac{-7+4\sqrt{2}}{17})\approx 94.53158079820^{\circ }$. The ratio between the lengths of the long edges and the short ones equals $\sqrt{2}$.

References

• uniform polyhedra and duals