Great rhombihexacron
# Great rhombihexacron

Great rhombihexacron :

| Type | Star polyhedron |
| - | - |
| Face | |
| Elements | F = 24, E = 48 V = 18 (χ = −6) |
| Symmetry group | $O_h,$ [4,3], *432 |
| Index references | $DU_{21}$ |
| 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 $(U_{21}).$ 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 $\arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }$ 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 ${\sqrt {2}}$.

