Great rhombihexacron

Template:Short description Template:Uniform polyhedra db 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

• Template:Citation
• uniform polyhedra and duals