Great rhombihexacron explained

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.^[1]

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(	1	+
	2

	1
	4

\sqrt{2}) ≈ 31.39971480992^\circ

and two angles of

\arccos(-	1	+
	4

	1
	2

\sqrt{2}) ≈ 62.79942961984^\circ

. The diagonals of each bow-tie intersect at an angle of

\arccos(	1	-
	4

	1
	8

\sqrt{2}) ≈ 85.80085557024^\circ

. The dihedral angle equals

\arccos(	-7+4\sqrt{2

})\approx 94.531\,580\,798\,20^.The ratio between the lengths of the long edges and the short ones equals

\sqrt{2}