In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the great rhombihexahedron is a nonconvex uniform polyhedron

Nonconvex uniform polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...

, indexed as U₂₁. Its dual

Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...

 is the great rhombihexacron

Great rhombihexacron

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

. Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 is a crossed quadrilateral.

Related polyhedra

It shares the vertex arrangement

Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....

 with the convex truncated cube

Truncated cube

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron (having 12 square faces in common), and with the great cubicuboctahedron

Great cubicuboctahedron

In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14.- Related polyhedra :It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra...

 (having the octagrammic faces in common).

Truncated cube Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

Nonconvex great rhombicuboctahedron

Great cubicuboctahedron Great cubicuboctahedron In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14.- Related polyhedra :It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra...

Great rhombihexahedron

It may be constructed as the exclusive or of three octagrammic prism

Octagrammic prism

In geometry, the octagrammic prism is one of an infinite set of nonconvex prisms formed by rectangular sides and two regular star polygon caps, in this case two octagrams.An octagrammic prism's octagrammic bases may be in either form below:...

s.