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 rhombihexacron is a nonconvex isohedral polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

. It is the dual

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

 of the uniform great rhombihexahedron

Great rhombihexahedron

In geometry, the great rhombihexahedron is a nonconvex uniform polyhedron, indexed as U21. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...

 (U₂₁). It has 24 identical bow-tie

Antiparallelogram

An antiparallelogram is a quadrilateral in which, like a parallelogram, the pairs of nonadjacent sides are congruent, but in which two opposite sides intersect and are therefore not parallel.-Properties:Every antiparallelogram has an axis of symmetry through its crossing point...

-shaped faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

, 18 vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

, and 48 edges

Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

.

It has 12 outer vertices which have the same 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....

 as the cuboctahedron

Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

, and 6 inner vertices with the vertex arrangement of a octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

.

As a surface geometry, it can be seen as visually similar to a Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

, the disdyakis dodecahedron

Disdyakis dodecahedron

In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...

, with much taller rhombus-based pyramids joined to each face of a rhombic dodecahedron

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

.