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 octahemioctahedron 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 U₃. 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

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...

.

It is one of nine hemipolyhedra

Hemipolyhedron

In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other polyhedron - hence the "hemi" prefix.The prefix "hemi"...

 with 4 hexagonal faces passing through the model center.

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....

 and edge arrangement with 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,...

 (having the triangular faces in common), and with the cubohemioctahedron

Cubohemioctahedron

In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.A nonconvex polyhedron has intersecting faces which do not represent new edges or faces...

 (having the hexagonal faces in common).

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,...

Cubohemioctahedron Cubohemioctahedron In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.A nonconvex polyhedron has intersecting faces which do not represent new edges or faces...

Octahemioctahedron

By construction it has tetrahedral symmetry

Tetrahedral symmetry

150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...

 (T_d), like the cantellated tetrahedron construction for the cuboctahedron, , with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry

Octahedral symmetry

150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

 (O_h).

Orientability

It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 of zero (a topological torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

).

The topological net of faces can be arranged as a rhombus Rhombus In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...  divided into 8 triangles and 4 hexagons. All vertex angle defects are zero.

The net represents a region of the trihexagonal tiling Trihexagonal tiling In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex...  plane, with Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....  3 3 > 3 and Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... .