# Octahemioctahedron

Octahemioctahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymOho
Coxeter diagramx3/2o3x3*a ()
Elements
Faces8 triangles, 4 hexagons
Edges24
Vertices12
Vertex figureBowtie, edge lengths 1 and 3
Measures (edge length 1)
Volume0
Dihedral angle${\displaystyle \arccos\left(\frac13\right) ≈ 70.52878°}$
Central density0
Number of pieces32
Level of complexity4
Related polytopes
ArmyCo
RegimentCo
DualOctahemioctacron
ConjugateNone
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Genus1
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The octahemioctahedron, or oho, is a quasiregular polyhedron and one of 10 uniform hemipolyhedra. It consists of 8 equilateral triangles and 4 "hemi" hexagons passing through its center, with two of each joining at a vertex. It also has 6 square pseudofaces. Its triangular faces, as well as its hemi hexagonal faces, are parallel to those of an octahedron: hence the name. It can be derived as a rectified petrial cube.

Unlike the other quasiregular hemipolyhedra, the octahemioctahedron is orientable. This is because the triangles can be grouped into two sets of four, one of which is seen as positive-density and the other seen as negative-density. In addition, while it has cubic symmetry, its Coxeter diagram also makes it uniform under tetrahedral symmetry.

The visible portion of this solid resembles a cuboctahedron with six square pyramids carved out. In fact the triangular faces are the same ones as from the cuboctahedron, while the hexagons are those of the cuboctahedron's equatorial planes.

The octahemioctahedron is topologically a torus, and is isomorphic to a quotient of the trihexagonal tiling.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the cuboctahedron.

## Representations

An octahemioctahedron has the following Coxeter diagrams:

• x3/2o3x3*a
• ß3o3x
• ß3/2x3x

## Related polyhedra

The icosidisicosahedron is a uniform polyhedron compound composed of 5 octahemioctahedra.