# Cubohemioctahedron

Cubohemioctahedron Type Uniform star polyhedron
Elements F = 10, E = 24
V = 12 (χ = −2)
Faces by sides 6{4}+4{6}
Wythoff symbol 4/3 4 | 3 (double-covering)
Symmetry group Oh, [4,3], *432
Index references U15, C51, W78
Dual polyhedron Hexahemioctacron
Vertex figure 4.6.4/3.6
Bowers acronym Cho

In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral.

It is given Wythoff symbol 43 4 | 3, although that is a double-covering of this figure.

A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders.

It is a hemipolyhedron with 4 hexagonal faces passing through the model center. The hexagons intersect each other and so only triangular portions of each are visible.

## Related polyhedra

It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the square faces in common), and with the octahemioctahedron (having the hexagonal faces in common). Cuboctahedron Cubohemioctahedron Octahemioctahedron

### Tetrahexagonal tiling

The cubohemioctahedron can be seen as a net on the hyperbolic tetrahexagonal tiling with vertex figure 4.6.4.6.

### Hexahemioctacron

Hexahemioctacron

Type Star polyhedron
Face
Elements F = 12, E = 24
V = 10 (χ = −2)
Symmetry group Oh, [4,3], *432
Index references DU15
dual polyhedron Cubohemioctahedron

The hexahemioctacron is the dual of the cubohemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the octahemioctacron.

Since the cubohemioctahedron has four hexagonal faces passing through the model center, thus it is degenerate, and can be seen as having four vertices at infinity.

In Magnus Wenninger's Dual Models, they are represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is convenient for the maker.

## See also

• Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.