Cubohemioctahedron

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.[1] Its vertex figure is a crossed quadrilateral.

Cubohemioctahedron
TypeUniform star polyhedron
ElementsF = 10, E = 24
V = 12 (χ = −2)
Faces by sides6{4}+4{6}
Wythoff symbol3 (double-covering)
Symmetry groupOh, [4,3], *432
Index referencesU15, C51, W78
Dual polyhedronHexahemioctacron
Vertex figure
4.6.4/3.6
Bowers acronymCho
3D model of a cubohemioctahedron

It is given Wythoff symbol ​4⁄3 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.

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
TypeStar polyhedron
Face—
ElementsF = 12, E = 24
V = 10 (χ = −2)
Symmetry groupOh, [4,3], *432
Index referencesDU15
dual polyhedronCubohemioctahedron

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.

References
  1. Maeder, Roman. "15: cubohemioctahedron". MathConsult.


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