geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the cubohemioctahedron is a

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. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

, indexed as U₁₅. It has 10 faces (6

squares In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

and 4 regular hexagons), 24 edges and 12 vertices. Its

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

is a

crossed quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

. It is given Wythoff symbol 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 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 po ...

with 4

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

al 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 A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...

(having the square faces in common), and with the octahemioctahedron (having the hexagonal faces in common).

Tetrahexagonal tiling

The ''cubohemioctahedron'' can be seen as a net on the hyperbolic tetrahexagonal tiling with vertex figure 4.6.4.6. Uniform tiling 64-t1

Hexahemioctacron

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 Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to ...

'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.

References

* (Page 101, Duals of the (nine) hemipolyhedra)

External links

* *
Uniform polyhedra and duals
Uniform polyhedra {{polyhedron-stub

Related polyhedra

Tetrahexagonal tiling

Hexahemioctacron

See also

References

External links