Octahemioctahedron

Octahemioctahedron
Type	Uniform star polyhedron
Elements	F = 12, E = 24 V = 12 (χ = 0)
Faces by sides	8{3}+4{6}
Wythoff symbol	3/2 3 | 3
Symmetry group	Oh, [4,3], *432 Td, [3,3], *332
Index references	U03, C37, W68
3.6.3/2.6 (Vertex figure)	Octahemioctacron (dual polyhedron)

In geometry, the octahemioctahedron is a nonconvex uniform polyhedron, indexed as U₃. Its vertex figure is a crossed quadrilateral.

It is one of nine hemipolyhedra with 4 hexagonal faces passing through the model center.

Related polyhedra

It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common).

Cuboctahedron

Cubohemioctahedron

Octahemioctahedron

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

Orientability

It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero (a topological torus).

The topological net of faces can be arranged as a rhombus divided into 8 triangles and 4 hexagons. All vertex angle defects are zero.

The net represents a region of the trihexagonal tiling plane, with Wythoff symbol 3 3 | 3 and Coxeter-Dynkin diagram .

See also

• Compound of five octahemioctahedra

External links

• Eric W. Weisstein, Octahemioctahedron (Uniform polyhedron) at MathWorld.
• Uniform polyhedra and duals

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