Octahemioctahedron

Octahemioctahedron
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
Octahemioctahedron
3.6.3/2.6
(Vertex figure)
Hexahemioctacron.png
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.

Contents

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.png
Cuboctahedron
Cubohemioctahedron.png
Cubohemioctahedron
Octahemioctahedron.png
Octahemioctahedron

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

Orientability

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

Octahemioctahedron topo net.png
The topological net of faces can be arranged as a rhombus divided into 8 triangles and 4 hexagons. All vertex angle defects are zero.
Uniform tiling 333-t01.png
The net represents a region of the trihexagonal tiling plane, with Wythoff symbol 3 3 | 3 and Coxeter-Dynkin diagram CDel branch 11.pngCDel split2.pngCDel node.png.

See also

External links