Octahemioctahedron

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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
Index references U03, C37, W68
Dual polyhedron Octahemioctacron
Vertex figure Octahemioctahedron
3.6.3/2.6
Bowers acronym Oho


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.

Orientability[edit]

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.


Related polyhedra[edit]

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).

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

Cuboctahedron Cubohemioctahedron Octahemioctahedron
Octahedral symmetry Tetrahedral symmetry Octahedral symmetry Tetrahedral symmetry
Cuboctahedron.png Cantellated tetrahedron.png Cubohemioctahedron.png Octahemioctahedron.png Octahemioctahedron 3-color.png
2 | 3 4 3 3 | 2 4/3 4 | 3
(double cover)
3/2 3 | 3
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel nodes 11.pngCDel split2.pngCDel node.png CDel label4-3.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel label3-2.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png



Octahemioctacron[edit]

Octahemioctacron
Octahemioctahedron
Type Star polyhedron
Face
Elements F = 12, E = 24
V = 12 (χ = 0)
Symmetry group Oh, [4,3], *432
Index references DU03
dual polyhedron Octahemioctahedron


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

Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.[1] In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.

The octahemioctacron has four vertices at infinity.

See also[edit]

References[edit]

External links[edit]