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Octahemioctahedron

Uniform star polyhedron with 12 faces From Wikipedia, the free encyclopedia

Octahemioctahedron
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In geometry, the octahemioctahedron or octatetrahedron[1] is a nonconvex uniform polyhedron, indexed as U3. It has 12 faces eight triangles and four hexagons, 24 edges, and 12 vertices. Its vertex figure is an antiparallelogram.

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3D model of an octahemioctahedron
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Construction and properties

An octahemioctahedron can be constructed from four diagonals of a cube that bisect the interior into four hexagons, and the edges form the structure of a cuboctahedron. The four hexagonal planes form a polyhedral surface when eight triangles are added. Thus, the resulting polyhedron has 12 faces, 24 edges, and 12 vertices. If six squares replace the triangular faces, the resulting polyhedron becomes cubohemioctahedron.[2] It is a uniform polyhedron, with the vertex figure being an antiparallelogram.[3]

It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero, a topological torus.[1]

An octahemioctahedron is the family of a concave antiprism.[4]

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Octahemioctacron

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The dual of an octahemioctahedron

The dual of the octahemioctahedron is the octahemioctacron, with its four vertices at infinity. 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. Wenninger (2003) stated that 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, Wenninger also suggested that strictly speaking, they are not polyhedra because their construction does not conform to the usual definitions.[5]

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See also

References

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