Octahemioctahedron

Octahemioctahedron
Type	Non-convex uniform polyhedron
Faces	8 triangles 4 hexagons
Edges	24
Vertices	12
Wythoff symbol	${\textstyle {\frac {3}{2}}\,3\mid 3}$[1]
Dual polyhedron	octahemioctacron

3D model of an octahemioctahedron

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.

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]

Octahemioctacron

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]

See also

• Compound of five octahemioctahedra
• Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.

References

1. Coxeter, Longuet-Higgins & Miller (1954), p. 417.
2. Pisanski, Tomaz; Servatius, Brigitte (2013), Configurations from a Graphical Viewpoint, Springer, p. 107, doi:10.1007/978-0-8176-8364-1, ISBN 978-0-8176-8363-4
3. Coxeter, H. S. M.; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 246 (916): 401–450, Bibcode:1954RSPTA.246..401C, doi:10.1098/rsta.1954.0003, JSTOR 91532, MR 0062446, S2CID 202575183
4. Obradović, M.; Popkonstantinović, B.; Mišić, S. (2013), "On the Properties of the Concave Antiprisms of Second Sort", Faculty of Mechanical Engineering Transactions, 41: 256–263
5. Wenninger, Magnus (2003) [1983], Dual Models, Cambridge University Press, p. 101, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208

External links

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