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Uniform polyhedra and duals
Toroidal polyhedra {{Polyhedron-stub
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensiona ...
and 4 hexagons), 24 edges and 12 vertices. Its vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is a crossed quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
.
It is one of nine hemipolyhedra, with 4 hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
al faces passing through the model center.
Orientability
It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with anEuler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of zero (a topological torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
).
Related polyhedra
It shares the vertex arrangement and edge arrangement with thecuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...
(having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common).
By Wythoff construction it has tetrahedral symmetry
image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that co ...
(Td), like the ''rhombitetratetrahedron'' construction for the cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...
, with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
(Oh). In this respect it is akin to the Morin surface, which has fourfold symmetry if orientation is ignored and twofold symmetry otherwise. However the octahemioctahedron has a higher degree of symmetry and is genus 1 rather than 0.
Octahemioctacron
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 thereal projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
at infinity. In Magnus Wenninger
Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction.
Early life and education
Born to ...
'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
* Compound of five octahemioctahedra * Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.References
* (Page 101, Duals of the (nine) hemipolyhedra)External links
* *Uniform polyhedra and duals
Toroidal polyhedra {{Polyhedron-stub