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 ...

, a hemi-octahedron is an

abstract regular polyhedron In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

, containing half the faces of a

regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

. It has 4 triangular faces, 6 edges, and 3 vertices. Its

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

is the

hemicube Hemicube can mean: * Hemicube (computer graphics), a concept in 3D computer graphics rendering *Hemicube (geometry) In abstract geometry, a hemicube is an abstract, regular polyhedron, produced by cutting a cube in half with a plane that passes ...

. It can be realized as a

projective polyhedron In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Proje ...

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

of the

real 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 ...

by 4 triangles), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into four equal parts. It can be seen as a

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

without its base. It can be represented symmetrically as a hexagonal or square

Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the ori ...

: :

It has an unexpected property that there are two distinct edges between every pair of vertices – any two vertices define a

digon In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...

References

* {{citation , last1 = McMullen , first1 = Peter , author1-link = Peter McMullen , first2 = Egon , last2 = Schulte , chapter = 6C. Projective Regular Polytopes , title = Abstract Regular Polytopes , edition = 1st , publisher = Cambridge University Press , isbn = 0-521-81496-0 , date=December 2002 , pages
162–165

External links

The hemioctahedron
Projective polyhedra

See also

References

External links