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In abstract
The hemicube
{{DEFAULTSORT:Hemicube (Geometry) Projective polyhedra
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 hemicube is an abstract, regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...
, produced by cutting a cube in half with a plane that passes through 2 opposite corners and the midpoints of 2 edges.
A hemicube is also sometimes called a square hemiprism.
Realization
It can be realized as a projective polyhedron (atessellation
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 three quadrilaterals), which can be visualized by constructing the projective plane as a hemisphere
Hemisphere may refer to:
In geometry
* Hemisphere (geometry), a half of a sphere
As half of Earth or any spherical astronomical object
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemi ...
where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
It has three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets.
From the point of view of graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
the skeleton
A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...
is a tetrahedral graph, an embedding of ''K''4 (the complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
with four vertices) on a projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
.
The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demicube is an ordinary polyhedron (in Euclidean space). While they both have half the vertices of a cube, the ''hemi''cube is a ''quotient'' of the cube, while the vertices of the ''demi''cube are a ''subset'' of the vertices of the cube.
Related polytopes
The hemicube is the Petrie dual to the regulartetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, with the four vertices, six edges of the tetrahedron, and three Petrie polygon quadrilateral faces. The faces can be seen as red, green, and blue edge colorings in the tetrahedral graph:
:
See also
*hemi-octahedron
In geometry, a hemi-octahedron is an abstract polytope, abstract regular polyhedron, containing half the faces of a regular octahedron.
It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the Hemicube (geometry), hemicube ...
* hemi-dodecahedron
In geometry, a hemi-dodecahedron is an abstract polytope, abstract, regular polyhedron, containing half the Face (geometry), faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective pla ...
* hemi-icosahedron
In geometry, a hemi-icosahedron is an abstract polytope, abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), ...
* Hemitesseract
Footnotes
References
*External links
The hemicube
{{DEFAULTSORT:Hemicube (Geometry) Projective polyhedra