Hypercell
Encyclopedia
In geometry
, a hypercell is a descriptive term for an element of a polytope
or tessellation
, usually representing an element one dimension higher than a cell
. The most generally accepted term is 4-face
because it contains a 4-dimensional interior. Another proposed name in use is teron, shortened from tetron, constructed from the prefix tetra- meaning four.
A 5-dimensional polytope
(5-polytope
) or 4-dimensional tessellation
can be considered constructed of 4-dimensional hypercells, 3-dimension
al cells, 2-dimensional faces, 1-dimensional edges, and 0-dimensional vertices.
For example the 5-dimensional penteract
(5-hypercube) is constructed from 10 tesseract
ic hypercells. Also the 4-dimensional tesseractic honeycomb is constructed from tesseract
hypercells. In the context of these figures, hypercells can also be called facets
representing the highest dimensional elements of the figures.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a hypercell is a descriptive term for an element of a polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
or tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
, usually representing an element one dimension higher than a cell
Cell (geometry)
In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
. The most generally accepted term is 4-face
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
because it contains a 4-dimensional interior. Another proposed name in use is teron, shortened from tetron, constructed from the prefix tetra- meaning four.
A 5-dimensional polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
(5-polytope
5-polytope
In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:...
) or 4-dimensional tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
can be considered constructed of 4-dimensional hypercells, 3-dimension
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
al cells, 2-dimensional faces, 1-dimensional edges, and 0-dimensional vertices.
For example the 5-dimensional penteract
Penteract
In five dimensional geometry, a 5-cube is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells....
(5-hypercube) is constructed from 10 tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
ic hypercells. Also the 4-dimensional tesseractic honeycomb is constructed from tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
hypercells. In the context of these figures, hypercells can also be called facets
Facet (mathematics)
A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
representing the highest dimensional elements of the figures.