Cell (geometry)
Encyclopedia
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 cell is a three-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al element that is part of a higher-dimensional object.
In polytopes
A cell is a three-dimensionDimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
element that is part of the boundary of a higher-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...
, such as a polychoron
Polychoron
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
(4-polytope) or honeycomb
Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
(3-space tessellation)
For example, a cubic honeycomb
Cubic honeycomb
The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron....
is made of cubic
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
cells, with 4 cubes on each edge. A 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...
is also made of cubic cells, but only has 3 cubes on each edge.
In polychoron names
Regular convex polychora are sometimes named by how many cells they contain, just like n-gon and n-hedron are used as a shorthand for polygonPolygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
al and polyhedral
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
names. For example, the 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...
can also be called an octachoron or an 8-cell because it contains 8 cubic cells.
See also
- Face (geometry)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...
- the two-dimensional element analogue of cells for polyhedraPolyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
and planar tilings. - Facet (geometry) as the highest dimensional subelements in a 4-polytope or 3-space tessellation, and 3-faces more systematically.
- HypercellHypercellIn 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...
s, or more clearly 4-faces, are four-dimensional elements (5-polytope5-polytopeIn 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:...
s and higher). Systematically n-faces are elements in (n+1)-polytopes and higher. - Cell complex