Ridge (geometry)
Encyclopedia
In geometry
, a ridge is an (n − 2)-dimensional element of an n-dimensional polytope
. It is also sometimes called a subfacet for having one lower dimension than a facet.
By dimension, this corresponds to:
Exactly two facets
meet at any ridge in a polytope.
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 ridge is an (n − 2)-dimensional element of an n-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...
. It is also sometimes called a subfacet for having one lower dimension than a facet.
By dimension, this corresponds to:
- a vertexVertex (geometry)In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
of a polygonPolygonIn 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...
; - an edgeEdge (geometry)In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
of a polyhedronPolyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
; - a faceFace (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...
of a polychoronPolychoronIn 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); - a cellCell (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...
of a 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:...
; - a 4-face of a 6-polytope6-polytopeIn six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....
; - and so forth.
Exactly two 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:...
meet at any ridge in a polytope.