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Edge, a 1-dimensional element; Face, a 2-dimensional element; Cell, a 3-dimensional element; Hypercell or Teron, a 4-dimensional element; Facet, an (n-1)-dimensional element; Ridge, an (n-2)-dimensional element; Peak, an (n-3)-dimensional element; For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and ...
This polyhedron is topologically related as a part of a sequence of cantellated polyhedra with vertex figure (3.4.n.4), which continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry .
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. [1] In a polygon, an edge is a line segment on the boundary, [2] and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces (or polyhedron sides ...
Such a figure is called simplicial if each of its regions is a simplex, i.e. in an n-dimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an n-dimensional cube.
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides , and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners .
A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.
Consider a line segment AB as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires ...
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.In particular, all its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.