Ingeometry, anedge is a particular type ofline segment joining twovertices in apolygon,polyhedron, or higher-dimensionalpolytope.[1] In a polygon, an edge is a line segment on the boundary,[2] and is often called apolygon side. In a polyhedron or more generally a polytope, an edge is a line segment where twofaces (or polyhedron sides) meet.[3] A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called adiagonal.
Anedge may also be an infiniteline separating twohalf-planes.[4]Thesides of aplane angle are semi-infinitehalf-lines (or rays).[5]
Ingraph theory, anedge is an abstract object connecting twograph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment.However, any polyhedron can be represented by itsskeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges.[6] Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized bySteinitz's theorem as being exactly the3-vertex-connectedplanar graphs.[7]
Anyconvex polyhedron's surface hasEuler characteristic
whereV is the number ofvertices,E is the number of edges, andF is the number offaces. This equation is known asEuler's polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces. For example, acube has 8 vertices and 6 faces, and hence 12 edges.
In a polygon, two edges meet at eachvertex; more generally, byBalinski's theorem, at leastd edges meet at every vertex of ad-dimensional convex polytope.[8]Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge,[9] while in higher dimensional polytopes three or more two-dimensional faces meet at every edge.
In the theory of high-dimensionalconvex polytopes, afacet orside of ad-dimensionalpolytope is one of its (d − 1)-dimensional features, aridge is a (d − 2)-dimensional feature, and apeak is a (d − 3)-dimensional feature. Thus, the edges of a polygon are its facets, the edges of a 3-dimensionalconvex polyhedron are its ridges, and the edges of a4-dimensional polytope are its peaks.[10]
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