Edge (geometry)

Relation to edges in graphs

In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theorem as being exactly the 3-vertex-connected planar graphs.{{citation | editor-last = Gorini | editor-first = Catherine A.

Number of edges in a polyhedron

Any convex polyhedron's surface has Euler characteristic

:V - E + F = 2,

where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges.

Incidences with other faces

In a polygon, two edges meet at each vertex; more generally, by Balinski's theorem, at least d edges meet at every vertex of a d-dimensional convex polytope. Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge.

Alternative terminology

In the theory of high-dimensional convex polytopes, a facet or side of a d-dimensional polytope is one of its (d − 1)-dimensional features, a ridge is a (d − 2)-dimensional feature, and a peak is a (d − 3)-dimensional feature. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, and the edges of a 4-dimensional polytope are its peaks.{{citation

References

References

1. Ziegler, Günter M.. (1995). "Lectures on Polytopes". Springer.
2. Weisstein, Eric W. "[http://mathworld.wolfram.com/PolygonEdge.html Polygon Edge]". From Wolfram MathWorld.
3. Weisstein, Eric W. "[http://mathworld.wolfram.com/PolytopeEdge.html Polytope Edge]". From Wolfram MathWorld.
4. Wylie, C. R. Jr.. (1964). "Foundations of Geometry". McGraw-Hill.
5. Senechal, Marjorie. (2013). "Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination". Springer.
6. Balinski, M. L.. (1961). "On the graph structure of convex polyhedra in ''n''-space". Pacific Journal of Mathematics.
7. Wenninger, Magnus J.. (1974). "Polyhedron Models". Cambridge University Press.