Edge (geometry)

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Complete graph K2.svg
An edge between two vertices
Square (geometry).svg
A polygon is bounded by edges, like this square has 4 edges.
Hexahedron.png
Every edge shares two faces in a polyhedron, like this cube.
Hypercube.svg
Every edge shares three or more faces in a 4-polytope, as seen in this projection of a tesseract.

In geometry, an edge is a line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.[1]

Relation to edges in graphs[edit]

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.[2] 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.[3]

Incidences with other faces[edit]

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.[4] Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge,[5] while in higher dimensional polytopes three or more two-dimensional faces meet at every edge.

Alternative terminology[edit]

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 polychoron are its peaks.[6]

See also[edit]

References[edit]

  1. ^ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, Definition 2.1, p. 51 .
  2. ^ Senechal, Marjorie (2013), Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 81, ISBN 9780387927145 .
  3. ^ Pisanski, Tomaž; Randić, Milan (2000), "Bridges between geometry and graph theory", in Gorini, Catherine A., Geometry at work, MAA Notes 53, Washington, DC: Math. Assoc. America, pp. 174–194, MR 1782654 . See in particular Theorem 3, p. 176.
  4. ^ Balinski, M. L. (1961), "On the graph structure of convex polyhedra in n-space", Pacific Journal of Mathematics 11 (2): 431–434, doi:10.2140/pjm.1961.11.431, MR 0126765 .
  5. ^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 1, ISBN 9780521098595 .
  6. ^ Seidel, Raimund (1986), "Constructing higher-dimensional convex hulls at logarithmic cost per face", Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (STOC '86), pp. 404–413, doi:10.1145/12130.12172 .

External links[edit]