Edge (geometry)

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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) meet. [3] A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.

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An edge may also be an infinite line separating two half-planes. [4] The sides of a plane angle are semi-infinite half-lines (or rays). [5]

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. [6] 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. [7]

Number of edges in a polyhedron

Any convex polyhedron's surface has Euler characteristic

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. [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.

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. [10]

See also

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References

  1. ↑ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, Definition 2.1, p. 51, ISBN   9780387943657 .
  2. ↑ Weisstein, Eric W. "Polygon Edge". From Wolfram MathWorld.
  3. ↑ Weisstein, Eric W. "Polytope Edge". From Wolfram MathWorld.
  4. ↑ Wylie Jr., C.R. (1964), Foundations of Geometry, New York: McGraw-Hill, p. 64, ISBN   0-07-072191-2
  5. ↑ Wylie Jr., C.R. (1964), Foundations of Geometry, New York: McGraw-Hill, p. 68, ISBN   0-07-072191-2
  6. ↑ Senechal, Marjorie (2013), Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 81, ISBN   9780387927145 .
  7. ↑ Pisanski, Tomaž; Randić, Milan (2000), "Bridges between geometry and graph theory", in Gorini, Catherine A. (ed.), Geometry at work, MAA Notes, vol. 53, Washington, DC: Math. Assoc. America, pp. 174–194, MR   1782654 . See in particular Theorem 3, p. 176.
  8. ↑ 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 .
  9. ↑ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 1, ISBN   9780521098595 .
  10. ↑ 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, S2CID   8342016 .
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