In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, 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. 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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Three edges AB, BC, and CA, each between two vertices of a triangle. -
A polygon is bounded by edges; this square has 4 edges. -
Every edge is shared by two faces in a polyhedron, like this cube. -
Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract.
An edge may also be an infinite line separating two half-planes. The sides of a plane angle are semi-infinite half-lines (or rays).
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.
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. 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.
See also
- Base (geometry)
- Extended side
References
- Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, Definition 2.1, p. 51, ISBN 9780387943657.
- Weisstein, Eric W. "Polygon Edge". From Wolfram MathWorld.
- Weisstein, Eric W. "Polytope Edge". From Wolfram MathWorld.
- Wylie Jr., C.R. (1964), Foundations of Geometry, New York: McGraw-Hill, p. 64, ISBN 0-07-072191-2
- Wylie Jr., C.R. (1964), Foundations of Geometry, New York: McGraw-Hill, p. 68, ISBN 0-07-072191-2
- Senechal, Marjorie (2013), Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 81, ISBN 9780387927145.
- 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.
- 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.
- Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 1, ISBN 9780521098595.
- 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, ISBN 0-89791-193-8, S2CID 8342016.
External links
- Weisstein, Eric W. "Polygonal edge". MathWorld.
- Weisstein, Eric W. "Polyhedral edge". MathWorld.
In geometry an edge is a particular type of line segment joining two vertices in a polygon polyhedron or higher dimensional polytope In a polygon an edge is a line segment on the boundary 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 A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal Three edges AB BC and CA each between two vertices of a triangle A polygon is bounded by edges this square has 4 edges Every edge is shared by two faces in a polyhedron like this cube Every edge is shared by three or more faces in a 4 polytope as seen in this projection of a tesseract An edge may also be an infinite line separating two half planes The sides of a plane angle are semi infinite half lines or rays Relation to edges in graphsIn 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 Number of edges in a polyhedronAny convex polyhedron s surface has Euler characteristic V E F 2 displaystyle 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 facesIn 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 terminologyIn 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 See alsoBase geometry Extended sideReferencesZiegler Gunter M 1995 Lectures on Polytopes Graduate Texts in Mathematics vol 152 Springer Definition 2 1 p 51 ISBN 9780387943657 Weisstein Eric W Polygon Edge From Wolfram MathWorld Weisstein Eric W Polytope Edge From Wolfram MathWorld Wylie Jr C R 1964 Foundations of Geometry New York McGraw Hill p 64 ISBN 0 07 072191 2 Wylie Jr C R 1964 Foundations of Geometry New York McGraw Hill p 68 ISBN 0 07 072191 2 Senechal Marjorie 2013 Shaping Space Exploring Polyhedra in Nature Art and the Geometrical Imagination Springer p 81 ISBN 9780387927145 Pisanski Tomaz Randic 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 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 Wenninger Magnus J 1974 Polyhedron Models Cambridge University Press p 1 ISBN 9780521098595 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 ISBN 0 89791 193 8 S2CID 8342016 External linksWeisstein Eric W Polygonal edge MathWorld Weisstein Eric W Polyhedral edge MathWorld
