Edge (geometry)















Triangle.TrigArea.svg



Three edges AB, BC, and CA, each between two vertices of a triangle.

Square (geometry).svg
A polygon is bounded by edges; this square has 4 edges.

Hexahedron.png
Every edge is shared by two faces in a polyhedron, like this cube.

Hypercube.svg
Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract.

For edge in graph theory, see Edge (graph theory)

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 side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet.[3] A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.




Contents






  • 1 Relation to edges in graphs


  • 2 Number of edges in a polyhedron


  • 3 Incidences with other faces


  • 4 Alternative terminology


  • 5 See also


  • 6 References


  • 7 External links





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.[4] 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.[5]



Number of edges in a polyhedron


Any convex polyhedron's surface has Euler characteristic


V−E+F=2,{displaystyle V-E+F=2,}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.[6]
Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge,[7] 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.[8]



See also


  • Extended side


References





  1. ^ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer, Definition 2.1, p. 51.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.


  2. ^ Weisstein, Eric W. "Polygon Edge." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PolygonEdge.html


  3. ^ Weisstein, Eric W. "Polytope Edge." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PolytopeEdge.html


  4. ^ Senechal, Marjorie (2013), Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 81, ISBN 9780387927145.


  5. ^ 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.


  6. ^ 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.


  7. ^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 1, ISBN 9780521098595.


  8. ^ 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



  • Weisstein, Eric W. "Polygonal edge". MathWorld.

  • Weisstein, Eric W. "Polyhedral edge". MathWorld.




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