edge (geometry)
{{Short description|Line segment joining two adjacent vertices in a polygon or polytope}}
{{distinguish|Edge (graph theory)}}
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File:Triangle.TrigArea.svg|Three edges AB, BC, and CA, each between two vertices of a triangle.
File:Square (geometry).svg|A polygon is bounded by edges; this square has 4 edges.
File:Hexahedron.png|Every edge is shared by two faces in a polyhedron, like this cube.
File:Hypercube.svg|Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract.
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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.{{citation|first=Günter M.|last=Ziegler|author-link=Günter M. Ziegler|title=Lectures on Polytopes|at=Definition 2.1, p. 51|url=https://books.google.com/books?id=xd25TXSSUcgC&pg=PA51|volume=152|series=Graduate Texts in Mathematics|publisher=Springer|year=1995|isbn=9780387943657}}. In a polygon, an edge is a line segment on the boundary,Weisstein, Eric W. "[http://mathworld.wolfram.com/PolygonEdge.html Polygon Edge]". From Wolfram MathWorld. 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.Weisstein, Eric W. "[http://mathworld.wolfram.com/PolytopeEdge.html Polytope Edge]". From Wolfram MathWorld. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.
An edge may also be an infinite line separating two half-planes.{{citation |last=Wylie Jr. |first=C.R. |title=Foundations of Geometry |year=1964 |at=p. 64 |place=New York |publisher=McGraw-Hill |isbn=0-07-072191-2}}
The sides of a plane angle are semi-infinite half-lines (or rays).{{citation |last=Wylie Jr. |first=C.R. |title=Foundations of Geometry |year=1964 |at=p. 68 |place=New York |publisher=McGraw-Hill |isbn=0-07-072191-2}}
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.{{citation|title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination|first=Marjorie|last=Senechal|author-link=Marjorie Senechal|publisher=Springer|year=2013|isbn=9780387927145|page=81|url=https://books.google.com/books?id=kZtCAAAAQBAJ&pg=PA81}}. 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
| last1 = Pisanski | first1 = Tomaž | author1-link = Tomaž Pisanski
| last2 = Randić | first2 = Milan | author2-link = Milan Randić
| editor-last = Gorini | editor-first = Catherine A.
| contribution = Bridges between geometry and graph theory
| location = Washington, DC
| mr = 1782654
| pages = 174–194
| publisher = Math. Assoc. America
| series = MAA Notes
| title = Geometry at work
| volume = 53
| year = 2000}}. See in particular Theorem 3, [https://books.google.com/books?id=Eb6uSLa2k6IC&pg=PA176 p. 176].
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.{{citation|title=On the graph structure of convex polyhedra in n-space|first=M. L.|last=Balinski|author-link=Michel Balinski|journal=Pacific Journal of Mathematics|volume=11|issue=2|year=1961|pages=431–434|mr=0126765|url=http://projecteuclid.org/euclid.pjm/1103037323|doi=10.2140/pjm.1961.11.431|doi-access=free}}.
Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge,{{citation|title=Polyhedron Models|first=Magnus J.|last=Wenninger|publisher=Cambridge University Press|year=1974|isbn=9780521098595|page=1|url=https://books.google.com/books?id=N8lX2T-4njIC&pg=PA1}}. 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
| last = Seidel | first = Raimund | author-link = Raimund Seidel
| contribution = Constructing higher-dimensional convex hulls at logarithmic cost per face
| doi = 10.1145/12130.12172
| pages = 404–413
| title = Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (STOC '86)
| year = 1986| isbn = 0-89791-193-8 | s2cid = 8342016 }}.
See also
References
{{reflist}}
External links
- {{mathworld |urlname=PolygonEdge |title=Polygonal edge}}
- {{mathworld |urlname=PolyhedronEdge |title=Polyhedral edge}}