Edge (geometry)

Source: Wikipedia, the free encyclopedia.
Not to be confused with
Edge (graph theory)
.
  • Three edges AB, BC, and CA, each between two vertices of a triangle.
    Three edges AB, BC, and CA, each between two vertices of a triangle.
  • A polygon is bounded by edges; this square has 4 edges.
    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 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.
    Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract.

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.

Relation to edges in graphs

In

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

where V is the number of

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 polyhedron are its ridges, and the edges of a 4-dimensional polytope are its peaks.[8]

See also

References

  1. ISBN 9780387943657
    .
  2. ^ Weisstein, Eric W. "Polygon Edge". From Wolfram MathWorld.
  3. ^ Weisstein, Eric W. "Polytope Edge". From Wolfram MathWorld.
  4. ISBN 9780387927145
    .
  5. MR 1782654. See in particular Theorem 3, p. 176
    .
  6. MR 0126765
    .
  7. ISBN 9780521098595
    .
  8. S2CID 8342016
    .

External links
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