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Generalized Polygon
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DEFINITION

Generalized n-gons (n \geq 2), are Incidence Structure s ''(P,B,I)'', with I\subseteq P imes B an Incidence Relation , satisfying certain conditions. These are best expressed by use of the (bipartite) incidence graph :

  • There is a ''s'' (s\geq 1) such that on every line there are exactly s+1 points. There is at most one point on two distinct lines.

  • There is a ''t'' (t\geq 1) such that through every point there are exactly t+1 lines. There is at most one line through two distinct points.

  • The Diameter of the graph is n.

  • The Girth of the graph is 2n.



EXAMPLES

Every Polygon in the usual sense of the term is a trivial example of a generalized ''n''-gon with s=t=1.


PROPERTIES

Walter Feit and Graham Higman proved that if we assume

:s\geq 2,t\geq 2,,

and both of them finite then ''n'' can only be

:2, 3, 4, 6 or 8.

More specifically,

  • If ''n'' = 2 the structure is trivial.

  • If ''n'' = 3, the assumption of only s\geq 2, already implies the structure is a Projective Plane

  • If ''n'' = 4, the structure is, without any assumptions on the parameters, a Generalized Quadrangle .


If ''s'' and ''t'' are both infinite then generalized ''n''-gons exist for each ''n'' greater or equal to 2. Whether or not there exist generalized ''n''-gons with one of the parameters finite and the other infinite is not known (these cases are called semi-finite).


REFERENCE

W. Feit and G. Higman, The nonexistence of certain generalized polygons, ''J. Algebra'' 1 (1964) 114-131.