Generalized n-gon
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In combinatorial mathematics, generalized n-gons are geometric structures introduced by Jacques Tits. They are a generalization of the projective planes, which form the most complex kind of axiomatic projective spaces, and generalized quadrangles, which form the most complex kind of polar spaces.
[edit] Definition
Generalized n-gons (
), are incidence structures (P,B,I), with 
an incidence relation, satisfying certain conditions. These are best expressed by use of the (bipartite) incidence graph :
- There is a s (
) 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 (
) 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.
[edit] Examples
Every usual-sense polygon is an example of a generalized n-gon, but they are trivial with s = t = 1.
[edit] Properties
Walter Feit and Graham Higman proved that if we assume
,
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
, 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).
