Concept
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss. Every generalized n-gon with n even is also a near polygon. A generalized 2-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For a generalized n-gon is an incidence structure (), where is the set of points, is the set of lines and is the incidence relation, such that: It is a partial linear space. It has no ordinary m-gons as subgeometry for . It has an ordinary n-gon as a subgeometry. For any there exists a subgeometry () isomorphic to an ordinary n-gon such that . An equivalent but sometimes simpler way to express these conditions is: consider the bipartite incidence graph with the vertex set and the edges connecting the incident pairs of points and lines. The girth of the incidence graph is twice the diameter n of the incidence graph. From this it should be clear that the incidence graphs of generalized polygons are Moore graphs. A generalized polygon is of order (s,t) if: all vertices of the incidence graph corresponding to the elements of have the same degree s + 1 for some natural number s; in other words, every line contains exactly s + 1 points, all vertices of the incidence graph corresponding to the elements of have the same degree t + 1 for some natural number t; in other words, every point lies on exactly t + 1 lines. We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.
