# Generalized polygon

A **generalized polygon** is a type of incidence geometry.^{[1]} Generalized polygons seem to be studied entirely as combinatorial objects. A generalized n -gon has two types of elements, "points" and "lines", and has the following rules:

- Two different elements can only be incident if one is a point and the other is a line. In other words, two points are incident iff they are the same point, and two lines are incident iff they are the same line. Thus, a generalized polygon is a type of incidence plane.
- For any two elements e and f , there is a chain of length at most n from e to f .
- For any two elements e and f , there is at most one irreducible chain of length less than n (see below for definition).

A length-n chain from e to f is a sequence of n + 1 elements that starts with e , ends with f , and every consecutive pair of elements is incident. A chain is irreducible if, for every sequence a , b , c in the chain, a and c are not incident. Irreducible chains must therefore alternate between points and lines, and cannot immediately backtrack. A generalized polygon is considered nondegenerate iff every element e has an element f such that the shortest chain from e to f has length n .

Generalized 2-gons connect every point to every line, and are not particularly interesting. Generalized 3-gons are projective planes.

## External links[edit | edit source]

- Wikipedia contributors. "Generalized polygon".
- Weisstein, Eric W. "Generalized Polygon" at MathWorld.

## References[edit | edit source]

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