---
tag: GQJCPFSYVK
kind: concept
name: Generalized quadrangle
msc: ["51E12", 05B25]
specializes: [PCT2HY9AAZ]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_quadrangle
same_as:
  wikidata: Q2754558
  wikipedia: Generalized quadrangle
---

## Summary

A generalized quadrangle is an [[incidence structure|FGKNAFNHA3]] of points and lines with no triangles.
In it, a point off a line is joined to exactly one point of that line.
It is the case $n = 4$ of a [[generalized polygon|PCT2HY9AAZ]].
Examples include the polar spaces of rank $2$ attached to quadrics and to symplectic and Hermitian forms.
They are basic building blocks in finite geometry.

## Explanations

### L1 human

In a generalized quadrangle, two points share at most one line, and there are no triangles.
Instead, given a point and a line not through it, there is exactly one way to walk from the point along a line to the given line.
A simple picture is a grid: the points of a square grid with its rows and columns as lines.
A point not on a given row lies on exactly one column, and that column meets the row in exactly one point.