---
tag: Q2QFXPP9EE
kind: concept
name: Configuration
msc: [05B30, 51A20, "51E30"]
sources:
  - url: https://en.wikipedia.org/wiki/Configuration_(geometry)
same_as:
  wikidata: Q1866689
  wikipedia: Configuration (geometry)
---

## Summary

A configuration is a finite system of points and lines in which two points share at most one line.
Every point lies on the same number of lines, and every line contains the same number of points.
A configuration with $p$ points on $\gamma$ lines each and $\ell$ lines with $\pi$ points each is written $(p_\gamma\ \ell_\pi)$, and $(p_k)$ when $p = \ell$ and $\gamma = \pi = k$.
Many classical theorems of projective geometry, such as those of Desargues and Pappus, are statements that certain configurations close up.

## Explanations

### L1 human

Draw some points and some lines so that the picture is completely regular: each point is on $3$ of the lines and each line passes through $3$ of the points.
Such a balanced picture is a configuration.
Counting the pairs (point, line through it) in two ways gives $p \gamma = \ell \pi$.
The Fano plane, with $7$ points and $7$ lines, is a $(7_3)$ configuration, but it cannot be drawn with straight lines in the real plane.