---
tag: N602W0WAVT
kind: concept
name: Hyperplane arrangement
msc: [52C35, 05B35, 32S22]
sources:
  - url: https://en.wikipedia.org/wiki/Arrangement_of_hyperplanes
same_as:
  wikidata: Q4795851
  wikipedia: Arrangement of hyperplanes
---

## Summary

A hyperplane arrangement is a finite set of hyperplanes in a vector space or affine space.
Over the real numbers the hyperplanes cut the space into convex pieces called regions.
One studies how many regions there are and how they fit together.
The combinatorics of an arrangement is captured by its intersection poset, the [[partially ordered|0QV2JS9RJ3]] set of all nonempty intersections of its hyperplanes.
Arrangements connect polytopes, [[matroids|PX31ZNYS3Z]], root systems and the topology of complements of hypersurfaces.

## Explanations

### L1 human

Draw a few straight lines on a sheet of paper.
They cut the sheet into pieces, and the question is how many pieces there are.
Three lines in general position give $7$ pieces, while three parallel lines give only $4$.
A hyperplane arrangement is the same picture in any dimension, with flat walls instead of lines.