---
tag: DCMP1A0138
kind: concept
name: Duality in projective geometry
msc: [51A10, 51A05]
sources:
  - url: https://en.wikipedia.org/wiki/Duality_(projective_geometry)
same_as:
  wikidata: Q735346
  wikipedia: Duality (projective geometry)
---

## Summary

In a [[projective plane|DANFNN8C4Y]], points and lines play symmetric roles: two points lie on one line, and two lines meet in one point.
Duality is the exchange of the words "point" and "line", together with "lies on" and "passes through".
The principle of duality says that the dual of a theorem valid in all projective planes is again a theorem valid in all projective planes.
In a projective space of dimension $n$, duality exchanges subspaces of dimension $d$ and $n - 1 - d$.

## Explanations

### L1 human

Take any true statement about points and lines of the projective plane and swap the two words.
"Two points determine a line" becomes "two lines determine a point", which is also true in the projective plane.
So every theorem comes with a free twin.
For example, [[Pascal's theorem|94AKWRHND0]] about six points on a conic has [[Brianchon's theorem|3THE3Q96KK]] about six lines tangent to a conic as its dual twin.