---
tag: 6DXF1T506B
kind: concept
name: Polar polytope
msc: [52B11, 52B05, 52A20]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_polyhedron
---

## Summary

Let $P$ be a [[convex polytope|300K6CB1KH]] with the origin in its interior.
Its polar polytope is the set $P^\circ$ of vectors $y$ with $x \cdot y \le 1$ for all $x \in P$.
It is again a convex polytope, and taking the polar twice gives back $P$.
Polarity reverses the order of faces: vertices of $P$ correspond to [[facets|DPDXN2FJC4]] of $P^\circ$ and facets of $P$ to vertices of $P^\circ$.
It turns [[simple polytopes|7E6FXTGH2Y]] into [[simplicial polytopes|YVJ15PGKDM]] and back, and it is the geometric form of the dual polyhedron.

## Explanations

### L1 human

Polarity swaps corners and walls.
The polar of a cube centred at the origin is an octahedron.
The $6$ faces of the cube become the $6$ corners of the octahedron, and the $8$ corners of the cube become its $8$ faces.
The polar of a regular dodecahedron is an icosahedron.
A large polytope has a small polar, and the other way round.