---
tag: DPDXN2FJC4
kind: concept
name: Facet
msc: [52B05, 52B11]
specializes: [KY7SB854VH]
sources:
  - url: https://en.wikipedia.org/wiki/Facet_(geometry)
same_as:
  wikidata: Q3847020
  wikipedia: Facet (geometry)
---

## Summary

A facet of a [[convex polytope|300K6CB1KH]] is a [[face|KY7SB854VH]] of the largest possible dimension below that of the polytope itself.
A polytope of dimension $d$ has facets of dimension $d - 1$.
The facets determine the polytope: a full-dimensional polytope is the intersection of the closed half-spaces bounded by the hyperplanes of its facets.
Counting facets and studying their shapes is a central theme of [[polytope combinatorics|879SBTVE7V]].

## Explanations

### L1 human

The facets of a solid are the flat pieces of its surface that are "one dimension down".
For a cube in space they are the six square faces.
For a polygon in the plane they are its edges.
For a four-dimensional cube they are eight ordinary cubes.
Each facet sits on a wall, and the polytope is what lies on the inner side of all the walls.