---
tag: QP7GEKDTYX
kind: concept
name: Polytope
msc: [52B11, 52B05]
sources:
  - url: https://en.wikipedia.org/wiki/Polytope
same_as:
  wikidata: Q747980
  wikipedia: Polytope
---

## Summary

A polytope is a geometric figure with flat sides: a polygon in the plane, a [[polyhedron|0325SGSJAR]] in space, and their analogues in any dimension.
The word is used with several meanings.
In convex geometry and combinatorics it usually means a [[convex polytope|300K6CB1KH]].
In the wider sense used here, a polytope is a finite union of convex polytopes, so it may be non-convex.
Further generalizations include star polytopes, whose faces may cross each other, and abstract polytopes, which keep only the incidence pattern of vertices, edges and faces.

## Explanations

### L1 human

Polytopes are the shapes built from flat pieces: triangles and squares, cubes and pyramids, and the four-dimensional hypercube.
A polytope need not be convex: an L-shaped region made of three unit squares is a polytope.
Curved shapes such as discs and balls are not polytopes.