---
tag: 879SBTVE7V
kind: concept
name: Polytope combinatorics
msc: [52B05]
sources:
  - url: https://en.wikipedia.org/wiki/Polyhedral_combinatorics
same_as:
  wikidata: Q8348314
  wikipedia: Polyhedral combinatorics
---

## Summary

Polytope combinatorics, also called polyhedral combinatorics, studies convex polytopes through their faces.
It asks how many faces of each dimension a polytope has and how the faces fit together.
Its typical questions ask which [[f-vectors|DCK64GFPD7]] are possible, as in [[Euler's polyhedron formula|09HT3P35YP]], the [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]].
The same name covers the study of the facets of polytopes defined by combinatorial optimization problems, a tool of integer programming.

## Explanations

### L1 human

A cube has $8$ vertices, $12$ edges and $6$ square faces, and any squashed cube with flat faces has the same pattern.
Polytope combinatorics forgets lengths and angles and keeps only this pattern of vertices, edges and faces.
It asks, for example, which lists of face numbers can occur, and in every dimension, not only in three.