---
tag: F60PD912D5
kind: concept
name: Regular polytope
msc: [51M20, 52B15]
specializes: [300K6CB1KH]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_polytope
same_as:
  wikidata: Q2745944
  wikipedia: Regular polytope
---

## Summary

A regular polytope is a polytope with the largest possible symmetry: its symmetries can carry any flag of faces to any other flag.
In the plane the regular polytopes are the [[regular polygons|01KJJ9A400]], and in space the convex ones are the five [[Platonic solids|017E5PRS1Q]].
Each regular polytope is described by its [[Schläfli symbol|MAE6BF4DFK]], and their full list is given by the [[classification of regular polytopes|XYCD53PB5K]].
Regular polytopes are the model objects of the theory of polytopes and of finite reflection groups.

## Explanations

### L1 human

A regular polytope looks the same from every corner, every edge, every face, and so on.
More precisely, pick a corner, an edge at that corner, and a face at that edge; any such choice can be moved to any other by a symmetry.
The cube is regular: any corner, edge and face that fit together can be rotated or reflected to any other such triple.
A box with sides of three different lengths is not regular, because a short edge cannot be moved onto a long one.