---
tag: BZXHGNGQAX
kind: concept
name: Uniform polytope
msc: [51M20, 52B15]
specializes: [300K6CB1KH]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_polytope
same_as:
  wikidata: Q3887718
  wikipedia: Uniform polytope
---

## Summary

A uniform polytope is a polytope whose symmetries can carry any vertex to any other vertex and whose facets are themselves uniform.
In dimension two the uniform polytopes are the regular polygons.
Every [[regular polytope|F60PD912D5]] is uniform, but uniform polytopes are far more numerous.
In dimension three the convex ones are the Platonic solids, the thirteen Archimedean solids, and the infinite families of prisms and antiprisms.
Most uniform polytopes arise from the [[Wythoff construction|J4EDDYCXMR]].

## Explanations

### L1 human

A uniform polytope is a solid with regular polygon faces in which every corner looks exactly like every other corner.
The faces need not all be the same: the cuboctahedron has squares and triangles, arranged the same way around each corner.
A triangular prism with square sides is uniform too.
A square pyramid is not, since its tip looks different from its base corners.