---
tag: J4EDDYCXMR
kind: concept
name: Wythoff construction
msc: [51M20, 20F55, 51F15]
sources:
  - url: https://en.wikipedia.org/wiki/Wythoff_construction
same_as:
  wikidata: Q2995289
  wikipedia: Wythoff construction
---

## Summary

The Wythoff construction builds a polytope from a finite reflection group and a chosen point.
One takes a point in the fundamental region of the group and reflects it in the mirrors over and over.
The convex hull of all the images is the polytope.
Choosing which mirrors the point lies on gives the regular polytopes, their truncations and most other [[uniform polytopes|BZXHGNGQAX]] with the same symmetry.
It ties the geometry of polytopes to the combinatorics of [[Coxeter groups|G6FND7X69J]].

## Explanations

### L1 human

Think of a kaleidoscope: a few mirrors meeting at angles like $90^\circ$ or $60^\circ$.
Put a single dot between the mirrors and look at all its reflections; then join up the dots into a solid.
If the dot sits on some of the mirrors, its images merge, and one gets for instance a cube or an octahedron.
If it sits at a well-chosen place between the mirrors, one gets a truncated shape such as the regular octagon from the mirrors of a square.