---
tag: S2JE8M3FMP
kind: concept
name: Schlegel diagram
msc: [52B05, 52B11, 52B10]
sources:
  - url: https://en.wikipedia.org/wiki/Schlegel_diagram
same_as:
  wikidata: Q2004228
  wikipedia: Schlegel diagram
---

## Summary

A Schlegel diagram is a picture of a $d$-dimensional [[convex polytope|300K6CB1KH]] inside one of its [[facets|DPDXN2FJC4]], one dimension lower.
It is obtained by looking at the polytope from a point just outside that facet and projecting the rest of the boundary onto it.
The result is a subdivision of the facet into convex polytopes, one for each other facet, that keeps the full face structure.
Schlegel diagrams make $4$-dimensional polytopes visible in space and give a quick proof that graphs of $3$-polytopes are [[planar|0NC9R64TXP]].

## Explanations

### L1 human

Put your eye very close to one face of a cube and look through it.
You see the far face as a small square inside the near one, joined to it by the four side faces seen as trapezoids.
That drawing, a square inside a square with the corners joined, is the Schlegel diagram of the cube.
The same trick shows a four-dimensional cube as a small cube inside a big cube, with six squashed cubes in between.