---
tag: MAE6BF4DFK
kind: concept
name: Schläfli symbol
msc: [51M20, 52B15]
sources:
  - url: https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol
same_as:
  wikidata: Q1347011
  wikipedia: Schläfli symbol
---

## Summary

The Schläfli symbol is a short notation $\set{p, q, r, \dots}$ for a [[regular polytope|F60PD912D5]].
The first entry gives the polygons that form the two-dimensional faces.
Each further entry says how many of the previous pieces meet around a face of lower dimension.
The symbol determines the regular polytope up to similarity, and reversing it gives the dual polytope.
It is the main bookkeeping tool in the [[classification of regular polytopes|XYCD53PB5K]].

## Explanations

### L1 human

The symbol $\set{4, 3}$ means: faces are squares, and three of them meet at each corner; this is the cube.
The symbol $\set{3, 4}$ means triangles with four at each corner; this is the octahedron.
In four dimensions, $\set{4, 3, 3}$ means cubes as the three-dimensional cells, with three cubes around each edge; this is the tesseract.
Reading the symbol backwards gives the dual shape, as the cube and the octahedron show.