---
tag: RPAVY0W0XS
kind: concept
name: Orbifold
msc: [57R18, 57M60]
sources:
  - url: https://en.wikipedia.org/wiki/Orbifold
same_as:
  wikidata: Q40460
  wikipedia: Orbifold
---

## Summary

An orbifold is a space that looks near each point like the quotient of an open subset of $\R^n$ by a finite group.
It generalizes a [[smooth manifold|R2TQ82PCNB]], which is the case where all these groups are trivial.
Orbifolds arise as quotients of manifolds by properly discontinuous actions of discrete groups, which may have fixed points.
They are central in the geometry of surfaces and $3$-manifolds.

## Explanations

### L1 human

Fold a sheet of paper into a cone: near the tip the cone is not flat, but it still looks like a flat disk divided by a rotation.
An orbifold is a space built from such pieces, where each piece is a flat piece of $\R^n$ divided by a finite group of symmetries.
For example, rotate the plane $\R^2$ by a third of a turn around the origin.
Points that are moved into each other are glued, and the result is a cone with one special point, the image of the origin.
The cone point remembers that the group of order $3$ fixed it.