Stir Your Coffee All You Like. One Point Ends Up Where It Started
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
A result guaranteeing that some point must map to itself sounds like a curiosity and turns out to underpin results in economics, physics and computer science.
What the result says
Take a shape with no holes, such as a filled disc or a solid ball, and apply any continuous transformation that sends every point of the shape to another point within the same shape. The theorem states that at least one point must end up exactly where it started. Continuity is the only requirement, meaning the transformation cannot tear or cut. Nothing else is assumed, so the transformation may stretch, squash, rotate and fold as violently as you like, and there is still a point that does not move.
The everyday illustrations
Several concrete versions make the claim tangible:
- •Stirring coffee leaves one molecule where it began, in principle
- •Crumpling a map and laying it on the original leaves one point above itself
- •Two identical maps of different scales share exactly one common point
- •At any moment there are two opposite points on Earth with the same temperature and pressure
- •The last of those follows from a related theorem
- •None of them tells you where the point actually is
Why the proof is unusual
The standard proof establishes that such a point exists without producing it, which is a feature of a whole family of arguments and is philosophically contentious. It proceeds by assuming no such point exists, using that assumption to construct a continuous map from the disc onto its own boundary circle that leaves the boundary fixed, and then showing such a map is impossible using topological invariants. The contradiction proves existence. It supplies no method for finding the point and no way to compute it, which is why a constructive school of mathematics rejects the argument as it stands.
Where the hole matters
The requirement that the shape has no holes is doing real work, and seeing why makes the result less mysterious. Take a ring rather than a disc and rotate it slightly about its centre. Every point moves and none returns to where it started, so the conclusion fails immediately. The same is true of a shape that is not bounded, since sliding an infinite plane sideways moves everything. What the proof actually uses is that the shape cannot be continuously shrunk onto its own boundary, which is a topological property that a ring does not have.
Where it gets used
The results in this family are heavily used precisely because existence is frequently all that is needed. General equilibrium theory in economics proves that a set of prices clearing every market simultaneously exists, and the proof is an application of exactly this, which is why economists learn it. Game theory uses a related version to prove that every finite game has an equilibrium, which is the result Nash received his prize for. Differential equations use a related theorem to prove solutions exist. Computer science uses fixed points to define the meaning of recursive definitions.
The takeaway
Any continuous transformation of a shape without holes into itself must leave at least one point exactly where it started, however violently it stretches and folds. The standard proof establishes existence by contradiction and supplies no way to find the point. Equilibrium prices in economics and equilibria in game theory are both proved to exist by applying this family of results.