Fixed Points

Brouwer’s theorem and the park map

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Brouwer: every continuous map of a disc (or ball) to itself has a fixed point.
• In one dimension it follows from the intermediate value theorem.
• Shapes with holes, like a ring, can be rotated with no fixed point.

Steps

1. Stir your coffee
2. One dimension
3. Why it must cross
4. You are here
5. Maps within maps
6. Finding the fixed point
7. When Brouwer fails