Worlds from a Square

Tori, Klein bottles and video games

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

• Gluing opposite edges of a square gives a torus, or with a flip, a Klein bottle.
• Straight paths on a flat torus close up exactly when the slope is rational.
• $\chi$ plus orientability classifies every closed surface.

Steps

1. A video-game world
2. What shape is the world?
3. Closing the loop
4. A world with a twist
5. Building a Klein bottle
6. The projective plane
7. Telling surfaces apart