The Euler Characteristic

V − E + F counts holes

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Euler characteristic $\chi = V - E + F$ is a topological invariant.
• Sphere: 2. Torus: 0. A surface with $g$ holes: $2 - 2g$.
• χ classifies closed orientable surfaces.

Steps

1. Beyond polyhedra
2. A grid on a doughnut
3. Any grid on a torus
4. A picture frame
5. An open box
6. Pretzels