🍩 Topology — Surfaces, Genus & Euler Characteristic
Explore topological surfaces — genus, orientability, and Euler characteristic in 3D
Surface
Display
Topology Stats
Formula
Sphere: g=0, χ=2
Orientable, 0 boundaries
About Topology of Surfaces
The Euler Characteristic
For any triangulated surface, the Euler characteristic χ = V − E + F (vertices minus edges plus faces) is a topological invariant — it doesn't change under continuous deformations. For a sphere χ=2; a torus χ=0; a double torus χ=−2. The formula χ = 2−2g relates it to genus g, the number of holes in the surface.
Orientability
A surface is orientable if you can consistently define an "outward" normal everywhere — like a sphere or torus. The Möbius strip is the simplest non-orientable surface: a bug walking along its middle edge returns to its start but mirrored. The Klein bottle is a closed non-orientable surface that cannot be embedded in 3D space without self-intersection.
Gaussian Curvature
Gaussian curvature K = κ₁·κ₂ (product of principal curvatures) is intrinsic to the surface. A sphere has K>0 everywhere (red); a saddle point has K<0 (blue); a cylinder or flat plane has K=0 (green). The Gauss-Bonnet theorem links total curvature to topology: ∬K dA = 2πχ — the integral of curvature equals 2π times the Euler characteristic.