Algebraic Topology & Euler Characteristic

Build simplicial complexes interactively. Compute χ = V − E + F and Betti numbers β₀ (connected components), β₁ (holes), β₂ (voids). Watch χ = β₀ − β₁ + β₂.

Click: add vertex | Click near vertex: select | Drag between vertices: add edge | Shift+click 3 vertices: add face

Mode

ADD VERTICES

Presets

χ = 00+0 = 0
β₀ = 0 (connected components)
β₁ = 0 (independent cycles / holes)
β₂ = 0 (enclosed voids)

χ = β₀ − β₁ + β₂ = 0
Gauss-Bonnet: χ = ∫K dA / 2π
Sphere: χ=2, Torus: χ=0
Klein bottle: χ=0
RP²: χ=1

β₁ = E − V + β₀ − F
(rank of H₁, cycles mod boundaries)

χ is a topological invariant — same for all homeomorphic spaces.