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