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Plateau's Laws
Soap films are minimal surfaces — every point has zero mean curvature (H = κ₁ + κ₂ = 0)
Film surfaces meet in threes along lines (Plateau borders), each pair at exactly 120°
Plateau borders meet in fours at vertices (tetrahedral angle ≈ 109.47°)
Steiner problem: connect N points with minimum total wire length (generalizes Plateau)
Mathematics
Young-Laplace equation:
ΔP = γ (1/R₁ + 1/R₂) = 2γH
Soap film: H = 0 everywhere
→ minimal surface
Steiner tree angle condition:
at every junction: ΣF_i = 0
⟹ all angles = 120°
Total length minimization:
δL = 0, L = ∫ γ ds
Network Stats
Vertices: 0
Steiner pts: 0
Total length: —