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Crumpled Membrane — Surface Roughness

Elastic sheets crumple into fractal ridges — roughness exponent H ≈ 0.64 from scaling theory

Height map h(x,y) — color = height
Cross-section h(x, y=0)
Height correlation C(r) = ⟨[h(x+r)-h(x)]²⟩^½ — log-log plot
Measured Hurst exponent:
Self-affine surfaces: The height field h(x) satisfies ⟨[h(x+r)-h(x)]²⟩ ∝ r^(2H) where H is the Hurst (roughness) exponent. For flat surfaces H→1, for rough membranes H≈0.64 (from renormalization group theory of elastic sheets with bending rigidity κ). The structure factor S(q) ∝ q^(-1-2H). Generated via spectral synthesis: assign random Fourier amplitudes ∝ |k|^(-H-1), inverse-FFT. H=0.5 is Brownian motion; H=0.64 is crumpled elastic sheet; H=1 is smooth surface.