Crumple — κ(s) → ∞ at fold lines

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Crumple

κ(s) → ∞ at fold lines

A smooth surface developing infinite curvature at fold lines — the geometry of crumpling paper.

Smoothness collapsing into sharp, irreversible folds

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Contraction — T(x) : ||T(x)-T(y)|| < ||x-y||

Contraction

T(x) : ||T(x)-T(y)|| < ||x-y||

Fold — f(x) = f(−x), x → 0

Fold

f(x) = f(−x), x → 0

$Shrink — A(t) = A₀ · e^{-λt}$

Shrink

A(t) = A₀ · e^{-λt}

Veil — f(x) = f(x) · (1 - g(x))

Veil

f(x) = f(x) · (1 - g(x))