Given a boundary curve, find the minimal surface — interactive wire frame solver
Bend a wire into any loop, dip it in soapy water, and the film that forms is the surface of least area spanning it. Proving such a surface always exists took nearly a century.
Watch a rough mesh spanning a circle relax, step by step, toward the minimal surface. Keep an eye on the area readout: it drops with every iteration.
Plateau's problem asks: given a closed curve in space, does there exist a surface of least area spanning that curve? The Belgian physicist Joseph Plateau studied this experimentally in the 1840s by dipping wire frames into soapy water — the films that formed were always minimal surfaces.
The mathematical proof of existence came nearly a century later, independently by Jesse Douglas and Tibor Radó in 1930–1931. Douglas received the first Fields Medal in 1936 for this work.
Watch a triangulated mesh spanning a circular boundary relax toward the minimal surface via Laplacian smoothing. The area decreases with each iteration as interior vertices slide toward their equilibrium positions.
Pick a boundary — circle, square, wavy loop, or two circles — and see how each wire shapes its own surface, from a flat disk to a saddle to a catenoid.
For any rectifiable Jordan curve in R³, there exists a disk-type surface of least area spanning that curve. The proof uses the direct method of the calculus of variations: take a minimizing sequence of surfaces, show it converges, and prove the limit is smooth.
The solution is not always unique — some boundary curves admit multiple minimal surfaces with different topologies. The question of regularity (smoothness of the solution) remained open for decades and was finally resolved by the deep work of Osserman, Alt, and others.
Click through 240 years of milestones, from Euler's catenoid in 1744 to Costa's genus-1 surface in 1984.
Joseph Plateau publishes his systematic experimental study of soap films, demonstrating that films spanning wire frames always form minimal surfaces.
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