Surfaces that minimize area - soap films and beyond
A minimal surface is a surface with zero mean curvature(H = 0) everywhere. This means that at every point, the surface curves equally in opposite directions — like a saddle.
Minimal surfaces model soap films: when a soap film spans a wire frame, it naturally finds the shape that minimizes surface area, which always has H = 0.
Explore famous minimal surfaces with iridescent soap-film-like shading. Each surface has zero mean curvature everywhere, despite their complex shapes.
The surface of revolution of a catenary curve. The only minimal surface of revolution.
A surface minimizes area (locally) if and only if its mean curvature vanishes. The mean curvature measures the “net bending” at each point:
The first variation formula shows that H = 0 is equivalent to being a critical point of the area functional.
The associate family: the catenoid and helicoid can be continuously deformed into each other through a family of minimal surfaces. Watch the smooth morphing animation!
The catenoid and helicoid are related by the associate family — a continuous 1-parameter deformation through minimal surfaces. They share the same Gauss map!
Every minimal surface belongs to a 1-parameter family of isometric minimal surfaces called the associate family. The transformation is:
where X* is the conjugate surface. The catenoid (θ = 0) and helicoid (θ = π/2) are conjugate pairs!
Compare minimal surfaces (H = 0) with non-minimal surfaces. Green indicates regions where H ≈ 0. Notice how minimal surfaces are uniformly green!
H = 0 everywhere
Discovered by Euler. The only minimal surface of revolution. Formed by rotating a catenary curve.
Discovered by Meusnier. A ruled minimal surface shaped like a spiral staircase. The only ruled minimal surface besides the plane.
A self-intersecting minimal surface with no boundary. Simple parametrization but beautiful saddle geometry.
Doubly periodic minimal surface. Can be viewed as a deformation of two infinite orthogonal planes joined by saddles.
Plateau's problem asks: given a closed curve in space, does there exist a minimal surface spanning that curve? This was answered affirmatively by Jesse Douglas and Tibor Radó in 1930.
Physical soap films solve this problem naturally: surface tension causes the film to minimize its area, always producing a surface with H = 0 (or forming bubbles where pressure differences create constant H ≠ 0).
Next: Theorema Egregium — Gauss's remarkable theorem that Gaussian curvature is intrinsic.