Minimal surfaces

thin film of soapy water enclosing air

right|thumb|350px|A helicoid with α = 1, −1 ≤ ρ ≤ 1 and − ≤ θ ≤ .

thumb|right|alt=three-dimensional diagram of a catenoid|A catenoid thumb|right|alt=animation of a catenary sweeping out the shape of a catenoid as it rotates about a central point|A catenoid obtained from the rotation of a catenary

surface that locally minimizes its area

minimal surface

a three-dimensional mathematical foam structure with equal-volume cells and low surface area

mathematical concept

set of mathematical rules governing the structure of soap films

in calculus of variations, the problem proving the existence of a minimal surface with a given boundary

Periodic minimal surface

thin layers of liquid surrounded by air

thumb|right|A gyroid minimal surface, coloured to show the Gaussian curvature at each point thumb|right|3D model of a gyroid unit cell

problem in differential geometry

Construction for minimal surfaces