Plateau's laws describe the structure of

soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...

s. These laws were formulated in the 19th century by the

Belgian Belgian may refer to: * Something of, or related to, Belgium * Belgians, people from Belgium or of Belgian descent * Languages of Belgium, languages spoken in Belgium, such as Dutch, French, and German *Ancient Belgian language, an extinct languag ...

physicist Joseph Plateau from his experimental observations. Many

patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, ...

are based on foams obeying these laws.

Laws for soap films

Plateau's laws describe the shape and configuration of soap films as follows: # Soap films are made of entire (unbroken) smooth surfaces. # The

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

of a portion of a soap film is everywhere constant on any point on the same piece of soap film. # Soap films always meet in threes along an edge called a Plateau border, and they do so at an angle of arccos(−) = 120°. # These Plateau borders meet in fours at a vertex, at the tetrahedral angle of arccos(−) ≈ 109.47°. Configurations other than those of Plateau's laws are unstable, and the film will quickly tend to rearrange itself to conform to these laws. That these laws hold for

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...

s was proved mathematically by

Jean Taylor Jean Ellen Taylor (born 1944) is an American mathematician who is a professor emerita at Rutgers University and visiting faculty at Courant Institute of Mathematical Sciences of New York University. Biography Taylor was born in Northern Califo ...

using

geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...

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* {{Patterns in nature Minimal surfaces

Laws for soap films

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