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Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau
Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repe ...
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, wave ...
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. # Themean 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.
T ...
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
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
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 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 ...
..
See also
*Young–Laplace equation
In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or ...
, governing the curvature of surfaces in a soap film
Notes
Sources
*External links
* {{Patterns in nature Minimal surfaces