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 Plat ...
s. 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 r ...
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.
# 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 that ...
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 surfac ...
.
[.]
See also
*
Young–Laplace equation
In physics, the Young–Laplace equation () is an 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 wall tensi ...
, governing the curvature of surfaces in a soap film
Notes
Sources
*
External links
*
{{Patterns in nature
Minimal surfaces
Bubbles (physics)