Math 2.0

Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Fact

Plateau's laws

Builds on Mean curvatureMinimal surface

Statement

Consider a stable soap film or cluster of soap bubbles in equilibrium.

  1. The film is made of smooth pieces of surface, and each piece has constant mean curvature.
  2. The pieces meet only in threes, along smooth curves, and any two of the three meet there at an angle of 120∘120^\circ.
  3. These curves meet only in fours, at isolated points, and any two of the four meet there at an angle of arccos⁡(−1/3)≈109.47∘\arccos(-1/3) \approx 109.47^\circ.

Remarks

Joseph Plateau stated these laws from experiments with soap films, in his 1873 treatise on liquids under molecular forces. The constant mean curvature in item 1 comes from the pressure difference across the film, by the Young–Laplace law. A film spanning a wire frame, with the same pressure on both sides, is a minimal surface. The angle in item 2 comes from a balance of forces. Three equal surface tensions pulling along a curve are in equilibrium only if they are at 120∘120^\circ to each other. The angle in item 3 is the angle between two segments joining the centre of a regular tetrahedron to two of its vertices. Other junctions, such as four surfaces meeting along one curve, are unstable: a film with such a junction can lower its area by splitting it into two triple junctions. Jean Taylor proved in 1976 that these are the only possible singularities for a mathematical model of soap films, the almost-minimizing sets of Almgren.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Mean curvature, concept, no evidence levelMean curvatureconceptMinimal surface, concept, no evidence levelMinimal surfaceconceptPlateau's laws, statement, no evidence levelPlateau's lawsstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Mean curvature concept
  2. Minimal surface concept
  3. Plateau's laws statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Plateau%27s_laws
About this page
Tag381RB634WS
URIm2://381RB634WS
Kindstatement, fact
WritingW0 stub
MSC49Q05, 53A10, 76A20
StatusThis copy of the site has no build data. Run m2 status 381RB634WS.
AboutMinimal surface
UsesMean curvature, Minimal surface
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Results about this

Used by