---
tag: 381RB634WS
kind: statement
role: fact
name: Plateau's laws
msc: [49Q05, 53A10, 76A20]
uses: [PA78NFEPCF, 41X0YW2R9N]
about: [41X0YW2R9N]
sources:
  - url: https://en.wikipedia.org/wiki/Plateau%27s_laws
same_as:
  wikidata: Q1974553
  wikipedia: Plateau's laws
---

## 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|PA78NFEPCF]].
2. The pieces meet only in threes, along smooth curves, and any two of the three meet there at an angle of $120^\circ$.
3. These curves meet only in fours, at isolated points, and any two of the four meet there at an angle of $\arccos(-1/3) \approx 109.47^\circ$.

## 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|41X0YW2R9N]].
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 $120^\circ$ 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.

## Explanations

### L1 human

Soap films always join in the same few ways.
Three films meet along an edge, at equal angles of $120^\circ$, and four such edges meet at a corner, at the angle seen at the centre of a regular tetrahedron.
You can see this in any cluster of soap bubbles: four films never meet along one edge.
These rules come from the fact that a soap film always tries to reduce its area.