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  "body": "\n## Statement\n\nConsider a stable soap film or cluster of soap bubbles in equilibrium.\n\n1. The film is made of smooth pieces of surface, and each piece has constant [[mean curvature|PA78NFEPCF]].\n2. The pieces meet only in threes, along smooth curves, and any two of the three meet there at an angle of $120^\\circ$.\n3. These curves meet only in fours, at isolated points, and any two of the four meet there at an angle of $\\arccos(-1/3) \\approx 109.47^\\circ$.\n\n## Remarks\n\nJoseph Plateau stated these laws from experiments with soap films, in his 1873 treatise on liquids under molecular forces.\nThe constant mean curvature in item 1 comes from the pressure difference across the film, by the Young–Laplace law.\nA film spanning a wire frame, with the same pressure on both sides, is a [[minimal surface|41X0YW2R9N]].\nThe angle in item 2 comes from a balance of forces.\nThree equal surface tensions pulling along a curve are in equilibrium only if they are at $120^\\circ$ to each other.\nThe angle in item 3 is the angle between two segments joining the centre of a regular tetrahedron to two of its vertices.\nOther 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.\nJean Taylor proved in 1976 that these are the only possible singularities for a mathematical model of soap films, the almost-minimizing sets of Almgren.\n\n## Explanations\n\n### L1 human\n\nSoap films always join in the same few ways.\nThree 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.\nYou can see this in any cluster of soap bubbles: four films never meet along one edge.\nThese rules come from the fact that a soap film always tries to reduce its area.\n",
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