![]() The solution is well-known to be the circle, so, a way for varying the problem is to add constraints preventing the circle as solution. Of fixed perimeter, which curve (if any) maximizes the area of itsĮnclosed region? This question can be shown to be equivalent to theįollowing problem: Among all closed curves in the plane enclosing aįixed area, which curve (if any) minimizes the perimeter? » Problem can be stated as follows: Among all closed curves in the plane Nachrichten Königlichen Gesellschaft Wissenschaften Göttingen, (1884) 1–13.« The classical isoperimetric problem dates back to antiquity. Schwarz, 1884: Beweis des Satzes, daß die Kugel kleinere Oberfläche besitzt, als jeder andere Körper gleichen Volumens. Hutchings, Michael Morgan, Frank Ritoré, Manuel Ros, Antonio (2002), “Proof of the double bubble conjecture”, Annals of Mathematics, 2nd Ser. Harrison, J., UC Berkeley, 2012: Soap film solutions to Plateau’s problem, ArXiv: Abstract. Surprisingly, this was just a conjecture until it was proved as recently as 2002. The double bubble comprising three spherical caps is the minimal surface enclosing two fixed volumes of air. But p1 is proportional to 1/r1 and p2 is proportional to 1/r3 which leads to the desired result. This result also follows from a physical argument: for balance of the forces at the interior interface, the pressure difference p1 – p2 must equal the surface tension, proportional to the curvature 1/r2. Using the sine rule, it is easy to show that The acute angles at point P are each 60º. The only way in which this can happen is if they meet at angles of 120º. At point P, three forces of equal magnitude in three different directions must sum to zero. For a static solution all forces must be in balance. A cross-section of a double bubble is shown below. For a double bubble, there are two spherical outer surfaces joined by a spherical inner interface. With just a single component, the solution is a spherical bubble. When we consider bubbles, an additional complication arises since there are variations in pressure, and the surfaces contain multiple components connected in complicated ways. ![]() Plateau’s original problem dealt with surfaces of minimum area without pressure differences (so H = 0) and with boundaries that are simple closed curves. Recently, Jenny Harrison of UC Berkeley has published a general proof of Plateau’s problem, including non-orientable surfaces and surfaces with multiple junctions. In 1936, Douglas was awarded the first Fields Medal for his work. Plateau’s problem was solved under certain conditions by the American mathematician Jesse Douglas in 1931, and independently at about the same time by Tibor Radó. Surfaces with zero mean curvature: catenoid (left) and helicoid (right) If the pressure differs on the two sides of a soap film film, there is a force normal to the surface, which must therefore be curved so that the surface tension can balance the pressure force.Įxamples of surfaces with zero mean curvature (H = 0) are the catenoid (generated by rotating a catenary about its directrix) and the helicoid (generated by the lines from a helix to its axis). Therefore, the mean curvature is also zero. If the pressure is equal on both sides, then Δp = 0, and there is no force normal to the surface. Thus, soap films assume a minimal area configuration. Surface energy is proportional to area, and surface tension acts to minimise the area. ![]() The curvature of a soap film is related to the pressure difference across the surface by a nonlinear partial differential equation called the Young-Laplace equation, Δp = 4 σ H, where Δp is the pressure difference, σ is the surface tension and H is the mean curvature. Soap films are tense: they act like a stretched elastic skin, trying to attain the smallest possible area. ![]() These laws hold for surfaces of minimal area, subject to constraints such as fixed enclosed volumes or specified boundaries. These curves meet in groups of four, at vertices with angles arccos(−1/3) ≈ 109.5°. Soap films connect in threes, along curves meeting at angles arccos(−1/2) = 120°.Ĥ. The mean curvature of each component of a soap film is constant.ģ. Soap films are made of components that are smooth surfaces.Ģ. Plateau formulated a set of empirical rules, now known as Plateau’s Laws, for the formation of soap films:ġ. Mathematically, the problem falls within the ambit of the calculus of variations. This problem was first posed by Joseph-Louis Lagrange in 1760, but it is named for Joseph Plateau, the Belgian physicist who experimented extensively with soap films. Given a simple closed curve C in three dimensions, prove that there is a surface of minimum area having C as its boundary. Plateau’s Problem: Soap Films, or Bubbles with Boundaries.
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