Double bubble conjecture

In the mathematical theory of minimal surfaces, the double bubble conjecture states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble — three spherical surfaces meeting at angles of 2π/3 on a common circle. It is now a theorem, as a proof of it was published in 2002.

According to Plateau's laws, the minimum area shape that encloses any volume or set of volumes must take a form commonly seen in soap bubbles in which surfaces of constant mean curvature meet in threes, forming dihedral angles of 2π/3. In a standard double bubble, these surfaces are patches of spheres, and the curve where they meet is a circle. When the two enclosed volumes are different from each other, there are three spherical surfaces, two on the outside of the double bubble and one in the interior, separating the two volumes from each other; the radii of the spheres is inversely proportional to the pressure differences between the volumes they separate, according to the Young–Laplace equation. When the two volumes are equal, the middle surface is instead a flat disk, which can be interpreted as a patch of an infinite-radius sphere.

The double bubble conjecture states that, for any two volumes, the standard double bubble is the minimum area shape that encloses them; no other set of surfaces encloses the same amount of space with less total area.

The same fact is also true for the minimum-length set of curves in the Euclidean plane that encloses a given pair of areas, and it can be generalized to any higher dimension.

The isoperimetric inequality for three dimensions states that the shape enclosing the minimum single volume for its surface area is the sphere; it was formulated by Archimedes but not proven rigorously until the 19th century, by Hermann Schwarz. In the 19th century, Joseph Plateau studied the double bubble, and the truth of the double bubble conjecture was assumed without proof by C. V. Boys in his 1896 book on soap bubbles.

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