Soap bubbles

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Soap bubbles are physical examples of
the complex mathematical problem of minimal surface.

They will assume the shape of least surface area possible containing a given volume.
A true minimal surface is more properly illustrated by a soap film,
which has equal pressure on inside as outside,
hence is a surface with zero mean curvature.

A soap bubble is a closed soap film:
due to the difference in outside and inside pressure,
it is a surface of constant mean curvature.


Soap films are thin layers of liquid (usually water-based) surrounded by air.
For example, if two soap bubbles come into contact,
they merge and a thin film is created in between.
Thus, foams are composed of a network of films connected by Plateau borders.
Soap films can be used as model systems for minimal surfaces,
which are widely used in mathematics.

Wiki

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Mathematically, the question of what shape the soap bubble will form
is a minimization problem:
the surface area seeks to be as small as possible under a constraint
(the volume is constant and the boundary spans a given countour).

This is known as Plateau's problem.

https://brilliant.org/wiki/math-of-s...nd-honeycombs/