Soap bubbles
 

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

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/