Optimal Bubbleology

This beautiful shape is simply the result of dipping a tetrahedral wire frame into soapy water. As you've probably noticed before, ordinary soap bubbles are round, but have you ever wondered why? Soap bubbles will assume the shape of least surface area possible to contain a given volume - this is known as a minimal surface. In a bubble, the inward surface tension forces of the water film are exactly balanced by the outward pushing pressure of the air inside.

Because of this lovely property, we can find these gorgeous and ephemeral solutions to problems in calculus of variations of finding the minimal surface of a boundary with specified constraints. This is known as Plateau's Problem, raised by Joseph-Louis Lagrange in 1760 but named after the Belgian physicist Joseph Plateau who solved some special cases of the problem experimentally using soap films and wire frames.

This one is a cube frame. Plateau noticed that only three smooth surfaces of a soap film can meet along a line (called a Plateau border) and the angle between any two of these surfaces is 2π/3 radians. In addition, only four Plateau borders may meet at a point and must meet at tetrahedral angles (arccos(1/3)) .

If you take the resulting minimal surface and pop several of the boundaries, you can get some really neat surfaces - here's one from the cube that turns out to have a saddle point.

Another great one is adding a bubble to the center of the cube, which gives a three dimensional representation of a 4D hypercube!

“Nature is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.” - Galileo Galilei