Why do soap bubbles meet at 120 degrees - Adam Oberman Math

A variational problem for a minimal curve

Suppose we wish to connect three points, A, B, C, by the shortest possible curve. (E.g. one point could be an electrical socket, and the two other points could be appliances). For some configurations, it might be shorter to have two separate straight lines.

But for others, it seems reasonable to expect that the shortest path is given by a Y shape, choosing a point x inside the triangle ABC, with lines connecting x to each point.

We are led to the problem:

Consider a triangle with vertices A, B, C. What is the shortest curve that connects the three vertices?

We can express this problem as

$\min_x f(x) = \| x - A\| + \| x - B\| + \| x - C\|$

looking for the critical point $\nabla f(x) = 0$ leads to

$\frac{x-A}{\| x - A\|} +\frac{x-B}{ \| x - B\|} + \frac{x-C}{\| x - C\|}$

This last equation says that we have three unit vectors which add up to zero. Connecting the vectors head to tail means that they form a cycle, which is a triangle. Since they all have the same length, the triangle is equilateral. Thus we are led to the conclusion that

The angle between the lines is 120 degrees.

This problem is related to the shape of soap bubbles, see the reference below.

A related variational problem for a triangle

If we minimize the sum of the distances squared, we get the centroid of the triangle

$\min_x f(x) = \| x - A\|^2 + \| x - B\|^2 + \| x - C\|^2$

looking for the critical point $\nabla f(x) = 0$ leads to

$x = \frac{A + B + C}{3}$

which is the centroid of the triangle, see wikipedia triangle. The centroid cuts every median in the ratio 2:1, which justifies the interpretation.

References

See the article, Mathematical Recreation, by Ian Stewart Double Bubble,Toil and Trouble http://www.fortunecity.com/emachines/e11/86/bubble.html