Mathematical Bubbles

We have all played with a soap solution and made soap bubbles at some point in our lives, but have you ever tried to make bubbles with other shapes?

When you blow on an area of soap film, the film closes up trapping the air you blew taking a spherical form. Most of us would find it ridiculous to expect it to take a different form, like a cube, but why? How are we so sure that it will be a sphere and why does the soap film prefer that form?

Well, it turns out that the soap film tries to save energy by stretching as little as possible. In other words, the film always forms the shape with the smallest area possible, so the stretching is reduced. Because of this smallest-area condition, mathematicians have been interested in the shapes formed by soap film and have named them “minimal surfaces.”

In this post, I will guide you through some experiments with soap bubbles to explore minimal surfaces.

Material:
• A big bucket
• Water
• Liquid dish soap
• Glycerin
• Wire
• Thread

The soap solution
To make the soap solution, mix the water with the liquid dish soap. Slowly, add the glycerine to the mix. The glycerin will make the solution more resistant, but too much glycerin will make it heavy and it won’t work, so you need to test your solution as you add the glycerin. The perfect solution will make it so that it is not too hard to blow the bubble and, once blown, it doesn’t break easily.

With every experiment, make sure you make a hypothesis about what will happen before dipping the frame into the solution. In making your hypothesis, remember to consider the smallest-area characteristic of the film.

Experiment 1
Use the wire to make a ring with a handle. Make other shapes with the wire, like a square or a star, and verify that you still get a spherical bubble.

Experiment 2
Make a square but without one of the sides, like a U shape. Tie a piece of thread to the two opposite sides of the U. Try it with a longer or shorter thread and see how that affects the result.

Experiment 3
Make a cube with the wire. Try a tetrahedron (triangular pyramid) and an octahedron as well. The octahedron is like two square pyramids glued together. What differences and similarities do you find in the soap film between these frames?

In the 19th century, a physicist named Joseph Plateau studied soap films and used them to solve a mathematical problem:

There is a need to build a highway that connects three cities. In order to save on building materials, the question is what is the shortest road possible?

To answer this question with soap bubbles, one can use two transparent plexiglass sheets, drill three holes on them and then fix them together using screws, leaving some space in between the two. When dipping this into the soap solution, the film will connect the screws.

Naturally, both the math problem and the experimental setup can be studied for four, five, and really any number of cities.

Though this gadget is harder to make at home, one can get a glimpse of it with the 3D shapes built before. Look at one face of the tetrahedron and sketch the path of the soap film on that face; that should give you an idea of what the shortest path between them is.
What frame can you use to get an idea of the solution to the 4-cities problem?

Experiment 4
Here are other interesting minimal surfaces you can study. Build them with wire and dip them into the soap solution.

• A helix with an axis through it.

• A cube minus two pairs of opposite sides.

• Two rings with handles that are about the same size. For this one, you will need to dip them into the solution and take them out together, then very slowly start pulling them apart but trying to keep them aligned. If the film breaks, start over.

Experiment 5
Make your own shapes and let us know what you discover!

This post was contributed by Buckeye Aha! Math Moments, the outreach program from OSU’s Department of Mathematics. Check out their website for more fun math activities.