# Wonderful soap skins

I wonder if you can also create “angular soap bubbles”? Sure you can! By dipping various edge models in soap suds, a wide variety of fascinating shapes are created. For example, it even proves quite difficult to wet the surface of a cube with soap skin. More often, a much more delicate-looking structure is created with a small square or even cube in the center, to which soap skins stretch from the edges of the cube:

Das Exponat lädt dazu ein, die sich ausbildenden Seifenhautstrukturen an Kantenmodellen wie z.B. Prismen, Zylindern oder Tetraedern zu untersuchen. Dabei wird so manch ein Besucher nicht nur einmal in Erstaunen versetzt, wenn die Seifenhaut eine ganz andere Form annimmt, als vielleicht zunächst intuitiv vermutet.

#### And now … the mathematics of it:

The following questions may arise for the visitor during experimentation:

1. “Certain stable surfaces” are formed which remain intact even during violent wobbling. What are these surfaces?

If you look closely — “under the magnifying glass”, as it were — you can discover fascinating regularities. Already the physicist Joseph Antoine Ferdinand Plateau (1801–1883) made (despite blindness) in the second half of the 19th century experiments with edge models in soap suds and found these two — consequently named after him — fascinating properties in the immediate vicinity of edges or points:

• In points, 4 edges always meet (see Figure 2).
• In edges, 3 surfaces always meet at an angle of 120° each (see Figure 3).

These soap skin structures are so-called (local) minimal surfaces, whose area is as small as possible for a given boundary. For each of the structures this means that the total area (of all soap skin surfaces together) is minimal. The mathematical description of such surfaces is very complicated (among other things, it is the subject of differential geometry). Also, many different minimal surfaces can be formed for a given edge model. Indeed, if the experimenter crushes a surface with a (preferably dry) finger, the soap skins immediately “jump” to the next stable state. Minimal surfaces are therefore — except for plane, closed edge curves — not unique.

2. Why do these minimal surfaces form? How can this be explained physically?

Because of the surface tension, the thin soap skins always contract to the smallest possible area. Everyone is familiar with the effect of surface tension from everyday life: Brush hairs, for example, form a small “tip” when dipped in water: At the surface, the liquid particles are “attracted” toward the liquid. A soap skin is a thin layer with two such surfaces, on which the same pressure — namely air pressure — acts from both sides. Therefore, shapes with the smallest possible “mean curvature” and thus minimal surface area are formed.

3. And why is the soap bubble round?

Closely related to the exhibit is the soap bubble, which has fascinated not only the smallest children for centuries. It is created when a soap skin is blown so hard that it breaks away from its ring. The soap skin jumps to the next possible stable state, enclosing a certain amount of air. And because the sphere has the smallest possible surface area with a fixed volume, the soap bubble is round – even if the bubble is blown from a frame with an angular shape, by the way.

4. Can the knowledge be put to good use?

One of the most breathtaking applications can be found in the field of architecture: The roof of the Munich Olympic Stadium was modeled after soap skins on specially designed edge models.

Aiming for minimal surfaces can not only reduce material costs and packaging waste, but also save space: Spiral staircases, for example, are modeled on a surface with the smallest possible content, the helical surface. The DNA double helix is also very similar to this. Nature has also made use of minimal surfaces to store as much genetic information as possible in the smallest possible space.

Different minimal areas can form, whose total area contents do not have to be the same. Rather, they are local minima, i.e. if the soap skins in the structure that has formed were only slightly different in shape, the area content would already be larger than that of the minimal area.

#### Literature

[1] Kühnel, W.: Differentialgeometrie: Kurven — Flächen — Mannigfaltigkeiten, 5. Auflage, Vieweg + Teubner, 2010.

[2] Arnez, A. und Polthier, K., Palast der Seifenhäute, Berlin / Bonn, 1995.