I wonder if you can also create “angular soap bubbles“? Of course! By dipping various edge models in soapy water, the most diverse, fascinating shapes are created. So it even proves to be 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 centre, to which soap skins stretch from the edges of the cube:
Figure 1: Soap skins on the cube
The exhibit invites visitors to examine the forming soap skin structures on edge models such as prisms, cylinders or tetrahedrons. In the process, many a visitor is astonished not only once when the soap skin takes on a completely different shape than perhaps initially intuitively suspected.
Visitors can ask themselves the following questions while experimenting:
1. “Certain stable surfaces” are formed which remain intact even during violent wobbling. What are these areas?
If you look closely — “under the magnifying glass“, so to speak — you can discover fascinating regularities. Already the physicist Joseph Antoine Ferdinand Plateau (1801–1883) made experiments with edge models in soap suds in the second half of the 19th century (despite blindness) and found these two — consequently named after him — fascinating properties in the immediate vicinity of edges or points:
Figure 2: Meeting of exactly four edges in points
Figure 3: Meeting of three soap skins at an angle of 120° each in edges
These soap skin structures are so-called (local) minimum surfaces whose area is as small as possible for a given edge. For each of the structures, this means that the total surface area (of all soap skin surfaces together) is minimal. The mathematical description of such surfaces is very complicated (it is the subject of differential geometry, among other things). Also, many different minimum surfaces can be formed for a given edge model. 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 flat, closed boundary curves — not unique.
2. why do these minimal surfaces form in particular? How can this be explained physically?
Due to 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 they are immersed in water: At the surface, the liquid particles are “attracted” towards 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 minimum surface area are formed.
3. And why is the bubble round?
Closely linked to the exhibit is the soap bubble, which has fascinated not only the youngest visitors for centuries. It occurs when a soap skin is blown so hard that it detaches 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 – by the way, even if the bubble is blown out of a frame with an angular shape.
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 modelled on soap skins on specially designed edge models.
Figure 4: The roof of the Munich Olympic Stadium (source)
Aiming for minimal surfaces can not only reduce material costs and packaging waste, but also save space: Spiral staircases, for example, are modelled on a surface with the smallest possible content, the screw surface. The DNA double helix is also very similar to this. Nature has also made use of minimal surfaces in order to store as much genetic information as possible in the smallest possible space.
Different minimum areas can be formed 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 would already be larger than that of the minimum area.
[1] Kühnel, W.: Differentialgeometrie: Kurven — Flächen — Mannigfaltigkeiten, 5. Auflage, Vieweg + Teubner, 2010.
[2] Arnez, A. und Polthier, K., Palast der Seifenhäute, Berlin / Bonn, 1995.