Giant soap film

Wires bent into different shapes lie in several containers of soapy water. If you take them out of the containers, shimmering soap films form in the wire forms, which have one decisive property in common: The skins strive to minimise their surface tension and therefore occupy the (locally) smallest possible areas between the wires.

Figure 1
Figure 2

The giant soap film in the Maths Adventure Land is formed between a tub of soapy water, which visitors climb into, and a large circular wire ring, which they pull up out of the tub on a rope. With a little skill, an initially cylindrical soap film can be pulled up so high that the visitors are surrounded by the gossamer film from foot to head.

However, the giant soap skin also strives to minimise its (local) surface area and thus the surface tension. That is why the cylinder shape tapers in the middle and contracts to an increasingly slender waist until the skin simply bursts after a few seconds at the latest.

And now … the mathematics:

As early as the middle of the 18th century, the mathematicians Leonhard Euler (1707–1783) and Pierre-Louis Moreau de Maupertuis (1698–1759) noted that nature strives for the greatest possible economy. It always strives for the conditions that require the smallest expenditure of energy and material. The soap films first studied a century later by the Belgian physicist and photographic pioneer Joseph A. Plateau (1801–1883) confirm this law of nature. The spherical soap bubbles also enclose a maximum volume with the minimum surface area. Such areas are called minimum areas.

For example, such minimal surfaces can be created in the following way. Consider the so-called chain line, which is created when you hang the two ends of a chain at the same height (see figure 3):

Figure 3: The catenary

Mathematically, this catenary is defined by the so-called cosine hyperbolicus:

    \[y=f(x)=\cosh(x)=\frac{e^x+e^ {-x} } {2} .\]

(x\in\mathbb R a real variable). If we now rotate this curve by 90°, we obtain the following diagram (Figure 4):

Figure 4: Rotated catenary

Now we let this curve rotate around a vertical axis on the “belly” side. This then creates a surface of rotation that corresponds to the ideal shape of the giant soap skin.

Figure 5: Catenoid

This surface, which was also described by Leonard Euler, is also called a chain surface or catenoid. It is defined by the equation

    \[\sqrt{x^2+y^2}=c\cdot\cosh(z/c)\]

with a real parameter c > 0.

Figure 6: Giant soap film in the Maths Adventure Land

Literature

[1] Beutelspacher, A.: Mathematik zum Anfassen, Gießen, 2005.

[2] Hildebrandt, St. A.: Tomba, Kugel, Kreis und Seifenblasen, Optimale Formen in Geometrie und Natur, Basel, 1996.

[3] Jacobi, J.: Minimalflächen, Universität zu Köln, 2007.

[4] Nitsche, J.C.C.: Vorlesungen über Minimalflächen, Berlin / Heidelberg, 1975.