Giant soap skin

In several containers of soap suds lie wires bent into different shapes. When they are taken out of the containers, shimmering soap skins form in the wire shapes, which have one crucial property in common: The skins strive to minimize their surface tension and therefore occupy the (locally) smallest possible areas between the wires.

Figure 1
Figure 2

The giant soap skin in MATHEMATICS ADVENTURE LAND forms between a tub of soapy water into which visitors climb and a large circular ring of wire that they pull up out of the tub on a rope. With a little skill, they manage to pull an initially cylindrical soap skin up so far that the visitors are surrounded all around by the gossamer film from foot to head.

However, the giant soap skin also strives to minimize its (local) surface area and thus the surface tension. That’s 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 of it:

As early as the middle of the 18th century, the mathematicians Leonhard Euler (1707–1783) and Pierre-Louis Moreau de Maupertuis (1698–1759) observed 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 skins first studied a century later by the Belgian physicist and photographic pioneer Joseph A. Plateau (1801–1883) confirm this law of nature. Spherical soap bubbles also enclose a maximum volume with the minimum surface area. Such surfaces are called minimal surfaces.

For example, such minimal surfaces can be generated in the following way. Consider the so-called chain line, which is created when the two ends of a chain are suspended 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: Twisted chain line

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, also already described by Leonard Euler, is also called 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 skin in the MATHEMATICS 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.