The surfaces of soap bubbles have a very important feature. These surfaces which have minimum surface-tension potential energy also have minimum areas. That is, soap bubbles or clusters have a natural tendency to minimize area for the volumes they enclose. Two different frames and the areas formed are shown in Figure 1. For a given closed frame, at least one such minimal area can be formed; however, mathematicians have had to strive to prove it.

The famous mathematician Richard Courant (1888–1972), together with his students, did soap bubble experiments with various frames.

Minimal areas can also be formed by using more than one closed frames. Figure 2 shows the minimal surfaces obtained by holding two circular frames parallel. If the frames are kept too far from each other, no surface will form. If they are kept sufficiently near to each other, surfaces similar to those shown in Figures 2a and 2b will be obtained. If they are kept close enough to each other, then three minimal surfaces adjacent to each other as shown in Figure 2c can form.

The minimum energy principle is commonly observed not only in living organisms, but also in lifeless matter. A chain will take the shape which produces the least potential energy of attraction when it is fastened at two points onto a rod as shown in Figure 3. This form (function) is called “catenary” in mathematics.

The areas which are formed as a result of rotating the catenary curve around an axis A are called catenoids. Two different types of catenoids are shown in Figures 4a and 4b. As presumed, catenoids are minimal areas and can be obtained by the use of soap bubbles. If such formations were selected and used in everyday utensils, such as glasses, dishes and so forth, ideal shapes which cause the least loss of heat could be designed.

If the katenoid shown in Figure 5a is cut from its edge as shown in Figure 5b and turned by being slightly extended, a helical form or a helicoid will be obtained as shown in Figure 5f. This helicoid also is a minimal surface. Architects have widely used this form in spiral-shaped staircase structures. See Fig. 6.

Fig. 1-Closed frames and soap film surfaces formed1 Fig. 2a&b–Single foam film surfaces over two parallel circular frames1 Plus, the perpetual screw system which is widely in use in technology is also in a form similar to this geometry.

If a cylinder of the smallest volume that can house a helicoid is drawn (Fig.7a) and the lines on which the surface and the cylinder intersect are marked, then a double helix structure (Fig. 7b) is obtained; this is used in modeling DNA molecules which are the genetic codes of living species.

There are two elementary principles related to soap bubbles. The first principle says that if a bubble touches a surface that supports it, it unites with that surface in a way to make 90° angles. The soap bubble on that plain surface forms into a semi-spherical shape and the angle between the bubble surface and the supporting surface will be 90° at every point of contact. The second principle says that if three soap bubble surfaces come together, they form 120° angles along a line. If soap films come together within a tetrahedron frame as in figure 8, then the angles between the lines will be 109° 28′ 16″.

The Steiner problem which is an elementary problem in mathematics can be solved by the application of the 90° and 120° principles. The Steiner problem investigates how n points over a surface can be united in the shortest way by a web. Two transparent surfaces are connected with thin and parallel pins of equal lengths and then dipped into a soapy solution. When it is taken out, soap films will form. These films have a 90° angle with the supporting transparent surfaces and when three soap films come together, they connect at 120° angles with one another.

When observed from above, the intersecting lines between the soap film and one of the surfaces give the shortest web which unites the points in n numbers. How four points are united is shown in Figure 9a and how five points are united is shown in Figure 9b. Someone seems to have equipped lifeless objects such as soap bubbles with the ability to solve complex problems like a math genius.

Periodically repeated minimal areas have been observed on walls separating organic and inorganic substances in the skeletons of certain sea animals like the sea urchin and the starfish. Figure 10 shows the micro structure of a sea urchin’s skeleton. It has calculated that the geometry of its skeleton has perfectly been shaped in such a way as to prevent the extension of possible cracks.

Experiments with soap bubbles have been a source of inspiration also for architects. Such experiments have yielded inspiration for roof and tent designs. The German architect Frei Otto is one of the most eminent names in this regard. Figure 11 shows the minimal areas which Frei Otto managed to obtain by dipping hair-thin threads in soapy water.

In order for such a soap bubble model to be converted into an architectural structure, it is carefully photographed and precisely measured. Later, solid models are made and tested in wind tunnels. The tensile pressures likely to form under loads of wind and snow are measured by special precision instruments. In real structures, thin steel cables having high tensile strengths replace the hairy threads, and transparent plastic and synthetic materials replace the soap bubble film.

Figure 12 shows roof of the Munich Olympic Stadium, Figure13 shows the roof of the Munich Olympic Athletic Arena and Figure 14 shows the roof of the Olympic Swimming Arena in the same city. All these roofs have been designed and erected using the minimal surfaces obtained from soap bubble experiments.

Children love playing with soap bubbles very much; they usually blow a round circle after dipping a wand into soapy water and then watch the bubbles flying out of it. However, it is not only children who play with soap bubbles and soap films. Scientists have, for hundreds of years, been doing experiments with soap bubbles, developing mathematical theories, obtaining various surfaces and transferring the compiled data into technology.

Experiments with soap bubbles have been a source of inspiration also for architects. Such experiments have yielded inspiration for roof and tent designs.

2) These roofs can easily be erected, dismantled and transported to elsewhere, whereas traditional buildings cannot easily be re-located.

3) These structures which are designed according to tensile strengths are very sturdy all over, whereas the tensile pressures of classical buildings are so high that extremely heavy materials such as concrete and brick are used in order to balance the pressure.

The structures of light and strong materials granted to living things are splendid. The lightness and endurance of our skeleton system, the perfect endurance in the stems of slender plants such as wheat and barley, the extremely thin and elastic structure of a fly’s wing, and thousands of similar examples can be given.2 The word of German architect Frei Otto in this subject are expressive: “Biology has become indispensable for architecture.” Witnessing similar perfections also in inanimate structures such as soap bubbles proves that laws in nature originate from the same hand.

Obtaining minimal surfaces has become much easier as a result of immense increases in computer capabilities. Extremely complicated minimal surfaces which can be obtained through computer-aided-designs and calculations have become easily available as alternatives to soap bubble experimentations. If we, as human beings, are aiming to realize developments in science and technology, we should look,more carefully and meditatively, at events which are seemingly simple and unimportant around us and we should also discover the beauties and perfections that God has granted us and put them into service of humanity. The more our designs are compatible with the laws of nature, the higher our chances of success will be.

References

• S. Hilderbrandt, A. Tromba, The Parsimonious Universe, Springer-Verlag, New York, 1996.
• M. S. Polatöz, Tabiatta Mühendislik (Engineering In Nature), Kaynak Publications, Izmir, 2003.
• A. B. Smith, The stereom microstructure of the echinoid test. Special Papers in Palaeontology, 25, 1–85, 1981.

In order to get a sharp image, ideally one point on the image (film) should receive light rays from only one point on the object. In practice, this ideal case can not be achieved in case of a large pinhole. If we have a large pinhole, as seen in Fig. 4, the light rays emerging from one point on the object reaches multiple points on the film. Also, multiple points on the object can arrive at a single point on the film. The result of both cases is a blurry image. For a small pinhole, however, there are light rays emerging from a point on the object in all directions, and only a very small amount of light is received on the film. To produce a decent image on the film, the exposure time needs to be very long, especially when a very small pinhole is used. Of course, we can make the pinhole a little larger to capture more light. But, wait! This would make the image blurry. Therefore, for pinhole photography, the size of the pinhole sets the quality of the image and the minimum exposure time. Also, we can take the picture of a still object, but a moving object cannot be captured easily by a pinhole camera due to the long exposure time. One important question is: Can we make reduce the size of the pinhole size to increase the quality of the image? Well, physics says: “No!” Fig. 5 shows a comparison of the images of a filament taken by a pinhole camera.⁶ As the size of the pinhole gets smaller and smaller, the effects of the diffraction phenomenon are more and more pronounced, and the image becomes blurry again. We also notice that the image is barely formed with a very small pinhole, indicating that very little light is received.

How can we gather more light and still make the image sharp? Can we achieve this with just the pinhole? Apparently not. That is why we mention lenses when talking about any imaging system. This is what a lens basically does. A lens gathers more light, and still preserves the one-to-one correspondence between the points on the object and those on the image. So, a lens is very useful for imaging. Most cameras have one or multiple lenses. Our eyes have lenses. We should remember, though, a pinhole camera takes a picture, but the compromise is the exposure time. Now, we see that a lens solves the exposure time problem, but is there a price to pay? To answer this question, let’s take a look at once again Fig. 2, where we see two pictures, one taken by a pinhole camera, and the other one by a lens camera. Look at the pictures carefully, and try to understand the difference before proceeding. As you probably observed, in the pinhole camera image everything in the picture is in sharp focus, from the close-by plants to the far distant beacon, and the clouds in the sky. However, in the lens camera image, only the closest daisy is in focus, and the other daisies, only a few meters away, are blurry. Indeed, we lose the depth of field in our images when using a lens camera. Depth of field can be defined as “The distance between the nearest and farthest points that appear in acceptably sharp focus in an image.” This is actually something we live with everyday. Try to focus your eyes on a mountain far away; the objects that are very close will not be in focus anymore. So, our eyes, consisting of lenses, also have limited depth of field. A nautilus eye, on the other hand, has an infinite depth of field, as it has a pinhole eye. Therefore, the price paid for gathering more light with a lens is that not everything will be in focus.

We can understand why lens cameras have a limited depth of field while a pinhole camera has nearly infinite depth of field if we consider the focusing mechanism of a lens. Given a pre-determined film position, a lens can only form sharp images of an object at a certain distance from the lens. Other points that are farther from or closer than same section of the object will be out of focus. However, if the size of the object is not large, we usually do not notice this effect. When we want to take pictures of close-by objects, then this effect is clearly seen, as Fig. 2 (b) shows.

Actually, we can correct this problem. The image of the farther points of the object forms at a farther point on the film. So, multiple light rays coming from one point will end up on the film. If we place an aperture next to the lens, then we can block some of these light rays. If we make the size of the aperture sufficiently small, we can get a sharp image of the farther point. Our original question about the lenses can be posed once again at this point: By using the aperture we make the image sharper, but what do we lose? Of course, since we block some of the light rays, we lose the light-gathering ability of the lens. Now, if we make the aperture size smaller and smaller, we finally reach the pinhole, and an almost infinite depth of field!

Our eyes also use similar mechanism of placing an aperture. The amount of light allowed to enter each eye is controlled by the iris, a circular diaphragm that opens wide at low light levels and closes to protect the pupil (the aperture) and retina (light detector of the eye) at very high levels of illumination. As illumination changes, the diameter of the pupil (positioned in front of the crystalline lens) reflexively varies between a size of about 2 to 8 millimeters. When illumination is very bright, the pupil narrows and light rays from the side are excluded from the optical pathway.⁷ The result is a sharper image on the retina. A very narrow pupil (approximately 2 millimeters) produces diffraction artifacts that spread the image of a point source on the retina, similar to the image of the filament captured by a very tiny pinhole (Fig. 5)

In conclusion, a pinhole camera is a very instructive tool for learning about imaging concepts. A pinhole camera has an infinite depth of field and the ability to produce sharp images regardless of the distance of the objects. This comes with a price, though: the small size of the pinhole limits the amount of light received by the film, so long exposure times are needed. A lens helps to gather more light, but compromises the depth of field. These concepts are used in imaging technologies, and can be found in the eyes of living organisms.

Notes

1. E. Hecht, Optics, 2nd edition, pp. 199. Addison-Wesley Publishing Co.
2. http://images.encarta.msn.com/xrefmedia/aencmed/targets/illus/ilt/T045986A.gif.
3. http://www.kosara.net/gallery/.
4. http://www.eyedesignbook.com/.
5. http://www.spylife.com/pinholecam.html.
6. http://www.umiacs.umd.edu/~ramani/cmsc426/Lecture3.pdf
7. http://www.olympusmicro.com/primer/lightandcolor/humanvisionintro.html.