If we try to fill a glass of soda without letting it overflow, we usually notice the bubbles or froth of the drink. As we fill the glass, bubbles form on the surface and these tiny bubbles grow. Reaching a certain size, the bubbles escape from the liquid surface, and vanish into the air. If we put our finger, or a straw into the soda-as most of us did as children- we immediately notice that tiny bubbles of gas form on the object immersed in the glass. Just like in the freezing of water or in the escape of gas from soda, a precise energy exchange occurs at the initial stage of any phase transformation. Completion of any phase transformation – freezing or condensation (clouds transforming to rain)- is impossible without such precise energy exchange. The fact that all these phase transformation occur with precise energy calculations in the best possible temperature ranges to support life is a clear proof that nothing in the universe was created by mere coincidence, and that everything occurs by the command of the Almighty.

We know that everything in the universe obeys the minimum energy principle. If we want to freeze water, all we have to do is to cool it to a temperature below 0°C, and the transition from water to ice begins. Water molecules tend to gather together to form clusters. When five to ten of these molecules bond together, however, a difficulty is encountered. The formation of solid-liquid, solid-gas, or liquid-gas interfaces requires a specific amount of energy. In the beginning, the surfaces of these clusters are quite large as compared to their volumes such that the energy they receive to form an interface is much greater than the energy they release; therefore the state of minimum energy is not reached. To explain this to you in another way: let us assume that we manufacture beads for the production of costume jewelry and garments, and the surface of the beads requires treatment. If the beads we manufacture are smaller than the specific size, they will be more expensive to treat, and therefore will not cover the costs, so only producing beads exceeding the specific size will be profitable to the manufacturer. The main aspect here is actually the size of the beads, so if manufacturing beads which exceed the specific size is simpler and more profitable, rejecting the beads smaller than these specifications would be inevitable.

As in this example, because of their high energy value, the molecular clusters formed initially (embryos) return to a liquid form. Then once again the particles begin to bond, but again the result is the same. An embryo must grow to a certain size for its surface area to decrease in comparison to its volume and thus reduce its energy. This is only feasible when many atoms bond, for only when a sufficient number of atoms join together does the embryo transform into a nucleus, and then begin to crystallize and eventually become solid. The process called homogeneous nucleation is only possible under certain conditions: the liquid must be at a temperature of around –40 C^o for both the transition in the balance of energy, and for the water molecules and atoms to become solid and bond to form a nucleus. If we contain pure water totally motionless in the deepfreeze at approximately –8 C^o, we will have supercooled water that has not yet transformed into ice; the temperature between the nucleation and the freezing points, is called supercooling. Supercooling is a metastable condition where liquid or gas remains supercooled without actually becoming frozen, but the slightest intervention or movement can cause the substance to transform into a solid. The tiny bubbles of carbon dioxide in soda is also in a metastable condition, for as soon as the bubbles have the opportunity, they escape from the liquid and vanish into the air. If we immerse a straw or finger into a glass of soda, this forms an added surface, which also facilitates a solid-gas interface, and if we add a teaspoon of sugar to the soda, this induces the drink to froth and bubble at great speed. Water boiled in a saucepan actually nucleates on the wall of the container.

Supercooling is a metastable form of the substance. Every substance or solution has a specific temperature value for cooling. For instance, liquid copper transforms into a solid at 1083 C^o. Homogeneous nucleation requires the bonding of 310 atoms, and supercooling to approximately 236 C^o.

Under normal conditions, substances which have more than one type of molecule undergo phase transformation known as heterogeneous nucleation. In this case, the atoms form primarily on the walls of a container on particles of impurity, or minute solid particles in the liquid, and this significantly reduces the surface energy barrier for nucleation. So for a moment let us return to the bead example. We have discovered that instead of directly manufacturing smaller beads, it would reduce the costs of decorating the surface of the beads to coat and treat larger beads, so the beads are being produced in this way, thus reducing losses.

Supercooling can occur at temperatures even as high as 2–3 C^o, and this is very important. The condensation of water or supercooled water droplets in clouds must reach a specific size and weight in order to fall to the earth as raindrops. Here, the solid microscopic particles combine to form nuclei. Even if the clouds are much lower in temperature, rain cannot form without nuclei. Particles of salt which escape from the sea, sand that rises from the desert, the sulphate released from the ashes of volcanic activity or minute atoms of dimethyl sulphate emitted by certain planktons are driven into the atmosphere by the wind and form nuclei. As the Almighty, the Creator of the universe revealed in Al-Hijr, verse 22 of the Qur’an: “And We send the winds to fertilize, and so We send down water from the sky, and give it to you to drink (and use in other ways)” indicating that one of the duties of the wind is fertilization. Even the particles in smoke released irresponsibly by humans from industrial chimneys, or from car exhausts form nuclei that eventually transform into rain.

During the foundry process, solid substances are added to liquid metals for certain purposes, such as enabling metal to set more rapidly, or increasing the metal’s durability. When liquid metal is cooled, its atoms form nuclei on microscopic solid impurities. These nuclei increase in size and assemble into groups called grains. The irregular zone between these groups is known as the grain boundary. The grain boundary forces the compressed atoms to move and weld, thus increasing the durability of the metal. This method known as infusion or grain contraction ensures an increase in the formation of nuclei, and also in the durability of the metal. Cloud seeding, a topic which mainly comes to light when there is a lack of rain, is actually inducing the clouds to form artificial nuclei that will in turn produce rain.

Some creatures on earth protect themselves with mechanisms bestowed by their Creator, and one of these creatures is the wood frog. As the water in its cells begins to freeze, the antigel protein found in its blood surrounds the formation of nuclei, and prevents the nuclei from increasing in size. The frog remains frozen and motionless until the temperature increases. If we touched a wood frog in this condition, its cells too would freeze suddenly, and the frog would die. It is impossible for a frog to know how to cool to the point of freezing, and nucleate. It is also impossible for a frog to adapt to such a mechanism because this would require practice and experience, which would of course be deadly. Therefore, is the frog’s ability to freeze, and its process of nucleation not a clear indication of the providence and blessing of God the Almighty?

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.