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.

One night, when sleeping, a man heard sounds on his roof. There were loud footsteps and he wondered what could be happening. He shouted out the window, Who is it? Is there a genie on my roof? An extraordinary, mysterious man looked into the window from above and said, We are going around by night, searching.

What are you searching for? the man asked.

Camels, he replied.

You must be joking! he exclaimed. Youre looking for camels on my roof?!

Thats right, camels. Just as you seek God on His throne. Is this so very different?

It was this and no more.

No one saw the man on the roof again.

He vanished like a genie in flight.

But did he not make his point about the madness of man,

That a camel which flies is no crazier than he

Who seeks God on the throne of delight?

Silence lies in the ocean,

While words flow through the river.

The ocean waits for you, do not wait for the river.

Look to the ocean and watch its message,

It will come, it will come

Death Makes Everybody Equal

A solitary dervish had taken up his station in the corner of a desert, when a king passed by. Inasmuch as contentment is the enjoyment of a kingdom, the dervish did not raise his head, nor show the king the least attention and, inasmuch as sovereignty is regal pomp, the king took offense, and said: “The tribe of ragged mendicants resemble brute beasts, and have neither grace nor good manners.” The vizier stepped up to the dervish, and said: “O generous man! The sovereign of this land has passed by you; why did you not do him homage, and discharge the duty of obeisance?” He answered and said, “Speak to your sovereign, saying: Expect service from that person who will court your favor; let him moreover know that the purpose of kings is the protection of the people; it is not the purpose of the people to be the subjects of kings. It is for their benefit that the kings glory is exalted, yet the king is but the shepherd of the poor. The sheep are not intended for the service of the shepherd, but the shepherd is

appointed to tend the sheep. Today you may observe one man proud from prosperity, another with a heart sore from adversity; have patience for a few days ’till the dust of the grave can consume the brain of that vain and foolish head. When the record of destiny comes into effect, the distinction of liege and subject disappears. Were a person to turn up the dust of the deceased, he could not distinguish that of the rich man from the poor.”

These words made a strong impression upon the king; he said, “Ask me for something.” The wise man replied, “What I desire is that you will not trouble me again!” The king said, “Favor me with a piece of advice.” He answered, “Now that the good things in life are in your hands, attend to them; wealth and dominion are passing from one hand into another.” 

The Unjust do not Remain Unpunished

A story is told of a profiteer who bought firewood from the poor at a low price, and sold it to the rich at a profit. A good and holy man went up to him and said, “You are a snake, which bites everybody it sees, or an owl, which digs up and fouls the place where it nests. Although your injustice may pass unpunished among us, it cannot escape God, the knower of secrets. Be not unjust with the people of this earth, so that their complaints may not rise up to heaven.” They say the unjust man was offended at these words, turned his face away from the holy man, and showed him no civility. But then, as it is expressed in the Qur’an: He, the glorified God, overtook him amidst his sins. One night, a spark from his kitchen fire flew onto a stack of wood, and caused a fire that burnt up all his belongings, laying him up in bed, tormenting him with the ashes of Hell. The good and holy man happened to be passing and saw that the profiteer was remarking to his friends, “I do not understand how this fire fell upon my dwelling.” The good man said, “It was from the smoke of the hearts of the poor! Guard against the smoke of the sore-afflicted heart, for a sore hidden away will at last gather to a head. Give no heart pain so long as you can avoid it, for one sigh may set the whole world into flames.”

It is said that the following inscription was engraved upon Kai-Khosraus* crown: “For how many years, and for what a continuance of ages, shall mankind walk over my head on this earth? As the kingdom came to me, from hand to hand, so it shall pass into the hands of others.”

* Kai Khosrau was one of the Seljuk Sultans of Anatolia, 1192-1196.

Daylighting is becoming more popular in commercial buildings and manufacturing facilities in the United States. After Thomas Edison invented the electric light, most architects changed their building plans and designed for more and more artificial lighting. As a result, only a very small percentage of the light used in major buildings and facilities came directly from the sun. Recently, the Europeans have realized the importance of daylighting and begun to use a reasonable percentage of sunlight in their buildings. The Rocky Mountain Institute, Snow-mass, Colorado, has investigated the benefits of daylighting and reported that daylighting increased productivity and reduced absenteeism by 15 percent. Also, sunlight reduces the heating and cooling bill.

In 1993, Wal-Mart Stores Inc. opened a prototype store in Lawrence, Kansas, with nine special skylights designed by Andersen Corp., Bayport, Minnesota. The architectural firm of Leo A. Daly, Omaha, Nebraska, opened an office building in 1983 with l5ft high window walls and a glazed roof. About 50% of its electricity bill for lighting was saved as a result (Reno Gazette-Journal, November 27, 1995). Daylighting is an inexpensive way of lighting interiors since the sun is a ‘free’ light source which can be further exploited by installing skylights on the roof. On an overcast day, the light entering through a 2 sq. ft skylight area is equivalent to three 100-watt light bulbs.

There are two additional reasons for installing skylights. First, the cooling load of a building can be reduced by using daylight. The reason for this is that whereas about 80% of the power of an electric light is converted to heat, sunlight has a far lower heat content and therefore requires far less air-conditioning. Second, just at the time when there is the heaviest demand for electricity (and other utilities) from manufacturing facilities, namely during the summer, sunlight is at its most plentiful and available: in short, skylights can significantly reduce peak load stresses and costs.

An illumination level of 50 footcandles (540 Lux) at the work site is the design standard in industrial facilities. A large portion of this illumination level comes from electric lights of fluorescent and incandescent lamps. Almost 5% of electricity consumption in the US is used up to provide adequate illumination in commercial and industrial buildings.

In production facilities generally, there is a lack of awareness about the energy conservation potential of skylights. Some manufacturers are so unaware about the cost savings that can be achieved by daylighting that they do not have skylights in their production areas, and leave lamps on in work areas throughout daylight hours. In other places which do have them, skylights have been neglected to the extent that they are so dirty they block the incoming sunlight.

Since heating and cooling load are increased with increased skylight surface area, there is a limitation associated with this measure. Some authorities have suggested that the optimum skylight surface area should be reckoned at between 2 and 4% of roof surface area. However, since the measure depends upon local climate conditions, the range should be allowed to vary between 2 and 10%. In many potential sites, heating only (and not cooling) is the principal consideration. Generally, therefore, building designers with heating costs in mind tend to prefer 2% for cold climates and 10% for warm climates.

Heating load and costs will increase when skylights are installed. However, the increase in heating cost is considerably smaller than the saving from reduced lighting cost. The average unit cost of electricity is three times greater than that of natural gas (typically preferred for heating). In any case, heat loss from the skylights can be minimized by double glazing them. In view of the favourable financial balance and the productivity improvements to be expected from daylight working, the benefits from installing skylights generally offset any negative consequences of doing so.

The amount of savings in electric lighting consumption and costs depends on climate and operating periods. The payback period for installing skylights ranges from one to five years. They can be an expensive roof aperture, but it is relevant to note that the lifetime of a skylight is more than twenty years. The skylights need to be cleaned at least once annually, which means that service and maintenance costs are negligible. Skylights are most cost- effective in uninsulated ceilings in climates, like that of southern California, which have no heating season. In such climates, the workplace roof is typically covered with corrugated metal sheets making skylights both easy and cheap to install: corrugated fibreglass sheets can be cut and fitted in place of the metal sheets wherever the skylights are required.

References 

• MURDOCH, B. J. (1985) Illumination Engineering: From Edison’s Lamp to the Laser, Macmillan Publishing Company, New York.
• NUTFER, D. W., BRITTON A. J. and HEFFINGTON W. M. (1993) ‘Conserve Energy to Cut Operating Costs’, Chemical Engineering, September, pp.126-37.
• PIERSON, J. (1995) ‘Natural light gets warm welcome’, Reno Gazette-Journal, November 27, pp.2ff.