XIII F:Role of Mathematics behind seemingly trivial but intricate soap bubbles.

XIII F: Role of Mathematics behind seemingly trivial but intricate soap bubbles.
           [Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.]
            Soap bubbles are no longer a matter of child`s play today but a gate way of an intricate mathematical world to be explored.

Nearly all of us have been enchanted sometimes in our lives by fragile smmetry of soap bubbles and soap films. It seems trivial and a matter of child’s play, but really not so. It is a form of natural pattern and has therapeutic, designing and architectural uses at present. But we have had no privilege of being introduced to them by a great scientist.

In this context, it may be said that art and science become inevitably intertwined in our appreciation of nature’s forms. Physics play a role in striking aesthetic impact of bubble patterns. Furthermore, the foams exhibit a balance between similarity and intricacy that symbolizes many of the complex systems that permeate nature and society.

Galileo Galilei’s famous declaration: “Nature’s great book is written in mathematical symbols”, suggests that key to unlocking nature’s secret lies in underlying science, quantifying the geometrical specifications behind nature’s pattern often provides step in discovering ‘how’ and ‘why’ of their formations. Interestingly, it might also serve as the springboard for explaining their aesthetic value. We might be drawn to nature’s pattern because they are direct manifestation of natural laws that dictate our lives. In this way, art and science become inevitably intertwined in our application of nature’s forms.

Now, let us come to the pattern of bubbles and foams. What rules drive bubble formation? Soap bubbles serve as the perfect illustration of physical objects striving to minimize their surface area. For example, the beautiful simplicity of a spherical soap bubble occurs because this is the optimal shape for enclosing a given volume of air, within a surface with a minimal area. Whereas this fundamental fact has been known for several centuries, science becomes more challenging as the number of bubbles increases. Consequently, foams consisting of many bubbles feature more intricate shapes compared to the simple sphere of isolated bubbles. [Pic. No. 20a: Foam bubbles]

 

       Pic.NO20a.

                                                                     Foam bubbles.

The search for the pattern that most efficiently packs bubbles into foam is known as “Kelvin Problem” after the name of the great Scottish physicist and also a mathematician and engineer Lord Kelvin also called Sir William Thomson (1824 – 1907), who profoundly influenced scientific thoughts of his generation. He was the first to focus on this challenge back in 1887. But what was that challenge?

The challenge is : to find an arrangement of cells or bubbles of equal volumes, so that total surface area of the walls between is as small as possible, in other words, to find the most efficient soap bubble foam. The problem is relevant to bone replacement materials because bone tissues has a honeycomb – like structure, called Kelvin’s Structure which you will get when you cut the corners off a three dimensional diamond shape. In this context it will be worthy to mention that this structure may lead to advancement in uses like hip replacement and replacement of bone tissues for bone cancer patients.

Kelvin’s structure believed to be the most efficient solution of over 100 years until physicists Denis Weaire of Trinity College, Dublin and his student Robert Phelan found a better structure than that of Kelvin’s in 1993 using computer Simulation which is now known as “Weaire- Phelan structure”.

This structure since become etched into public’s memory for reasons beyond its unique packing properties, fame spread into the field of architecture and its distinctive form appeared on the TV screen across the globe when the structure served as the inspiration for design of the Beijing National Aquatic Centre for 2008 Olympic swimming venue in Beijing, China.

The foam pattern of that water cube is shaped by more than 22,000 steel beams. It measures 177 meters long, 177 meters wide and 30 meters high and cover an area of 62,950 sq. meters. It has four floors. The resulting structural system is inherently strong and light weight. The frame work fills a large volume of space with a reduced amount of materials similar to a hexagon in two dimensions.

Many researchers after Weaire and Phelan have applied their minds for betterment of the foam pattern structures. The study of foams, seems to be a subject poised very delicately between complexity and symmetry and one in which final word has not yet been said. [Pic. No.-20b: Water cube of Beijing Olympic]

Pic. No.-20b

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                                                         Water cube in Beijing Olympic.
                                                                                                                                 [To continue]

 Reference Internet:
                                 Images are downloaded from Public Domain.