XIIIB: Role of Mathematics behind amazing self similarities in nature nature.

 XIII B: Role of Mathematics behind amazing self similarities in nature.
             [Contd. A Journey to the Wonderland of Math. by Ajay Kumar Chaudhuri.]

               " My life seemed to be a series of events and accidents.Yet when I look back,I feel a pattern."------Benoit B.Mandelbrot.

Another spectacular pattern we observe in the natural world around us is “fractals”. But what is it? It is essentially a new branch of mathematics and art propounded by Benoit Mandelbrot (1924 – 2010), a Polish born (in Warsaw), French and American mathematician with broad interests in practical Sciences, regarding what he labelled as “the art of roughness of physical phenomena” and “uncontrolled elements of life.” He published this innovation in his highly successful book “The Fractal Geometry in Nature” in 1982.  He also wrote many articles on this idea. Mandelbrot’s work is a stimulating mixture of conjecture and observation, both into mathematical process and their occurrence in nature. He is universally known as the father of fractals. This idea has been employed to describe diverse behavior in economics, finance, the stock market, astronomy and computer science.

Most physical systems of nature and many human artifacts are not regular geometric shapes of standard geometry derived from Euclid. Fractal geometry offers almost unlimited ways of describing, measuring and predicting these natural phenomena. But is it possible to define the whole world using mathematical equations?

Many people are fascinated by the beautiful images termed fractals. Extending beyond the typical perception of mathematics as a complicated, boring formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers what makes fractals even more interesting is that they are the best mathematical descriptions of many natural forms, such as coastlines, mountains or parts of living organisms.

Although the fractal geometry is closely connected with computer techniques, some people had worked on fractals long before the invention of computers. Those people were British Cartographers who encountered the problem in measuring the Britain coast. It is to be borne in mind that geographical curves are so involved in their detail that their lengths are often infinite or more accurately indefinable. However many are statistically “Self-similar” meaning each portion can be considered a reduced- scale image of the whole. This very problem was faced by those Cartographers in measuring the length of Britain’s coastline. The coastline measured on a large scale map was approximated half the length of coastline measured on a detailed map. The closer they looked, the more detailed and longer coastline became. They did not realize that they had discovered the main properties of fractals.

Without going to discuss elaborately the concepts and properties of fractal geometry, we can definitely assert that the most important characteristic of fractals is self–similarity. But what does self-similarity mean?

To get an idea of it in reality, let us look carefully at a fern leaf. We will notice that every little leaf, which is part of the bigger one, has the same shape as the whole fern leaf [Pic. No. 15a]. We can say that the fern leaf is self-similar. It is true for all fractals: We can magnify them many times and after every step we will see the same shape, which is the characteristic of that particular fractal.

 

                           Pic. No. 15a

                                                              Fern leaf.

There are a lot of different types of fractals, Fractal like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geological fault lines, mountains, coastlines, snowflakes, crystals, blood vessel branching and ocean waves and so many in the natural world.

Here are some amazing pictures showing fractals in nature:  as in Broccoli [Pic. No. 15b], Pineapples [Pic. No. 15c], common loosely fractal features of mountain ranges  and shorelines  [Pic.No15d(i) and d(ii) respectively]   Tree branches, cracks in the ground and lightning bolts possess same fractal [Pic. No.15e (i),(ii) and (iii) respectively. ] A river delta, human kidney and fractal image of deciduous tree in winter also possess same fractal [Pic. No.15f (i),(ii) and (iii) respectively.]                 

  

                                Pic. No.- 15b

                                                                                                                 

                                                 Fractals in Broccoli and Cauliflower.

               

                                  Pic. No.- 15c                                                             

                                                         Fractals in Pineapples.

                     

                      Pic.No15d(i)

                                                                  

                                                                     Mountain ranges.

                     Pic.No15d(ii)

                                                                      Shorelines.

                    pic.No15e(i)

                      Pic.No15e(ii)                                                   Tree branches.

                                                              Cracks in the ground.

          Pic.No15e(iii)

                                                                 Lightning bolts.

                     Pic.No15f(i)

               

                                                                           A river delta.

                      Pic.No15f (ii).

                                                             Section of a human kidney.
               Pic.No15f (iii).

                                                          A deciduous tree in winter.

Fractals, first named by its inventor Benoit Mandelbrot in 1975, are special mathematical sets of numbers that display similarity through the full range of scale, which means, they look same, no matter how big or how small they are,. Another characteristic of fractals is that they exhibit great complexity driven by simplicity, for some of the most complicated and beautiful fractals can be created with an equation containing with just a handful of terms.

Fractals are ubiquitous in nature. The laws that govern the creation of fractals seem to found throughout the natural world. Pineapple grows according to fractal laws and ice crystals form in fractal shapes, the same ones that show up in river deltas and veins of our bodies. In fact, no artist of world-fame, no great architect or famous designer can excel Mother Nature in their creation. So, it is often been said that Mother Nature is a nightmare for a good designer, and fractals can be thought of as the design principles follows when putting things together. Fractals are hyper efficient and allow plants to maintain exposure to sunlight and cardiovascular system to most efficient transport oxygen to all parts of the body. Fractals are beautiful wherever they pop up, so there are plenty of examples to share.

               Reference: Internet
                                 All the images (except otherwise stated ) in this article are downloaded from Public Domain.
                                Pic.No15f (ii) Kidney Anatomy----Attribution: (https://en.wikipedia.org/wiki/Kidney)
                                                                                                                                [To continue]