INTRODUCTION:

Fractals are tenuous spatial objects whose geometric characteristics include irregularity, scale-independence, and self-similarity. While natural spatial objects such as coastlines, plants, and clouds have long been treated as fractals of various dimensions [1]. The concept of "fractal" was firstly introduced by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature [2]. A fractal is a mathematical set that exhibits a repeating pattern that displays at every scale. It is also known as infinite symmetry or expanding symmetry. If the replication is exactly identical at every scale, then it is called a self-similar pattern. It is mainly the concept of self similarity repeated again and again in an ongoing loop. Actually while analyzing fractals, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Therefore, Fractals mainly include the idea of a detailed pattern that repeats itself. Fractals are generally different from other geometric figures because shape of fractals depend in which they scale.

For example, if we double the edge lengths of a any polygon, then its area get multiplied by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Similarly, if the we double the radius of sphere, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). But if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer [2]. This power is called the fractal dimension of the fractal. A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. Cantor set is an early example of a fractal geometry explained by Georg Cantor in 18th centaury. The Cantor set has a Lebesgue measure of zero, but, it is also countable infinite. Also, it has the property of being self-similar, meaning that if one magnifies a section of the set, and one obtains the whole set again showing the feature of fractals. In 1904, Swedish mathematician Helge von Koch published paper and constructed a curve by the using geometrical means known as von Koch curve and hence the Koch snowflake (see figure (i), which is three von Koch curves joined together. Koch curve just like the Cantor set is also example of fractal. It has also the property of self-similarity. He mainly aimed to provide an alternative way of proving that functions that were non-differentiable (i.e. functions that "have no tangents" in geometric parlance [3]. In doing so, von Koch expressed a link between these non-differentiable "monsters" of analysis and geometry. This also leads to understanding the fact, that fractals as mathematical equations are "nowhere differentiable. Two French mathematicians, Gaston Julia and Pierre Fatou, developed results in the 1918 that ended up being important to fractal geometry. They studied mappings of the complex plane and iterative functions. Their work with iterative functions led to the ideas of attractors, points in space which attract other points to them; and repellors, points in space that repel other points, usually to another attractor. These concepts are also important to chaos theory. The boundaries of the various basins of attraction turned out to be very complicated and are known today as Julia set as shown in figure (ii).

Figure (i)

Figure(ii)

Characteristics of Fractals:

Fractals are geometric shapes generally rough structures that can be divided into parts, each of which is diminished size copy of the original. The main characteristic of fractals is that they exhibit great complexity driven by simplicity. It is difficult to give the exact definition of fractals but can be elaborated on the idea of self-similarity. According to Falconer [4], the fractals have the following characteristics. Self-similarity, which may be manifested as:

(i)Exact self-similarity: identical at all scales; e.g. Koch snowflake

(ii)Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set, but not exact copies.

(iii)Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines, for example, the Koch snowflake.

Nature and Fractals:

The various fractals has been found in nature showing self-similarity over extended, but finite, scale ranges. Natural fractal include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes[5]. Mathematics is very helpful to explain fractals in nature at different levels. The various natural phenomenon having fractal features include river networks, mountain ranges, coastlines, trees, pineapple, earthquakes, snowflakes, crystals, ocean waves, proteins, heart rates and heart sounds etc.

Figure(iii).Fractals in Plants

Mathematical Examples of Fractal:

In real life, it is very difficult to draw fractals with infinite similar patterns. However, BY using the mathematics , we can draw shapes which is just like the fractals. Some examples of fractals which can be formed mathematically.

(a)Menger Sponge : Menger Sponge was first described by Karl Menger in 1926 to do studies of the concept of topological dimension in mathematics. It is an example of the fractal curve which is known as the Menger universal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet, although it is slightly different from a Sierpinski sponge. This fractal exhibits infinite surface area and zero volume. The construction of a Menger sponge can be done as described:

(i) Begin with a cube (first image).

(ii) Divide every face of the cube into 9 squares. This will sub-divide the cube into 27 smaller cubes.

(iii) Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes (second image). This is a level-1 Menger sponge .

(iv) Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate again and again Similarly, the second iteration gives a level-2 sponge (third image), the third iteration gives a level-3 sponge (fourth image), and so on. Therefore,the Menger sponge itself is the limit of this process after an infinite number of iterations.

Figure (iv). Construction of Menger Sponge

(b) Pascal's Triangle: Pascal triangle is one of mathematical example of fractals. Pascal triangle is a number pyramid in which every number is the sum of the two numbers above.

Figure (v). Construction of Pascal triangle

(c)The Mandlebrot Set:

The Mandelbrot set is the set of complex numbers c for which the function does not diverge when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value. is the square of the previous number plus c. In this we get the collection of points known as the Mandelbrot set. That is ,we can create infinite sequence of numbers according to the above equation. We start with 0. Every new number.

Figure (vi). The Mandelbrot set

REFERENCES:

[1]. Shen Guoqiang," Fractal dimension and fractal growth of urbanized areas int. j. geographical information science, 2002vol. 16, no. 5, 419± 437

[2]. Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5.

[3] Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics, Archived from the original on 4 February 2012.

[4] Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 0-470-84862-6.

[5] Stevens, Peter. Patterns in Nature, 1974. Page 3.

[6] Menger, Karl (1928), Dimensions theories, B.G Teubner Publishers.

[7] Menger, Karl (1926), "Allgemeine Räume und Cartesische Räume. I.", Communications to the Amsterdam Academy of Sciences. English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO, ISBN 978-0-8133-4153-8, MR 2049443.

Received on 10.06.2016 Accepted on 25.06.2016

Research J. Engineering and Tech. 2016; 7(2): 75-78.

DOI: 10.5958/2321-581X.2016.00016.7