FRACTALS

INTRODUCTIONA fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far. Fractals may be exactly the same at every scale, they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.

The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. Fractal art is not drawn or painted by hand. It is usually created indirectly with the assistance of fractal-generating software.The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, 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.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, 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.

HISTORY OF FRACTALSGaston Julia (1893-1978) was a French mathematician who published a book on the iteration of rational functions in 1918. Before computers, he had to draw the sets of functions by hand. These types of fractals are now called Julia sets.

Benoit Mandelbrot (1924-2010) is an emeritus professor at Yale University. He used a computer to explore Julia's iterated functions, and found a simpler equation that included all the Julia sets. This Mandelbrot set is named after him.

The mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). In his writings, Leibnitz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. It was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable. Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals. Also in the last part of that century, Felix Klein and Henri Poincar introduced a category of fractal that has come to be called "self-inverse" fractals.

CHAOS THEORY AND FRACTALSChaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy.Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect.

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.The Repeating Patterns looks Chaotic but when we find the Right Fractal behind the Order in Chaos we can know where the Path is Moving to (the Attractor).

Fractals immeasurably enhance this world-view byproviding a description of much around us that is rough andfragmented of objects that have structure on many sizeslike mountains, coastlines, rough surfaces, etc. So giventhat kind of details enough to describe dynamically createdobjects; fractals are considered the geometry of chaos. Fractals are used to describe attractors, texture and otherchaos related geometric objects. Chaotic attractors are defined as attractors with fractal structure or fractalsgeometric property. So we can clearly see the relationbetween chaos and fractalsFractals can help detect chaos