In the mid 70’s when people were playing around with early versions of computers, a mathematician was plotting the results of an equation containing a complex number, in a reiterative manner. And to his surprise, the results were quite fascinating: the patterns that emerged were beautiful to look at; but more interestingly, when you zoom into a small part of the pattern, it seemed to bring out a pattern that was so similar to the original. Similar, but different. And the mathematical discipline of fractals was born. It did not take long for people to recognize that this was an important discovery. Fractals had fractal dimensions. A river is not a straight line, mountains are not cones and clouds are not spherical. The new discipline brought out the fractal dimensions of natural structures. Mathematicians could paint trees, mountains, rivers and clouds with equations. And they looked more realistic than any painter could ever imagine. Scientists were suddenly seeing fractals everywhere. Terms such as scaling laws and power laws started appearing in almost all disciplines.
Soon mathematicians realized a hidden complexity: fractals that keep changing their dimensions in different directions. And the notion of multi-fractals was born. This too spread to other disciplines like wild fire. Patterns in time and space that hitherto seemed too complex to explain suddenly became simple enough. Even the complexity of literature, music, painting and other art forms came under the purview of this simplicity.
The study of science has always tried to model the present phenomena to predict the future. With different fields of the study of science evolving with the same goal, a modern branch of study has developed which finds its basics in the famous quote “The present determines the future, but the approximate present does not approximately determine the future”. Chaos Theory has recently turned fifty, celebrating more than half a century of flapping butterfly wings in Brazil and creating tornadoes in Texas. It was a meteorologist named Edward Lorenz who first outlined why seemingly consistent and knowable systems can still go wildly wrong.
Fractals are geometric shapes that are very complex and infinitely detailed. You can zoom in on a section and it will have just as much detail as the whole fractal. They are recursively defined and small sections of them are similar to large ones. One way to think of fractals for a function f(x) is to consider x, f(x), f(f(x)), f(f(f(x))), f(f(f(f(x)))), etc. Fractals are related to chaos because they are complex systems that have definite properties.
The word fractal was first introduced by Mandelbrot and Ness (1968)  and laid the foundations for fractal geometry. He also advanced fractals by showing that fractals cannot be treated as whole-number dimensions; they must instead have fractional dimensions. Calculation of fractal dimensions or rather measuring self-similarity has been a major area in the field of study of chaos.
- Fractional Brownian Motions, Fractional Noises and Applications, SIAM Review 10(1968), no. 4, 422–437. ,