A fractal, which is short for fractal dimension, is a branch of math that deals with irregular patterns made of parts that are in some way related to a whole. Some examples of this are twigs or tree branches and a property called self-similarity or self-symmetry.
This is different from conventional geometry, which deals with regular shapes and whole-number dimensions. Examples of this are lines and cones. Fractal geometry deals with shapes found in nature that have non-integer, or fractal, dimensions, or lines, like rivers and cone-like mountains. .
Fractal geometry developed from Benoit Mandelbrot's study of complexity and chaos. He published a series of fluctuations of the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on irregular shorelines on the English coast, all beginning in 1961. By 1975 Mandelbrot had developed a theory of fractals that became a serious subject for mathematical study. Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.
He proposed the idea of a fractal as a way to cope with problems of scale in the real world. He defined the fractal as any curve or surface that is independent of scale. This process is referred to as self-similarity and means that any portion of the curve, if it is blown up in scale, will appear identical to the whole curve. Therefore the transition from one scale to another can be represented as iterations of a scaling process.
Chaos is another part of fractal geometry. It is type of dynamical behavior related to fractals. The most obvious feature is sensitivity to initial conditions: tiny changes can grow to huge effects. While short-term prediction is possible in chaotic systems, long-term is not. Inevitable uncertainties in our knowledge of the initial conditions grow to overwhelm the prediction. Yet chaos is not randomness.