Fractals

• Authors: Dietmar Saupe
• Venue: Encyclopedia of Computer Science
• Year: 2003

Much scientific research of the past has analyzed human-made machines and the physical laws that govern their operation. The success of science relies on the predictability of the underlying experiments. Euclidean geometry-based on lines, circles, etc.--is the tool to describe spatial relations, where differential equations are essential in the study of motion and growth. However, natural shapes such as mountains, clouds or trees do not fit well into this framework. The understanding of these phenomena has undergone a fundamental change in the last two decades. Fractal geometry, as conceived by Mandelbrot, provides a mathematical model for many of the seemingly complex forms found in nature. One of Mandelbrot's key observations has been that these forms possess a remarkable statistical invariance under magnification. This may be quantified by a fractal dimension, a number that agrees with our intuitive understanding of dimension but need not be an integer. These ideas may also be applied to time-variant processes.