Abstract
Fractal geometry can help us to describe the shapes in nature (e.g., ferns, trees, seashells, rivers, mountains) exceeding the limits imposed by Euclidean geometry. Fractal geometry is quite young: The first studies are the works by the French mathematicians Pierre Fatou (1878-1929) and Gaston Julia (1893-1978) at the beginning of the 20th century. However, only with the mathematical power of computers has it become possible to realize connections between fractal geometry and other disciplines. It is applied in various fields now, from biology to economy. Important applications also appear in computer science because fractal geometry permits us to compress images, and to reproduce, in virtual reality environments, the complex patterns and irregular forms present in nature using simple iterative algorithms executed by computers. Recent studies apply this geometry to controlling traffic in computer networks (LANs, MANs, WANs, and the Internet). The aim of this chapter is to present fractal geometry, its properties (e.g., self-similarity), and their applications in computer science.
TopBackground: What Is A Fractal?
A fractal could be defined as a rough or fragmented geometric shape that can be subdivided in parts, each of which is approximately a reduced-size copy of the whole (Mandelbrot, 1988).
“Fractal” is a term coined by Benoit Mandelbrot (b. 1924) to denote the geometry of nature, which traces inherent order in chaotic shapes and processes. The term derived from the Latin verb “frangere”, “to break”, and from the related adjective “fractus”, “fragmented and irregular”. This term was created to differentiate pure geometric figures from other types of figures that defy such simple classification. The acceptance of the word “fractal” was dated in 1975. When Mandelbrot presented the list of publications between 1951 and 1975, date when the French version of his book was published, the people were surprised by the variety of the studied fields: linguistics, cosmology, economy, games theory, turbulence, noise on telephone lines (Mandelbrot, 1975). Fractals are generally self-similar on multiple scales. So, all fractals have a built-in form of iteration or recursion. Sometimes the recursion is visible in how the fractal is constructed. For example, Koch snowflake, Cantor set and Sierpinski triangle are generated using simple recursive rules. The self similarity, the Iterated Function Systems and the Lindenmayer System are applied in different fields of computer science (e.g., in computer graphics, in virtual reality, and in the traffic control of computer networks).