Fractals and L-Systems

Abstract

In geometry objects are often defined by explicit rules and transformations which can easily be translated into mathematical formulae. For example, a circle is the set of all points which are at a fixed distance r from a centre. In contrast to this, the objects of fractal geometry are usually given by a recursion. These fractal sets (fractals) have recently found many interesting applications, e.g., in computer graphics (modelling of clouds, plants, trees, landscapes), in image compression and data analysis. Furthermore, fractals have a certain importance in modelling growth processes.

Typical properties of fractals are often taken to be their non-integer dimension and the self-similarity of the entire set with its parts. The latter can frequently be found in nature, e.g. in geology. There it is often difficult to decide from a photo without a given scale whether the object in question is a grain of sand, a pebble or a large piece of rock. For that reason fractal geometry is often exuberantly called the geometry of nature.

In this chapter we exemplarily have a look at fractals in ℝ² and ℂ. Furthermore, we give a short introduction to L-systems and discuss, as an application, a simple concept for modelling the growth of plants.