Analysis for Computer Scientists pp 111-125 | Cite as

# 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 ℝ^{2} 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.

## Keywords

Fractal Dimension Image Compression Explicit Rule Substitution Rule Complex Derivative## References

## Further Reading

- 5.M. Barnsley: Fractals Everywhere, Academic Press, Boston 1988. MATHGoogle Scholar
- 9.
- 20.H.-O. Peitgen, H. Jürgens, D. Saupe: Fractals for the Classroom. Part One: Introduction to Fractals and Chaos. Springer, New York 1992. Google Scholar
- 21.H.-O. Peitgen, H. Jürgens, D. Saupe: Fractals for the Classroom. Part Two: Complex Systems and Mandelbrot Set. Springer, New York 1992. Google Scholar