Fractals and L-Systems

  • Michael Oberguggenberger
  • Alexander Ostermann
Part of the Undergraduate Topics in Computer Science book series (UTICS)


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.


Fractal Dimension Image Compression Explicit Rule Substitution Rule Complex Derivative 
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Further Reading

  1. 5.
    M. Barnsley: Fractals Everywhere, Academic Press, Boston 1988. MATHGoogle Scholar
  2. 9.
    J.P. Eckmann: Savez-vous résoudre z 3=1? La Recherche 14 (1983), 260–262. Google Scholar
  3. 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
  4. 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

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Institute of Basic Sciences in Civil EngUniversity of InnsbruckInnsbruckAustria
  2. 2.Department of MathematicsUniversity of InnsbruckInnsbruckAustria

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