Many popular fractal curves are a good example to illustrate the modus operandi of L systems. This is so because of their very recurring nature. Also, they demonstrate how a very compact set of data can generate large and detailed structures.
There are countless fractal curves and similar objects: Sierpinsky triangle, Hilbert curve, Koch islands, Penrose tilings, Peano-Gosper curve, Barnsley's fern leaves or Cantor dust. Many of them can be easily constructed using L systems.
Random walks
Random walks are one of the most simple fractals. They can be easily constructed by starting at a given point and moving on each succesive step to a randomly selected position in its neighborhood.
| Evolution of a random
walk in one dimensional space after 150 steps. Horizontal axis represnts
time and vertical axis position. L system: random walk 1d |
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Properties of random walks offer interesting insight in several physical, chemical and biological processes.
| 2d random walks reflecting
the diffusion of five substances in a plane, starting from the same
central point. L system: diffusion |
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Koch snowflake
In 1904, Swedish mathematician Helge von Koch defined a continuous curve that could not be differentiated. It was just another example of a discovery first made some years before by Karl Weierstrass, but it has lead to more general constructions.
| Outline of Koch island
or snowflake curve after five iterations. L system: Snowflake |
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Instead of using the same rule on every step, an element of chance can be introduced by allowing to switch to the opposite orientation. This simple effect leads to more irregular outlines resembling natural coastlines. However, the fractal dimensions of both figures remain the same: approximately 1.262.
| Outline of random snowflake
curve after five iterations. A couple of stochastic with opposite
orientation rules produces much closer to natural coastlines curves
than the original one. L system: RandomSnowflake |
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The Koch curve, and other similar constructions, have a remarkable self-similarity property. A feature that can be clearly noticed in the corresponding L system definitions, where development rules are self-referenced, i. e., each part contains small replicas of itself.
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More constructions based on Koch island. |
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Space-filling curves
Space-filling curves are continuous curves which completely fill up higher dimensional spaces, such as squares or boxes. When the first examples of those curves were introduced in 1890 by Giuseppe Peano, they seemed to be a paradox, since curves were thought to be inherently one-dimensional. Shortly after him, space-filling curves were also studied by David Hilbert. Today, these concepts are a basis of modern theory of fractals.
| First three iterations
of construction of a three dimensional Hilbert curve, made out of
of 5110 individual pieces, filling up a box. L system: Hilbert 3d |
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Other space-filling curves are: Peano curve, Peano-Gosper curve, Dekking's Church or Heighway's Dragon curve.
| A row of trees. L system: row of trees |
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Kolams
Many decorative patterns resemble fractal curves. Kolams are Indian ornamental motifs created by women to protect and purify their houses. They trace with rice symmetrical lines on the ground that may be wavy and rounded on the edges of the corners.
| Three consecutive stages
in the formation of a snake kolam pattern. L system: snake kolam |
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References
Chaos and fractals: new frontiers of
science
Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe
Springer-Verlag, 1992
Eric Weisstein's world of mathematics
http://www.math.smith.edu/~phyllo
Fractal cities
Michael Batty, Paul Longley
Academic Press, 1994
Mathematics and Art Project
http://www-uk.hpl.hp.com/brims/art/index.html
Classic iterated function systems
Larry Riddle
Department of Mathematics
Agnes Scott College
http://www.agnesscott.edu/aca/depts_prog/info/math/riddle/ifs/ifs.html
Geometría fractal
Departamento de Matemáticas
Universidad de Oviedo
http://coco.ccu.uniovi.es/geofractal
