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Althought Mandelbrot is known as the father of fractals, we must not forget the influence of the great populariser Martin Gardner. He is the one who got me and countless others interested in dragon curves. His articles appeared in Scientific American in March,1967, page 216, in April, 1967, pages 124-125; pages 118-120, and in July, 1967, pages 115 and 217. And these articles appeared a year before Lindenmayer published his paper "Mathematical Models for Cellular Interaction in Development", Journal of Theoretical Biology, Vol. 18, 1968, pages 280-315. The mathematical model he created is now known as an L-system.
Three years later the great mathematician and computer scientist Donald Knuth teamed up with C. Davis to put their spin on things in two articles in the Journal of Recreational Math. The first was "Number Representations and Dragon Curves, part I", vol. 3,1970, pages 61-81. part II appeared on pages 133-149 in the same issue.
I must have taken up the quest for dragon curves after I read Mandelbrot's first book on fractals. It was called Fractals: Form, Chance, and Dimension, 1977. For, as I recall, I was making dragon curves using a Sol 20 computer (which is still in working order with a 1k operating system in ROM and Bill Gate's 5k BASIC on cassette tape) and a dumb plotter. Home computers were not widely available before 1976.
Making dragon curves was really tough until I ran across an article on fractals by F. M. Dekking, "Recurrent Sets", Advances in Math, Vol. 44, 1982, pages 78-104. There I found the formal definition of an L-system, which he called a recurrent set. I also picked up some insight by examining the figures contained in the article (I ignored the mathematics of Hausdorf dimension).
The references to Dekking's article were also useful. Lindenmayer's article was referenced and another by A. Cobham ("Uniform Tag Sequences", Math Systems Theory, Vol. 6, 1972, pages 154-192) which taught me how to modify existing L-systems.
In August, 1987 BYTE magazine published some of my L-systems together with a crude program to draw them. I didn't realize how crude until a few months ago when I discovered FRACTINT on the internet, a truly marvelous program.
All the images used to help illustrate this tutorial can be found in the file, mcworter.l found at this site.
To those of you out there who like playing with L-systems, I'd love to see your
work and add it to the
collection of L-systems now available in the database
at SPANKY .
If you want to contribute to this growing library of interesting
L-Systems you can send them to me at:
By using graphic objects more robust than line segments, you can model more realistic plants. See Laurens Lapre's homepage or C.J.van der Mark's page on 3D Lsystems.
Here you will need the free program LPARSER;
FRACTINT is limited to line segments.
For tilings there is a remarkable program called QUASITILER which can be used online to construct tilings. This program constructs tilings by projecting an n-dimensional grid onto 2-space or 3-space.
Other tutorials are available at:
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