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Lindenmayer System

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LindenmayerSystem

A Lindenmayer system, also known as an L-system, is a parallel string rewriting system. It consists of an alphabet, an initial string called the axiom, and production rules. At each step, all symbols in the current string are replaced simultaneously according to the rules (Prusinkiewicz and Lindenmayer 1990, pp. 3-4). Lindenmayer (1968a, 1968b) introduced these systems to model cellular development.

LindenmayerRewriting

In a simple deterministic example, the axiom is F and the rule is F->F+F--F+F. Interpreting F as a forward step and + and - as turns through 60 degrees gives successive approximations to the Koch curve. Symbols without a specified rule, such as the turn symbols, remain unchanged. The geometric interpretation is separate from the rewriting rules (Prusinkiewicz and Lindenmayer 1990, pp. 6-8). Several fractals generated using Lindenmayer systems are illustrated above.

See also

Box Fractal, Dragon Curve, Exterior Snowflake, Fractal, Hilbert Curve, Koch Curve, Koch Snowflake, Peano Curve, Peano-Gosper Curve, Sierpiński Arrowhead Curve, Sierpiński Curve, String Rewriting System

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References

Bulaevsky, J. "L-System Based Fractals." http://ejad.best.vwh.net/java/fractals/lsystems.shtml.Bulaevsky, J. "A Process to Generate Fractals." http://ejad.best.vwh.net/java/fractals/process.shtml.Charpentier, M. "L-Systems in PostScript." https://web.archive.org/web/20120624210258/http://www.cs.unh.edu/~charpov/Programming/L-systems/.Dickau, R. M. "Two-Dimensional L-Systems." https://www.robertdickau.com/lsys2d.html.Lindenmayer, A. "Mathematical Models for Cellular Interactions in Development. I. Filaments with One-Sided Inputs." J. Theor. Biol. 18, 280-299, 1968a. https://doi.org/10.1016/0022-5193(68)90079-9.Lindenmayer, A. "Mathematical Models for Cellular Interactions in Development. II. Simple and Branching Filaments with Two-Sided Inputs." J. Theor. Biol. 18, 300-315, 1968b. https://doi.org/10.1016/0022-5193(68)90080-5.Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.Prusinkiewicz, P. and Hanan, J. Lindenmayer Systems, Fractal, and Plants. New York: Springer-Verlag, 1989.Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic Beauty of Plants. New York: Springer-Verlag, 1990.Rozenberg, G. and Salomaa, A. (Eds.). Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology. Berlin, Germany: Springer-Verlag, 1992. https://doi.org/10.1007/978-3-642-58117-5.Stevens, R. T. Fractal Programming in C. New York: Holt, 1989.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. §1.5.9. https://www.mathematicaguidebooks.org/.Wagon, S. "Recursion via String Rewriting." §6.2 in Mathematica in Action. New York: W. H. Freeman, pp. 190-196, 1991.

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Lindenmayer System

Cite this as:

Weisstein, Eric W. "Lindenmayer System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LindenmayerSystem.html

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