Lindenmayer systems

L-systems

Aristid Lindenmayer introduced L-systems in 1968 to model plant growth. A symbolic axiom is rewritten in parallel at every step by applying production rules; the resulting string is then interpreted by a turtle that walks and turns, drawing the fractal as it goes.

At a glance

How an L-system works

1. Axiom: a starting string, e.g. F or F++F++F.
2. Production rules: each symbol can be replaced by a string, e.g. F → F+F−F−F+F.
3. Iteration: apply every rule to every symbol in parallel. After n steps the string can be exponentially long.
4. Turtle interpretation: walk through the final string. F G A B step forward (drawing); + − turn by a fixed angle; [ ]push and pop the turtle’s position and heading on a stack so the curve can branch.

Classic examples

• Koch curve: axiom F, rule F → F+F−F−F+F, angle 90°.
• Koch snowflake: axiom F++F++F, rule F → F−F++F−F, angle 60°.
• Sierpinski triangle: axiom F−G−G, rules F → F−G+F+G−F, G → GG, angle 120°.
• Heighway dragon: axiom FX, rules X → X+YF+, Y → −FX−Y, angle 90°.
• Plant: axiom X, rules X → F+[[X]−X]−F[−FX]+X, F → FF, angle 25°.
• Hilbert curve: axiom A, rules A → +BF−AFA−FB+, B → −AF+BFB+FA−, angle 90°.

Why parallel rewriting matters

In a context-free grammar, derivation is sequential, one symbol at a time. L-systems rewrite all symbols simultaneously at each step, modelling biological cells that divide together. Mathematically this means an L-system step is a homomorphism on strings, and string length under iteration is governed by the rule’s growth matrix.

Try it

The L-system playground renders any of seven classic systems with adjustable iteration depth.

References

• Lindenmayer, A., “Mathematical models for cellular interactions in development,” J. Theoretical Biology, 1968.
• Prusinkiewicz, P. & Lindenmayer, A., The Algorithmic Beauty of Plants, Springer, 1990.
• Prusinkiewicz, P. & Hanan, J., Lindenmayer Systems, Fractals, and Plants, Springer, 1989.
• Koch snowflake · Dragon curve · L-system playground