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Home/Math Visualization/L-Systems (Turtle)

L-Systems (Turtle)

This interactive simulator explores L-Systems (Turtle) in Math Visualization. Lindenmayer string rewriting + turtle: Koch, Sierpinski, Hilbert, Heighway dragon, plant. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: Best once you already know the basic definitions and want to build intuition. Typical context: Math Visualization.

Key terms

  • systems
  • turtle
  • l systems
  • math
  • visualization

L-system

Systems

4
1.2 px
axiom: F--F--F
F → F+F--F+F
δ = 60°

Measured values

segments drawn0
expanded length0
turning angle δ60°
depth4

How it works

An L-system is a parallel string-rewriting grammar invented by Aristid Lindenmayer (1968) to model plant growth. Start with an axiom and repeatedly substitute every symbol by its production. A turtle then walks the resulting string: F = step forward, +/− = turn by δ, [/] = push/pop position. With just a few rules you can reproduce Koch snowflake, the Hilbert space-filling curve, the Heighway dragon, Sierpinski triangles and remarkably plant-like trees.

Key equations

stringₙ₊₁ = apply(rules, stringₙ)
turtle: F→step, +→turn(+δ), −→turn(−δ), [/]→push/pop
|stringₙ| = O(bⁿ) where b = max |rule|