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l-system

L-systems (Lindenmayer systems) are formal grammars that produce fractal patterns through string rewriting. Starting with an axiom (e.g. "F"), each iteration replaces every symbol with a rule (e.g. F → F+F-F-F+F). After N iterations, the resulting string drives a turtle graphics engine to draw the pattern. L-systems model many natural forms — trees, snowflakes, ferns, neural networks. Famous examples: Koch snowflake, Sierpinski triangle, dragon curve, fractal plants. The ZTools L-System Generator lets you define axiom + rules + angle, runs iterations, and renders the result in SVG.

Use cases​

Generative art​

Tweak rules to produce wildly different fractals. Each combination is unique.

Teach grammar / formal languages​

L-systems are simple context-free grammars. Visual feedback makes the abstract concrete.

Procedural generation in games​

Trees, plants, terrain features procedurally generated via L-systems.

Mathematical curiosity​

See the Koch snowflake, dragon curve, fractal tree all from a few rules.

How it works​

  1. Define axiom — Starting string (e.g. "F").
  2. Define rules — Replacement rules (e.g. F → F+F-F-F+F). Multiple rules per symbol if you want stochastic L-systems.
  3. Set parameters — Iterations (5-8 typical), angle (e.g. 90° for Koch), step length.
  4. Render — Tool runs iterations, then draws via turtle graphics: F = forward, + = turn left, - = turn right, [ = push state, ] = pop state.

Examples​

Input: Koch snowflake: axiom F, rule F → F+F--F+F, angle 60°, 4 iterations

Output: Iconic snowflake-edge fractal.

Input: Dragon curve: axiom FX, rules X→X+YF+, Y→-FX-Y, angle 90°, 12 iterations

Output: Self-similar curve resembling a dragon.

Input: Fractal plant: axiom X, rules X→F+[[X]-X]-F[-FX]+X, F→FF, angle 25°

Output: Tree-like plant. Resembles a real fern.

Frequently asked questions​

How many iterations?

Each iteration multiplies string length by the rule's expansion factor. After 8-10 iterations, strings get massive (millions of characters). Tool caps at reasonable rendering.

Privacy?

All in browser.

Tips​

  • Start with classic examples (Koch, dragon, plant) to understand the mechanics.
  • For your own designs, tweak one rule at a time — small changes produce dramatic differences.
  • For posters, use SVG export — vector scales infinitely.
  • L-system rules are notation-sensitive — extra spaces or capitalisation matter.

Try it now​

The full l-system runs in your browser at https://ztools.zaions.com/l-system — no signup, no upload, no data leaves your device.

Open the tool ↗

Last updated: 2026-05-06 · Author: Ahsan Mahmood · Edit this page on GitHub