An L-system (Lindenmayer system) is a formal grammar invented by biologist Aristid Lindenmayer in 1968 to model plant growth. It rewrites a starting string (axiom) using production rules, and a turtle interprets the result as drawing commands (F = move forward, +/− = turn, [ ] = branch).

Why does increasing iterations create more detail?

Each iteration applies the production rules again, replacing every symbol with a longer sequence. The string grows exponentially: a rule that triples string length at each step produces 3ⁿ characters after n iterations. The turtle draws more and finer segments, adding detail at every scale — the hallmark of self-similarity.

What is the fractal dimension of the Koch Snowflake?

The Koch Snowflake has a Hausdorff dimension of log(4)/log(3) ≈ 1.2619. At each iteration, each segment is replaced by 4 new segments each 1/3 the length, so the curve length grows by (4/3)ⁿ → ∞ while enclosing a finite area.

Can I create custom L-systems in this simulator?

The six built-in presets cover the most famous deterministic L-systems. Adjust the branch angle slider to explore variations: changing the Koch snowflake angle away from 60° creates entirely different fractal curves. Higher iterations (up to 8) reveal deeper self-similar structure.

Home▸Drawing with Maths▸L-System Fractal Generator | Interactive Lindenmayer System Simulator

🧪 L-System Fractal Generator | Interactive Lindenmayer System Simulator

Generate stunning fractal patterns with Lindenmayer systems and turtle graphics. Koch Snowflake, Sierpiński Triangle, Dragon Curve, Barnsley Fern, Hilbert Curve, and Fractal Plant. Adjust iterations and angle, pan and zoom, save PNG. Free.

Drawing with Maths2DModerate60 FPS

⚙ Under the hood

This simulation allows users to generate fractals using L-systems, a formal grammar for describing branching structures. By modifying the rules of the L-system, they can create diverse and complex fractal patterns, demonstrating the principles behind self-similarity.

L-SystemFractalLindenmayer

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install