L-System Playground

What Are L-Systems?

L-systems were introduced by biologist Aristid Lindenmayer in 1968 to model plant growth. They use recursive rewriting rules to generate surprisingly complex structures from simple initial conditions.

An L-system is defined by four components:

Axiom — the initial string
Production Rule — how symbols are replaced
Iterations — how many times rules are applied
Angle — rotation used during drawing

How It Works

A tiny rule can produce thousands of drawing instructions after only a few iterations.

Example

Axiom: F
Rule: F → F[+F]F[-F]F

String Expansion

Iteration 0
F

Iteration 1
F[+F]F[-F]F

Iteration 2
F[+F]F[-F]F[+F[+F]F[-F]F]F[+F]F[-F]F[-F]F[+F]F[-F]F

After expansion, the resulting string is interpreted by a turtle graphics renderer that draws the visual structure.

Symbol Reference

Symbol	Meaning
`F`	Move forward and draw
`A / B`	Alternative drawing symbols
`+`	Rotate right by angle
`-`	Rotate left by angle
`[`	Save current position and angle
`]`	Restore saved position and angle

The bracket system [ and ] is what enables branching structures like trees, bushes, roots, and coral forms.

Turtle Graphics Model

The renderer maintains the current position (x, y), direction, and a state stack to handle branching.

Execution Example

F[+F][-F]

Draw forward
Save position
Turn right and draw branch
Restore position
Turn left and draw another branch

This simple mechanism creates hierarchical growth patterns found in nature.

Try It Yourself

Modify the axiom, rule, angle, and iterations below. Watch how small changes create dramatically different structures.

Axiom (start)

Rule (F →)

Symbols:
F draw forward
+ turn right
- turn left
[ save position
] restore position

Parameters

Angle 25°

Iterations 4

Examples

Included Presets

Tree

Axiom: F
Rule: F[+F]F[-F]F
Angle: 25°

Classic symmetric branching structure. Each branch splits into two.

Fern

Axiom: F
Rule: F[+F][--F][-F]
Angle: 25°

Asymmetric branching creates leaf-like patterns.

Bush

Axiom: F
Rule: F[+F][-F]F[-F][+F]F
Angle: 20°

Dense multi-branch vegetation with complex structure.

Algae

Axiom: A
Rule: AB

Lindenmayer's original algae model. Shows pure string expansion (Fibonacci pattern).

Fractal

Axiom: F
Rule: F[+F]F[-F][F]
Angle: 22°

Balanced recursive branching pattern creating mathematical beauty.

Exponential Growth

L-systems grow exponentially. Even simple rules produce huge strings:

Iteration	Approximate Command Count
1	11
2	61
3	311
4	1,561
5	7,811

This is why even tiny rules quickly generate intricate, detailed structures.

Things To Experiment With

Try changing these in the playground:

Branch density (more brackets = more branches)
Rotation angle (smaller angles = tighter spirals)
Iteration count (more iterations = more detail)
Rule symmetry (asymmetric rules = organic forms)
Nested brackets (enable complex hierarchies)

Example Rules

F[++F][--F]
F[-F]F[+F]F
F[+F[-F]]
F[+F][+F][+F]

Small changes often produce dramatically different results.

Real-World Applications

Procedural Generation

Trees, vegetation, roots, coral structures, alien flora in video games and movies.

Computer Graphics

Fractal art, organic 3D modeling, environmental generation for VFX and games.

Scientific Simulation

Plant morphology research, branching system analysis, growth pattern studies in biology.

Procedural Content

SpeedTree uses L-systems to generate millions of unique trees with minimal memory overhead.

🌳 L-System Playground