Topic · Geometry

L-Systems

Letters that rewrite themselves into plants.

An L-system is a tiny grammar: a starting string, a handful of rewrite rules, and a turtle that turns the letters into lines. In a few generations, three letters become a fern. In the Explorer you edit the axiom, slide the depth and watch the turtle draw — Koch flakes, dragons, Hilbert curves, whole forests — out of a handful of characters.

Discover the story → Open the Explorer

First encounter

Replace letters. Draw the result. Get a fern.

The big idea

Pick a short string of letters. Pick a tiny dictionary that says what each letter becomes in the next round. Apply the dictionary to every letter, simultaneously. Repeat. Now hand the resulting string to a turtle that knows how to draw — F means forward, + means turn left, − means turn right. After a few rounds the turtle is drawing ferns, coral, lightning, and trees.

A concrete example

Lindenmayer's algae: axiom A, rules A → AB and B → A. After 0 steps: A. After 1: AB. Then ABA, ABAAB, ABAABABA, and at step 5 the string is ABAABABAABAAB — thirteen letters. The lengths 1, 2, 3, 5, 8, 13 are the Fibonacci sequence: a botanical pattern hiding in pure rewriting.

Why it matters

Aristid Lindenmayer was a Hungarian biologist in Utrecht who wanted a formal language for how plants grow. He published the formalism in 1968 to model the cell-by-cell development of algae. The same idea now grows the trees in your favourite video game, the ferns in animated films and the procedural cities of the demoscene. A grammar for cells became a grammar for worlds.

Below: step the rewrite by hand, then let a turtle draw the result.

Section 01 · The rewriting rule

Alphabet, axiom, parallel rewrite

An L-system has three pieces. An alphabet of symbols. An axiom — the starting string. A set of production rules, one per symbol, that say what each symbol becomes in the next generation. The defining trick is parallelism: at every step every symbol is rewritten at once, the way every cell in a young leaf divides at once. That single design decision is what separates an L-system from a Chomsky grammar, where one symbol is rewritten at a time. Parallelism gives you growth.

Interactive · rewrite by hand

Preset

Axiom

Rules

A → AB
B → A

Generation · 0

Length · 1

NoteLengths: 1, 2, 3, 5, 8, 13… — the Fibonacci sequence.

Section 02 · The turtle interpretation

F, +, −, [, ] — a pen with memory

Letters on their own are just text. The geometry appears when you hand the string to a turtle. F means draw forward by one unit. + turns left by a fixed angle (often 25°, 60°, 90°). − turns right by the same angle. Two more symbols give the turtle a memory: [ pushes its current position and heading onto a stack, ] pops them back. With just push and pop a one-dimensional string suddenly branches — bracket pairs become side-stems and twigs. Symbols outside the drawing alphabet (X, Y, A, B…) are silent variables: they carry information through the rewrites but the turtle ignores them on the page.

Section 03 · Famous L-systems

Koch, Sierpiński, dragon, Hilbert, fern

Four rules, four wildly different organisms. Koch snowflake: axiom F++F++F, rule F → F−F++F−F, angle 60°. Sierpiński arrowhead: A → B−A−B, B → A+B+A, angle 60°, alternating parity. Heighway's dragon: axiom FX, rules X → X+YF+, Y → −FX−Y, angle 90° — twelve rewrites and a folded paper strip falls out. Hilbert curve: A → +BF−AFA−FB+, B → −AF+BFB+FA−, angle 90°, sweeps every point of a square. Fractal plant: X → F+[[X]−X]−F[−FX]+X, F → FF, angle 25°, the canonical fern with side-branches and tips. Same engine, different worlds.

Interactive · let the turtle draw

Preset

Axiom

Rules

X → X+YF+
Y → -FX-Y

Angle

90°

Depth

Section 04 · Branching with brackets

[ pushes, ] pops — that is how plants grow

Real plants are not curves; they branch. An L-system gets branching for free from two characters. When the turtle reads [ it saves its current position and direction on a stack, then keeps drawing. When it reads ] it restores them. Everything drawn between the matching brackets becomes a side-branch and the turtle then continues the trunk. Nesting brackets give you branches on branches on branches — the same way a sequoia's twigs grow off its branches, which grow off its limbs, which grow off its trunk. Two characters, infinite botany.

Section 05 · Stochastic and context-sensitive

Beyond strict determinism

The simple version above is context-free and deterministic: each symbol has exactly one rule. Two extensions make L-systems genuinely powerful. Stochastic L-systems give a symbol several possible rewrites, each with a probability — so two ferns from the same grammar look like siblings, not clones. Context-sensitive L-systems let a rule fire only when a symbol sits between specific neighbours, which is how biologists model signalling: a hormone in the parent cell triggers a particular division in the daughter. There are also parametric L-systems, where every symbol carries continuous parameters (length, angle, age), and timed L-systems where rules fire at different speeds. The whole family models real morphogenesis with surprising fidelity.

Section 06 · Real biology, real graphics

From Lindenmayer 1968 to every animated forest

Lindenmayer was modelling the cell-by-cell growth of Anabaena catenula, a filamentous blue-green alga, when he introduced the formalism in 1968. He kept refining it — for tobacco growth, for compound inflorescences, for the topology of branching axes. Przemyslaw Prusinkiewicz at Calgary turned the idea into a working pipeline with the 1990 book "The Algorithmic Beauty of Plants," still the standard reference. From there it leaked into computer graphics: SpeedTree (used in countless films and games) is essentially an industrial L-system; Pixar's vegetation, the trees in The Witcher and Horizon, the wheat fields in Red Dead Redemption — all descend from Lindenmayer's parallel rewrite. A grammar for cells became a grammar for worlds.

Lindenmayer · 1968

A → AB, B → A

The two-line algae grammar that started a whole field of mathematical botany — published in the Journal of Theoretical Biology, 18(3): 280–315.

Now grow your own

Three letters. One turtle. A whole forest.

Edit the axiom. Add a rule. Slide the depth and watch a grammar bloom into geometry. The Explorer is one click away.

→ Open the Explorer

Where you meet it

How and where this technique lives in the world today.

Procedural plants in games & film

Trees, ferns and grass in titles like The Lion King (1994), Avatar (2009) and countless modern games are generated from L-systems via SpeedTree and similar middleware.

Architecture & CAD

Generative architecture tools (Grasshopper for Rhino) use L-systems to grow branching structures, façades and street networks.

Plant biology research

Plant biologists fit L-systems to real species (e.g. apple tree topology) to study growth dynamics, light competition and yield optimisation.

Music composition

Composers map L-system strings to MIDI events to algorithmically grow themes that develop fractal self-similarity over time.