Fractals from Simple Rules

In 1968, Hungarian biologist Aristid Lindenmayer introduced a formal grammar to model the growth patterns of algae. He observed that a single cell could be described by a character, and its division by a rewriting rule: A becomes AB, B becomes A. Applied repeatedly, this simple substitution produces strings of increasing complexity. When those strings are interpreted as drawing instructions, intricate branching structures emerge: trees, ferns, corals, and abstract fractals, all from rules small enough to write on a napkin.

How String Rewriting Works

An L-system has three components: an axiom (the starting string), one or more production rules (character substitutions), and an iteration count (how many times to apply the rules). Each iteration replaces every character in the string simultaneously according to its rule. Characters without rules pass through unchanged. After all iterations complete, the final string is interpreted as turtle graphics: F means "draw forward," + and - mean "turn," and brackets [ ] mean "save your position and return to it later." That bracketing mechanism is what creates branching. Each [ opens a new branch, and ] returns to the fork.

The Connection to Nature

The branching structures L-systems produce are strikingly similar to real biological forms. This is because actual plant growth follows a parallel rewriting process: every cell in a growing tip divides simultaneously according to genetic rules, and the tip elongates while side branches sprout from saved positions. Lindenmayer's insight was that this biological parallelism maps directly to formal string rewriting. The resemblance between an L-system "tree" and a real tree is a consequence of shared mathematical structure, discovered independently by evolution and by mathematics.

Famous L-Systems

The Koch snowflake (1904, Helge von Koch) replaces each line segment with a peaked zigzag, creating an infinite-perimeter curve enclosing finite area. The Sierpinski triangle subdivides triangles recursively into ever-smaller copies. The dragon curve (discovered by NASA physicists John Heighway, Bruce Banks, and William Harter) folds a strip of paper in half repeatedly and unfolds to 90-degree angles, producing a space-filling fractal. Each of these can be expressed as an L-system with a single axiom and one or two rules.

Discover Award-Winning Generative Design

The branching fractals demonstrated above at dice83 showcase the power of simple rules to generate astonishingly complex structures. L-systems, or Lindenmayer systems, use recursive string rewriting to transform a starting axiom into elaborate forms like trees, ferns, or abstract fractals. Each iteration interprets characters as drawing instructions, creating branching, self-similar patterns from rules small enough to fit on a napkin. This approach mirrors natural growth processes: every cell in a plant tip divides and elongates according to local rules, producing forms that resemble real trees, corals, and leaf patterns. By applying similar algorithmic principles, contemporary generative designers explore fractal geometry, recursive structures, and parametric forms in art, architecture, and computational graphics. If our L-system experiment inspires you, the A' Generative, Algorithmic, Parametric and AI-Assisted Design Award category showcases projects where architects and designers leverage recursion, formal grammars, and computational rules to produce innovative products, projects and visual outcomes.

Above is today's featured Generative Design. Discover more through the A' Design Awards.

In the Classroom

L-systems make abstract mathematics visible. Have students write a production rule on paper and apply it by hand for three iterations, then compare the string length growth. The exponential explosion becomes tangible. For a cross-disciplinary exercise, generate plant L-systems and compare them to photographs of real trees and ferns. Students identify which structural features the L-system captures (branching angle, branch density) and which it misses (leaf shape, bark texture). The tool requires no accounts, collects no student data, and sets no cookies.