L-System Fractal Garden (3D)

Lindenmayer systems

An L-system is a string-rewriting game: start from an axiom and, in every iteration, replace all symbols at the same time using the production rules (for the Koch curve F -> F+F--F+F). A turtle then reads the final string: F moves forward and draws, + and - turn by the fixed angle, [ saves the position and heading and ] returns to them, which is how branches are made. In 3D the turtle also pitches (&, ^) and rolls (\, /) with the heading, left and up vectors H, L, U rotated by H' = H cos d + L sin d and so on. The symbol ! thins the branches (width times a ratio), the double quote shortens the step, and L marks a leaf.

Growth counts

For a rule that replaces one segment by m segments the count after n iterations is m^n: the Koch curve has 4^n segments, and with the step shrunk to 3^-n its length is (4/3)^n, which diverges although the curve fits in a finite box. The Hilbert curve has 4^n - 1 unit steps and visits every cell of a 2^n x 2^n grid exactly once. The first and third checks verify exactly these facts on the generated strings.

Fractal dimension

Each iteration multiplies the segment count by N = 4 while the figure grows by s = 3 (fixed step), so the similarity dimension is D = ln N / ln s = ln 4 / ln 3 = 1.2619. The same number is measured independently by box counting: count the boxes of size e that the curve touches, N(e) ~ e^-D, and fit the slope on a log-log plot (second check). Space-filling curves approach D = 2; a tree's apparent dimension depends on the length and width ratios.

Plants

Branching systems such as A -> F[+A][-A] double the number of tips every iteration; tapering (!) and shortening (") make daughter branches thinner and shorter than their parents, as in real trees where branch length falls off roughly geometrically. Random angle jitter is a simple way of imitating the variation between real branches.

Honest simplifications

These are geometric models, not growth physics: no light, gravity, water transport or competition between branches, so they illustrate recursion and self-similarity rather than predict how a particular species grows. The drawn segment count is capped at 60,000 for speed; leaves are simple discs or cards at branch tips.

What to try

• Pick Koch snowflake, raise n step by step and watch the segment count quadruple and the box-count dimension settle near 1.26.
• Change the turn angle of the Koch curve from 60 to 72 degrees: the self-similarity (and the dimension) changes because the rule no longer closes up.
• Switch to Hilbert curve: the dimension rises towards 2 as the curve fills the square.
• In the tree presets lower the width ratio to thin the twigs, or raise the jitter to make a more natural, asymmetric crown.
• Set the growth speed to 0 to show the final shape at once, or press New random jitter for another individual.