L-System Fractal Tree

Potting the bonsai...

Presets

Grammar

Growth, wind, light

Foliage

Auto-orbit camera

Live readouts

Graphs

Symbol F count per iteration: measured (bars) vs substitution-matrix prediction (dots), log

Branches by bracket nesting depth

Pipe model: branch radius vs distance from the tip (each dot one branch)

Frame time (ms), green line = 16.7 ms

Numerical checks

running...

How to use

Pick a grammar (or write your own), set the rewriting iterations and the turn angle. The turtle draws F, turns with + - (yaw), & ^ (pitch), \ / (roll), and saves or restores its state at [ ]. Planar grammars become spatial when you add an extra roll at each branch. Drag the Growth slider or press Play to watch the tree grow outward from the trunk; add wind, gravity droop and blossoms. Orbit with drag, zoom with the wheel.

The science

An L-system rewrites every symbol of a string in parallel: w(n+1) = P(w(n)). With the substitution matrix M (M[a][b] = number of symbols b produced by symbol a) the symbol-count vector obeys c(n+1) = M^T c(n), so counts grow like the dominant eigenvalue of M: for the grammar F -> FF+[+F-F-F]-[-F+F+F] the number of F is exactly 8^n, for F -> F[+F]F[-F]F it is 5^n.

The 3D turtle keeps an orthonormal frame (H, L, U) and applies rotation matrices about U (yaw), L (pitch) and H (roll). Gravity droop follows Prusinkiewicz and Lindenmayer's tropism: H' = normalise(H + e T) with T = (0,-1,0). Branch radii use the pipe model attributed to Leonardo and to Shinozaki: the cross-section is conserved at every fork, sum of r_child^alpha = r_parent^alpha (alpha = 2), with a fixed tip radius. Growth reveals segments by path length from the root, so thick trunks appear first and twigs last.

Simplifications: no light competition, no secondary thickening over time, wind and droop are geometric offsets (not mechanics), leaves and blossoms are decorative markers at branch tips, and the scene scale (1 unit = 12 cm) only gives the readouts a size. Bark, leaves, blossoms, ceramic pot, moss and wood are baked in Blender 5.2.

What the checks mean

Counts compare the number of F, [ and ] in the expanded string with the substitution-matrix prediction for every iteration up to the current one. Turtle closure sends the real turtle round a square, a pitched loop and a hexagon and tests the return to the origin; frame orthonormality applies 100,000 random rotations and tests that H, L, U stay orthonormal. Connectivity and pipe model are read back from the rendered instance matrices.

Keyboard and accessibility

Space pauses, R resets, C re-runs the checks (while the page has focus).
All controls are native inputs with labels.