About

Fractal geometry describes structures where self-similar patterns repeat at every scale. A coastline measured with a 1km ruler yields a different length than one measured at 1m resolution. This non-integer scaling is captured by the fractal dimension D, where D = log Nlog s for N self-similar pieces at scale factor s. This generator implements two families: L-Systems (Lindenmayer grammars) that produce Koch curves, Sierpinski gaskets, dragon curves, and botanical branching via string rewriting and turtle graphics; and escape-time fractals (Mandelbrot and Julia sets) computed per-pixel in the complex plane. Misconfiguring iteration depth wastes computation or produces featureless blobs. Setting L-System depth above 8 on complex grammars can produce strings exceeding 10⁷ characters. The tool caps parameters to keep rendering under 30 seconds on typical hardware.

Limitations: L-System rendering assumes uniform segment length. Mandelbrot coloring uses cyclic HSL mapping. The tool approximates floating-point arithmetic at 64-bit IEEE 754 precision, so deep zooms beyond 10¹³× magnification will show pixelation artifacts. For research-grade exploration, arbitrary-precision libraries are required. Pro tip: for botanical fractals, stochastic angle variation (the jitter parameter) prevents the artificial symmetry that makes computer plants look synthetic.

Formulas

L-System string rewriting applies production rules iteratively. Given axiom w and rule set P, the string at generation n is:

S₀ = w , S_n+1 = P(S_n)

String length grows as |S_n| = O(kⁿ) where k is the expansion ratio of the dominant production. Koch curve: k = 4. Turtle graphics interprets the string character by character. F draws forward by step length d, + rotates by angle δ, − rotates by −δ, [ pushes state, ] pops state.

For escape-time fractals, each pixel maps to a point c in the complex plane. The iteration is:

z_n+1 = z_n² + c

For the Mandelbrot set, z₀ = 0 and c varies per pixel. For Julia sets, c is fixed and z₀ varies. A point escapes if |z_n| > 2 (bailout radius). The escape iteration count determines color via smooth coloring:

μ = n − log₂(log₂(|z_n|))

Where μ is the smoothed iteration count, n is the discrete escape iteration, and |z_n| is the modulus at escape. The fractal dimension of an L-System curve with N self-similar copies at scale 1s is D = log Nlog s.

Reference Data

Fractal	Type	Dimension D	Axiom / Formula	Typical Iterations	Discovery
Koch Snowflake	L-System	1.2619	log 4log 3	4 - 6	Helge von Koch, 1904
Sierpinski Triangle	L-System	1.5850	log 3log 2	5 - 8	Wacław Sierpiński, 1915
Dragon Curve	L-System	2.0	Space-filling	10 - 15	Heighway, 1966
Hilbert Curve	L-System	2.0	Space-filling	4 - 7	David Hilbert, 1891
Fractal Plant	L-System	≈1.5	Bracketed grammar	4 - 7	Prusinkiewicz, 1990
Lévy C Curve	L-System	2.0	Space-filling	10 - 16	Paul Lévy, 1938
Gosper Curve	L-System	1.1292	Hexagonal tiling	3 - 5	Bill Gosper, 1970s
Sierpinski Carpet	L-System	1.8928	log 8log 3	3 - 5	Sierpiński, 1916
Mandelbrot Set	Escape-time	2.0 (boundary)	z_n+1 = z_n² + c	100 - 1000	Mandelbrot, 1980
Julia Set	Escape-time	Variable	z_n+1 = z_n² + c (fixed c)	100 - 500	Gaston Julia, 1918
Burning Ship	Escape-time	2.0 (boundary)	z_n+1 = (\|Re\| + i\|Im\|)² + c	100 - 500	Michelitsch, 1992
Barnsley Fern	IFS	≈1.45	4 affine transforms	50000 - 500000 points	Michael Barnsley, 1988

Frequently Asked Questions

L-System string length grows exponentially. A Koch curve at depth n produces 4ⁿ segments: depth 6 = 4096 segments, depth 10 = 1,048,576. The generator caps iterations per fractal type and uses a Web Worker for string expansion, but Canvas line-drawing is still single-threaded. Reduce iterations or increase step length to compensate.

The escape-time algorithm counts how many iterations n it takes for |z_n| to exceed the bailout radius of 2. Smooth coloring uses μ = n − log₂(log₂(|z_n|)) to eliminate banding. This μ value maps to an HSL hue cycle. Points that never escape (inside the set) are colored black.

The turning angle δ defines how much the turtle rotates per + or - command. Koch curves require exactly 60°, Sierpinski uses 120°, Dragon curves use 90°. Changing the angle from its canonical value produces entirely different geometries. The presets lock angles to their mathematically correct values, but custom mode allows experimentation.

Both use the same recurrence z_n+1 = z_n² + c. In the Mandelbrot set, z₀ = 0 and each pixel represents a different c. In a Julia set, c is a single fixed complex number and each pixel represents a different z₀. Every point in the Mandelbrot set corresponds to a connected Julia set. Points outside the Mandelbrot set produce disconnected (dust-like) Julia sets.

JavaScript uses 64-bit IEEE 754 floating-point arithmetic, providing approximately 15 - 17 significant decimal digits. Zoom factors beyond roughly 10¹³ exhaust this precision, causing adjacent pixels to map to the same complex coordinate. Overcoming this requires arbitrary-precision arithmetic (perturbation theory or BigFloat libraries), which is beyond the scope of this browser-based tool.

The canvas renders at the displayed resolution (up to 1200 × 1200 pixels by default). At 300DPI, that yields a print size of approximately 4 × 4 inches. For larger prints, increase iterations and adjust your browser zoom to enlarge the canvas before exporting. L-System fractals, being line-based, scale better than pixel-based escape-time fractals.