Draw Mcworter Dendrite Fractal

William McWorter (active 1990s) is a mathematician who published several fractal constructions on his personal website and in Conway-style recreational mathematics communities. His most-cited contribution is the Pentigree – a self-similar fractal built from 6 copies at scale 1/(φ²) = 1/2.618 around a pentagonal layout, where φ is the golden ratio. Its dimension is log(6)/log(φ²) ≈ 1.862. There’s no widely-recognized “McWorter Dendrite” fractal – that term appears to have been applied to a different curve (the Heighway dragon, as implemented here) by mistake. If you want the genuine Pentigree, search for “McWorter Pentigree IFS code”.

Same engine (Heighway dragon L-system). Sister tool has more features: symmetry mode (alternating red/blue per segment to show recursive structure), fire gradient (red → amber along traversal), separate Reset and Clear, loading state, complete history disclosure including the NASA 1966 discovery context. This tool is the bare-bones version. Pick the sister tool unless you specifically want the historical misnomer preserved (e.g. for a textbook chapter discussing fractal name confusions).

L-system (Lindenmayer system) is a mathematical formalism introduced by Aristid Lindenmayer in 1968 to model plant growth. It works by recursive string rewriting: start with an axiom string, apply replacement rules to every character per iteration. After N iterations the string can be interpreted as turtle-graphics commands to draw a curve. F = forward + draw, + = turn right, – = turn left, X/Y = non-drawing state symbols for the recursion. Used in fractal generation, plant modelling, procedural city generation, and DNA folding analogies.

For self-similar sets: dim = log(N)/log(S) where N copies at scale 1/S. Heighway: 2 copies at scale 1/√2. dim = log(2)/log(√2) = log(2)/(½ log(2)) = 2. Dimension 2 means the limit curve covers a 2D region with positive area – it’s “space-filling.” The Heighway region (called the “Heighway dragon” shape) has a fractal boundary but a positive Lebesgue measure interior.

Each iter applies the rules to EVERY character. The rule expands X to 5 characters and Y to 5 characters, and after a few iterations roughly half the string is X+Y (then re-expanded). String length grows by factor ~2 per iter; F-segment count exactly doubles. Iter 10: ~120k F-segments, L-system string ~1.5M chars. Iter 12 would exceed practical memory limits (browser memory, not the math).

Axiom: FX. Iter 1: replace X. FX → F + X+YF+ = FX+YF+. Iter 2: apply rules to both X and Y. FX+YF+ → F · (X+YF+) + (-FX-Y)F+ = FX+YF+-FX-YF+. Each iter doubles F count: iter 1 = 2 F’s, iter 2 = 4, iter 3 = 8, … iter N = 2^N.

At iter 5-8, the shape resembles a Chinese dragon with a serpentine body and zigzag scales. Martin Gardner popularised the name in his 1967 Scientific American “Mathematical Games” column where he wrote about NASA’s discovery. The name stuck. “Dendrite” – meaning tree-like branching – does NOT really apply to this curve.

Yes – 100% client-side. L-system expansion and Canvas drawing run in your browser. No parameters or images leave your device. Verify via your browser’s Network tab.