Draw Triangle Dragon Fractal

The terdragon (triangle dragon curve) is a space-filling fractal curve introduced by Chandler Davis and Donald E. Knuth in their 1970 paper “Number representations and dragon curves” (Journal of Recreational Mathematics). It is built from the L-system with axiom F and rule F → F+F-F at 120° turns. An older version of this FAQ called it a “variant of the Heighway dragon” – that is incorrect; the terdragon is its own independent Davis-Knuth construction, distinct from John Heighway’s 1966 dragon curve.

The similarity dimension is log(3) divided by log(√3), which equals exactly 2.0. Solve directly: log(3) / log(3 to the 1/2) = log(3) / (0.5 × log(3)) = 2. The terdragon is therefore space-filling: its image in the limit is a closed region of nonzero area. An older version of this FAQ wrote “approximately 1.5850” – that figure is actually log(3)/log(2), the Sierpinski triangle’s dimension, accidentally pasted into the terdragon description. The displayed stats line has always been correct (2.0000); only the FAQ text was wrong, and is now corrected.

Axiom F, rewrite rule F → F+F-F, turn angle 120°. F means “draw forward one unit”; + and − mean “turn left/right 120°”. Iterating the rewrite n times produces a string with 3ⁿ F characters; interpreting it as turtle graphics traces the terdragon polyline.

The completed curve has 3-fold rotational symmetry: rotating it about its center by 120° yields the same figure. This follows from the symmetric L-system rule (3 forward moves balanced by one + and one − turn) and the 120° angle. The curve tiles the plane in triples – three terdragons fit together at their start point to fill a hexagonal region exactly.

Iteration 10 generates 3¹⁰ = 59,049 segments. Each segment is a separate stroke operation (or a separate color in rainbow mode), so the per-segment overhead dominates. On a modern machine it takes 2-4 seconds. For most exploration, iter 6-8 (729 to 6,561 segments) is plenty.

Each segment is colored by its index in the curve’s traversal order, mapped to a hue from 0° (red, first segment) to 360° (purple, last segment). This reveals how the curve winds through the plane and which sub-regions are visited early vs late.

Only 120° produces the canonical terdragon. Other angles (e.g. 60°, 90°) generate non-canonical figures from the same L-system rule – sometimes interesting, sometimes overlapping into a thick blob. The stats line marks the figure as “non-canonical angle” when you leave 120° so you know you are no longer drawing the named fractal.

No. The tool loads three static files (HTML, CSS, JS) and then runs entirely in your browser – L-system expansion, drawing, PNG export, clipboard copy. You can disconnect from the internet after the page loads. No analytics, no tracking, no cookies.

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