Draw Moore Fractal

Eliakim Hastings Moore’s 1900 variant of David Hilbert’s 1891 space-filling curve. Same dimension (2.0 – truly space-filling), same 90° turns, same recursive subdivision structure. Key difference: Moore’s curve is a CLOSED LOOP – the start and end of the curve are adjacent in 2D. Hilbert’s curve is open – start at one corner, end at the diagonally-opposite corner. The closed-loop property makes Moore useful for cyclic traversal problems where you need to come back to the start.

American mathematician (1862-1932). Founded the Mathematical Department at the new University of Chicago in 1892. Trained an unusual number of influential 20th-century American mathematicians – students included G.D. Birkhoff (ergodic theorem), Leonard Dickson (number theory), Oswald Veblen (topology), R.L. Moore (his namesake-but-not-related, point-set topology), and the Bourbaki group’s influence. His 1900 closed-curve construction appeared as a footnote in a paper extending Hilbert’s work – he didn’t see it as a major contribution. The curve is sometimes called “Moore’s variant” or “Hilbert-Moore closed curve.”

Axiom: LFL+F+LFL. Rule: L → -RF+LFL+FR- and R → +LF-RFR-FL+. Both L and R are interpreted as draw-forward (along with F). + = turn right 90°. – = turn left 90°. This is a non-canonical formulation – some references treat L/R as non-drawing state symbols (like with Hilbert). Both formulations produce visually-identical Moore curves; this one is just simpler to render (fewer special cases in the renderer).

Self-similarity: dim = log(N)/log(S) where N copies at scale 1/S. Moore: 4 copies at scale 1/2 (the 2×2 subdivision pattern), so dim = log(4)/log(2) = 2. Dimension 2 confirms space-filling: the limit set covers a 2D region with positive Lebesgue measure (a 4-fold-symmetric square in the Moore case, slightly larger than the Hilbert curve’s region due to the closed-loop overhead).

It shows the closed-loop property. Enable the toggle – green dot at start, red dot at end. At iter 0 they’re far apart (separate corners of the axiom shape). At iter 1-2 they get closer. At iter 5+ they’re nearly overlapping – visually adjacent. The stats line shows “endpoints distance” in pixels: a small number confirms closure. Compare to Hilbert curve: there, start and end are at opposite corners – distance is roughly the canvas diagonal.

Sister tool. That one is a multi-pattern explorer with both Hilbert and Moore curves selectable side-by-side. THIS tool is Moore-only, with start/end-marker emphasis to demonstrate the closed-loop property. Pick this one if you specifically want to study Moore’s closed-loop variant. Pick the sister tool if you want to compare Hilbert and Moore curves visually side-by-side.

Iter 7 ≈ 82,000 drawing operations. Iter 8 would be ~330,000 – at which point Canvas path stroke takes seconds and the L-system string itself is ~100 KB. Iter 7 stays under 2 s on desktop, 4 s on mobile. Going higher would require multi-frame rendering (requestAnimationFrame yielding) and offers no new visual information – segments become sub-pixel.

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.