Draw Peano Fractal

The Peano curve is a continuous mapping from the unit interval onto the unit square – its image fills the square completely. It was published by the Italian mathematician Giuseppe Peano in 1890 in the paper “Sur une courbe, qui remplit toute une aire plane” (Mathematische Annalen, vol. 36). It was the first such “space-filling curve” ever constructed and overturned the prevailing belief that a continuous curve must remain one-dimensional. Hilbert’s better-known curve appeared the following year (1891) as a simpler 2×2 variant of the same idea.

The L-system used here is the standard one: axiom L, rule L → LFRFL-F-RFLFR+F+LFRFL, rule R → RFLFR+F+LFRFL-F-RFLFR. L and R are non-drawing recursion tokens; only F draws a segment. Each rule contains exactly 8 F characters and 9 L/R children. Solving the recurrence: F-count after n iterations is 8 · (9ⁿ − 1)/(9 − 1) = 9ⁿ − 1. So iter 1 has 8 segments (not 9), iter 2 has 80, iter 3 has 728, iter 4 has 6,560, iter 5 has 59,048. An earlier version of this FAQ claimed 9ⁿ – that was wrong; the displayed stats have always shown the true count.

The Peano curve is genuinely space-filling, so its similarity dimension is exactly log(9) over log(3) = 2.0. At each iteration the pattern is replaced by 9 copies scaled by 1/3, so dim = log(N) / log(1/r) = log(9) / log(3) = 2. The displayed value is this limit; at any finite iteration the rendered polyline is still 1-dimensional in the strict sense.

Both are space-filling, both have dimension 2.0, both came out of late-19th-century work on the continuity of dimension. Peano (1890) uses a 3×3 subdivision with 90° turns and visits 9 sub-cells per iteration. Hilbert (1891) uses a 2×2 subdivision with 90° turns and visits 4 sub-cells. Hilbert’s curve has better locality preservation (consecutive index points stay close in 2D), which is why modern spatial indexes prefer it; Peano’s curve is the historical first.

The continuous limit curve is non-self-crossing – the L-system rules are designed so adjacent 3×3 sub-cells meet at shared boundary points but the path never doubles back. At any finite iteration the rendered polyline may share endpoints between adjacent unit segments (that is the L-system’s job), but it does not loop back through previously visited interior points. Peano explicitly proved the non-self-intersection property in his 1890 paper.

Each F-segment is drawn with an HSL color whose hue is set by the segment’s position along the curve. The first segment is red (hue 0°), the last segment is purple (hue ~330°), and the spectrum in between traces the traversal order. This makes it visually clear which sub-cell is visited first, which is visited next, and so on – useful for understanding how Peano’s traversal order differs from Hilbert’s or Moore’s.

The overlay draws the cell boundaries at the current iteration depth: an 9×9 grid at iter 2, a 27×27 grid at iter 3, etc. The Peano curve enters each cell, traverses its interior recursively, and exits at a specific boundary point. The grid helps you see exactly which cells are being filled and in what order.

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