Draw Hilbert Fractal

This tool is a focused Hilbert-curve viewer with 4 starting orientations. The sister draw-hausdorff-fractal tool (originally misnamed as “Hausdorff” – see its own FAQ) is a Hilbert + Moore (closed-loop variant) multi-pattern explorer. Pick this tool if you want the canonical Hilbert curve with rotation variants. Pick that one if you want to compare Hilbert vs Moore curves side-by-side.

One of the most influential mathematicians of the late 19th and early 20th centuries (1862-1943). Founded modern functional analysis, formalised geometry (Hilbert’s axioms), proposed the 23 “Hilbert problems” (1900) that shaped 20th-century math. The Hilbert space-filling curve was a short note (1891) demonstrating that Peano’s earlier space-filling curve wasn’t unique. Less philosophical, more constructive: just here’s another one. Today the curve is named after him because his construction is cleaner – uses a simple 2×2 subdivision instead of Peano’s 3×3.

At iteration N, the curve visits each cell of a 2^N × 2^N grid exactly once. As N → ∞, the grid resolution becomes arbitrarily fine, so the curve passes arbitrarily close to every point in the unit square. The Hausdorff dimension of the LIMIT set is 2 – confirming it covers the square with positive area. This violates the early-1800s intuition that a 1D curve can only have 1D “extent” – Cantor (1877) and Peano (1890) shocked the math world by proving otherwise.

If two 2D points are NEARBY, then their 1D Hilbert-curve indices (positions along the curve) tend to be nearby too. The converse isn’t quite true (some 1D-close points are 2D-far apart due to curve boundaries between sub-blocks), but locality-preservation is good ENOUGH to be useful. This is why Hilbert curves are used for spatial database indexes (R-tree, Hilbert R-tree), GPU memory access patterns, and image-dithering orders.

A spatial index for 2D rectangles. Compute the Hilbert index of each rectangle’s centre point, then insert into an R-tree sorted by Hilbert index. Result: rectangles that are CLOSE in 2D end up in the same B-tree node, giving fast range queries. Used in PostGIS, Oracle Spatial, SQL Server, MongoDB GeoJSON indexes. Original paper: Kamel & Faloutsos 1994. Alternative: Z-curve (Morton order) – simpler but worse locality.

Iter N has 4^N − 1 segments. Iter 9 = 262,143. Iter 10 = 1,048,575. Each segment is one lineTo Canvas call in a single beginPath…stroke batch. At ~1M lineTo calls, path data is tens of MB and stroke takes seconds. Iter 9 stays under 3 s on desktop. The L-system string at iter 9 is ~3 MB of characters – generating IT alone takes a few hundred ms.

Color lerps from your chosen line color (start) to orange (#f97316) at the end. Shows the TRAVERSAL ORDER. You’ll see the curve start in one corner, work through one quadrant, jump to the next quadrant, etc. The locality-preserving property is visually obvious: spatially-close segments tend to be similar colors (close in the lerp). Adjacent quadrant boundaries show as sharp color jumps where the curve teleports across the figure.

The canonical Hilbert curve starts at one corner and ends at an adjacent corner. Rotation 0° puts the start at one corner; 90°/180°/270° put it at others. Useful when fitting the result into a specific page layout, or when you want to visualise the curve from a specific viewpoint. Doesn’t change the fractal mathematically – just rotates the rendered image.

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.