Draw Morton Fractal

The Morton curve (also called the Z-order curve) is a space-filling curve that maps a 1D index to 2D coordinates by interleaving the bits of x and y. It was introduced by Guy Macdonald Morton at IBM in a 1966 technical report on file sequencing for geodetic databases – it predates the published quadtree paper and is one of the earliest spatial indexing schemes on record.

If x = 101 in binary and y = 110 in binary, the Morton code is built by interleaving: y2-x2-y1-x1-y0-x0 = 111001 in binary = 57 in decimal. Incrementing the Morton code visits cells in the canonical NW, NE, SW, SE order recursively – that is the “Z” shape this curve is named for.

These are the well-known Morton discontinuities. When the curve finishes one 2×2 block and starts the next, consecutive Morton indices can correspond to cells that are far apart in 2D – for example, index 3 sits at (1,1) and index 4 sits at (2,0). The tool counts these as “long jumps” in the stats. It is a genuine property of the curve, not a render bug. The Hilbert curve avoids these jumps at the cost of more complex computation.

The Morton curve is a true space-filling curve, so its fractal (Hausdorff) dimension is exactly 2.0 in the limit as iterations approach infinity. At any finite iteration the rendered figure is still 1-dimensional in the strict sense – the displayed value reflects the theoretical limit, not the finite drawing.

Each iteration subdivides every cell into 4 sub-cells (NW/NE/SW/SE), so the grid grows to 2 to the n by 2 to the n cells. That gives 4 to the n points total: iter 5 = 1,024; iter 7 = 16,384; iter 9 = 262,144. (Note: an earlier version of this FAQ incorrectly stated iter 9 was 16,777,216 – that figure is actually 4 to the 12. This has been corrected.)

The bit-interleave encoding is O(1) per coordinate with simple bitwise ops, making Morton order ideal for: database B-tree indexing on 2D spatial data, GPU texture swizzling (better than row-major for 2D locality), octree and quadtree linearization, and cache-friendlier matrix layouts. The tradeoff: it preserves some locality on average but breaks badly at quadrant boundaries.

It draws the 2 to the n by 2 to the n cell boundaries at the current iteration. At iter 3 you see an 8×8 grid; at iter 7 you see a 128×128 grid. The grid is most useful at iter 1 through 5 where you can still see individual cells; at iter 7 and above it becomes a solid mesh.

No. The page loads three static files (HTML, CSS, JS), then everything – Morton encoding, drawing, PNG export, clipboard copy – runs in your browser. You can disconnect from the internet after the page loads and the tool keeps working. No analytics, no telemetry, no cookies.

Yes – free, unlimited, no signup, no watermark, no usage cap. Use the PNGs in lectures, papers, blog posts, or art freely. Attribution to is appreciated but not required.