🌀 Space-Filling Curves

The 3D Hilbert curve recursively subdivides a cube into 8 sub-cubes and visits them in a Gray-code order so that consecutive sub-cubes share a face. Each sub-cube is itself a smaller Hilbert curve, but rotated and reflected so that its entry point touches the exit of the previous sub-cube and its exit lines up with the next. The standard implementation applies these per-octant rotations recursively (or equivalently uses Skilling's transpose/Gray-code algorithm) to generate the index → (x, y, z) mapping for all 8ⁿ cells.

Both are space-filling curves, but they subdivide differently. The Hilbert curve divides each square into 4 sub-squares (base-2 grid) and uses a U-shaped path with two reflections at each step. The Peano curve divides into 9 sub-squares (base-3 grid) with an S-shaped scan, and was the first space-filling curve published (Giuseppe Peano, 1890 — one year before Hilbert). Hilbert's version is generally preferred in computing because the power-of-2 grid aligns naturally with binary addressing.

The Z-order or Morton curve is constructed by interleaving the binary representations of the coordinates. It is extremely fast to compute — requiring only bit-interleaving operations — and is used in many spatial indexes including S2 geometry and some GPU texture layouts. However, it has a locality defect: it makes diagonal jumps between quadrants, so clusters of nearby points can map to widely separated 1D indices more often than the Hilbert curve does.

The Hausdorff (fractal) dimension of the 2D Hilbert curve is exactly 2; for the 3D Hilbert curve it is exactly 3. This is computed from the self-similarity: in 3D each step replaces one segment with 8 scaled copies at factor 1/2, so D = log(8)/log(2) = 3. A dimension equal to the embedding space means the curve is as "space-filling" as that region, though it still has zero volume in the Lebesgue measure sense.

At order n the 3D Hilbert curve visits every cell of a 2ⁿ × 2ⁿ × 2ⁿ grid, so it has 8ⁿ points connected by 8ⁿ − 1 segments. Order 1 has 8 points, order 2 has 64, order 3 has 512, order 4 has 4,096, and order 5 has 32,768 points. This simulation caps the order at 5 to keep the vertex count and rendering performance reasonable in the browser.

Yes — the Hilbert curve is a mathematically proven continuous function from [0, 1] onto the unit square (or cube in 3D). The proof uses the fact that each finite-order approximation is uniformly continuous and the sequence converges uniformly, so the limit function is also continuous by the uniform limit theorem. However, the limiting curve is not differentiable anywhere — it has no tangent at any point — making it an example of a continuous-but-nowhere-differentiable function.