A 1D curve that passes through every point in a 2D square — Hilbert's construction and the path to fractal dimension 2
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Level: 3
Segments: 0
Points visited: 0
Fractal dim ≈ 2.000
Hilbert (1891): At level n, the unit square is divided into 4ⁿ cells, each visited exactly once. The curve length grows as 4ⁿ · (1/2ⁿ) → ∞.
Peano (1890): The first example, divides into 9 cells per level using straight lines.
Key insight: As n→∞, every point in [0,1]² is a limit point of the curve. The map [0,1]→[0,1]² is continuous and surjective — a "space-filling" homeomorphism onto its range.
Dimension: Hausdorff dim = 2 (exactly fills the square).