Space-filling

Peano curve

In 1890 Giuseppe Peano stunned mathematicians by constructing a continuous surjection from [0, 1] onto [0, 1] × [0, 1], the first space-filling curve. The result forced the reformulation of dimension theory and led directly to Hilbert’s, Lebesgue’s, and Sierpinski’s constructions.

At a glance

Construction

1. Subdivide the unit square into a 3 × 3 grid of nine sub-squares.
2. Traverse the nine sub-squares in a serpentine order.
3. Recurse: replace each sub-square with a smaller 3 × 3 serpentine, oriented so the curve remains continuous.

Why it shocked mathematics

Pre-Peano, “curve” meant 1-dimensional, “area” meant 2-dimensional, and the two were thought to be distinct. Peano’s map is continuous, surjective, and yet maps a one-dimensional interval onto a two-dimensional region. The notions of topological vs. Hausdorff dimension were created in part to disentangle the paradox.

Properties

• Not injective: some points of the square have many pre-images.
• Not differentiable: no tangent direction at any point.
• Nowhere monotone: in both coordinates, oscillates infinitely often.

References

• Peano, G., “Sur une courbe, qui remplit toute une aire plane,” Math. Ann., 1890.
• Sagan, H., Space-Filling Curves, Springer, 1994.
• Hilbert curve

Try it

Run an interactive playground at /tools/lsystems.