Space-filling curves
A space-filling curve is a continuous map whose image covers a region of higher dimension. It is not a wire with physical thickness; the result belongs to the infinite mathematical limit.
Peano’s surprise
In 1890 Giuseppe Peano gave the first construction of a continuous surjection from an interval onto a square. Hilbert soon supplied a geometrically transparent recursive version whose polygonal approximations visit an increasingly fine grid.
Peano’s map is continuous and surjective from the unit interval onto the square. It does not contradict the intermediate value theorem; it contradicts the naive expectation that continuity preserves dimension. Finite approximations visit increasingly fine cells in a prescribed order, and only their uniform limiting map reaches every point. The map must also fail to be one-to-one: multiple parameter values may describe the same point in the square.
Dimension depends on the question
The domain remains a one-dimensional interval in the topological sense, while the image covers a two-dimensional square. Space-filling curves therefore demonstrate why topology, measure and scaling dimension cannot be treated as interchangeable.
The parameter domain has topological dimension one while its image is a two-dimensional region. Different dimensions are therefore correct depending on the definition and object under discussion. Every finite polygonal approximation still behaves as an ordinary line; only the limit fills. The example warns against naming “the dimension” without specifying object and definition, especially when parameter, image and approximation appear in one diagram.
Locality as a practical benefit
Finite Hilbert and related orders tend to keep nearby grid cells close along the one-dimensional traversal. This makes them useful in spatial indexing, cache-aware processing and image algorithms, even though practical implementations stop far before the mathematical limit.
Hilbert and Morton orders have practical value because nearby positions along the ordering often correspond to spatially nearby cells. The preservation is imperfect but useful for caches, databases and image processing. A two-dimensional tile set can be processed as a one-dimensional sequence without repeatedly jumping across memory. A historical curiosity thus becomes a concrete strategy for modern storage hierarchies.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Space-Filling CurveWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
When curves became ‘pathological’
Why continuous nowhere-smooth and space-filling curves disturbed classical intuition—and later became central examples.
The dragon curve
Meet the Heighway dragon, a recursive paper-folding curve whose limit has a space-filling interior without crossing itself.
The Menger sponge
Explore the three-dimensional relative of the Sierpiński carpet, with zero limiting volume and unbounded surface complexity.