Peano curve
The first space-filling curve (1890): a snake through a 3ⁿ × 3ⁿ grid whose limit covers the whole square.
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Formulas and key values
- Construction rule
Visit the nine blocks of the square column by column, up, down, up; inside each block repeat the whole snake, mirrored so neighbours join.
- Arc lengthIn the graph ≈ 26.96
- Cells visitedIn the graph = 729
- Dimension of the limit
n is the order; the curve lives in the unit square.
Explainer video
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Opens the Creator with a ready-made brief for this curve. Free-form generation is in beta.
Start the video01 · Definition
What is the Peano curve?
In 1890 Giuseppe Peano surprised his contemporaries with a continuous curve that passes through every point of a square. His construction splits the square into nine, visits the nine blocks in a snake (up, down, up), and inside each block repeats the same snake, mirrored so neighbouring blocks join.
The order-n path visits all 9ⁿ cells of a 3ⁿ × 3ⁿ grid. Its limit is a continuous map of a segment onto the square, which showed that a curve, in this sense, can have dimension 2. Hilbert’s simpler two-by-two version came a year later.
02 · Properties
Key properties
- Order n visits 9ⁿ cells, each exactly once.
- In the unit square the order-n path has length 3ⁿ − 3⁻ⁿ.
- The limit is a continuous surjection from [0, 1] onto the square.
- Its image has dimension 2.
- Each block is the whole pattern, mirrored horizontally, vertically, or both.
03 · Questions
Peano curve: common questions
- How is the Peano curve constructed?
- Construction rule: Visit the nine blocks of the square column by column, up, down, up; inside each block repeat the whole snake, mirrored so neighbours join.. n is the order; the curve lives in the unit square.
- What is the arc length of the Peano curve?
- Arc length: L_n = 3^n - 3^(-n). n is the order; the curve lives in the unit square.
04 · Keep exploring