What makes the Peano curve historically significant?

Published by Giuseppe Peano in 1890, it was the first known example of a continuous curve that passes through every point of a filled square, a discovery that upended the 19th-century assumption that a one-dimensional curve couldn't cover a two-dimensional area.

Why does the curve fill a square in a boustrophedon pattern?

Each iteration replaces one straight segment with nine shorter segments arranged in a back-and-forth, ox-plowing sweep across a 3-by-3 sub-grid. Repeating that replacement recursively is what lets the curve eventually pass arbitrarily close to every point in the square.

What does the Iterations input control?

It sets how many times the nine-segment replacement rule is applied. Enter a whole number from 0 to 4; each increase multiplies the segment count by nine and packs the sweeping pattern into a finer grid.

Can I get the curve as an image file?

Yes. Once a valid iteration count renders a curve, the Download SVG button saves the vector path as-is, and Download PNG rasterizes the same view to a PNG image.

Peano Curve Generator - Classic Space-Filling Curve Online

Overview

Introduction

The Peano curve was the first space-filling curve ever published, a single continuous line that, in the limit, touches every point of a square. This tool renders its classic order-3 construction as SVG line art.

Enter an iteration count and the tool expands the standard Peano L-system rule that many iterations, then draws the resulting path so you can see how a simple back-and-forth replacement rule fills more and more of the square as it repeats.

What Is Generate a Peano Curve?

A real, named mathematical curve, first described by Giuseppe Peano in 1890 as a proof that a continuous curve can fill an entire two-dimensional area, contradicting the intuition that curves are one-dimensional.

This tool uses its standard base-3 L-system definition: a single forward segment expands into nine segments arranged in a boustrophedon (back-and-forth) sweep, applied recursively.

How Generate a Peano Curve Works

The generator starts from the axiom "F" and repeatedly applies the rule F becomes "F+F-F-F-F+F+F+F-F" for as many iterations as you enter, where + and - are 90-degree left and right turns.

That nine-segment replacement is the classic order-3 Peano construction, each straight segment turns into a sweep through a 3-by-3 sub-grid, alternating direction row by row so the resulting path never crosses itself.

The expanded instruction string is walked by a turtle to produce line segments, which are then rescaled to fit the SVG viewbox and drawn as one continuous path.

When To Use Generate a Peano Curve

Use it to see a real space-filling curve in action, for teaching continuity and dimension in a way that's easier to grasp visually than algebraically.

It's also useful as generative-art source material, the dense, evenly-spaced sweeping pattern makes distinctive geometric backgrounds and prints.

For a closed-loop space-filling curve instead of Peano's open one, use the Moore Curve Generator.

Often used alongside Generate a Moore Curve, Generate a Gosper Curve and Generate a Sierpinski Arrowhead Curve.

Features

Advantages

A well-documented, classic mathematical curve with a single, unambiguous construction, not an invented variant.
Fills its bounding square evenly at every iteration, producing a visually balanced pattern at any depth.
Downloadable as both SVG (for further vector editing) and PNG (for quick use anywhere).

Limitations

Segment count grows nine-fold per iteration, so the iteration range is capped at 4 to keep rendering and downloads responsive; higher orders exist mathematically but aren't offered here.
As a genuinely space-filling curve, high iteration counts produce very dense line art where individual segments become difficult to distinguish visually.

Examples

Zero iterations

Input

Iterations: 0

Output

A single straight line segment, the bare axiom before any Peano sweeping is applied.

At 0 iterations there's no replacement yet, just the starting segment.

Default iteration count

Input

Iterations: 3

Output

A dense, evenly-spaced sweeping pattern that visibly fills most of its square bounding area.

By order 3 the boustrophedon sweeping pattern that defines the Peano curve is clearly visible at multiple scales.

Best Practices & Notes

Best Practices

Start at order 1 or 2 to see the basic nine-segment sweep before jumping to order 3 or 4, where the recursive self-similarity becomes the dominant visual feature.
Download the SVG rather than the PNG if you plan to recolor, scale, or further edit the curve, vector output has no resolution limit.

Developer Notes

Implemented as the standard L-system preset ("peano-curve" in l-system-presets.ts): axiom "F", rule F: "F+F-F-F-F+F+F+F-F", 90-degree turn angle, capped at 4 iterations. Expansion and turtle interpretation both go through the shared generateLSystemCurve/drawLSystem engine used by every other L-system tool in this category, before geometry normalization and SVG rendering.

Generate a Peano Curve Use Cases

Teaching space-filling curves, continuity, and fractal dimension with a real, citable example
Generative art and print patterns built from a dense, evenly-swept geometric curve
Downloadable SVG line art for design projects referencing a classic mathematical fractal

Common Mistakes

Assuming higher iteration counts always look "better", past order 3 or 4 the sweep becomes so dense that the self-similar structure is hard to see without zooming into the downloaded SVG.
Confusing this with the Hilbert or Moore curves, all three are space-filling curves but each uses a different production rule and produces a visually distinct sweep pattern.

Tips

Compare against the Moore Curve Generator and Gosper Curve Generator to see how different space-filling and plane-filling constructions produce very different visual textures.
Zoom into the downloaded SVG in a vector editor to trace the boustrophedon sweep row by row at higher iteration counts.

Generate a Peano Curve