How is this different from the Sierpinski arrowhead curve?

Both are single unbroken paths built with L-systems, but they trace different shapes with different turn angles. The Sierpinski Arrowhead Curve Generator uses 60-degree turns and traces the triangular Sierpinski gasket. This tool uses 45-degree turns and traces a curve that fills a square-ish region, closer in spirit to a space-filling curve like the Peano or Hilbert curve than to the triangle itself.

Why is it sometimes called the Sierpinski square curve?

Because its right-angle-turn construction weaves back and forth to progressively fill a square region, rather than tracing the outline of a triangle. The name distinguishes it from the arrowhead curve, which is also linked to Sierpinski's work but stays triangular.

What does the Iterations input control?

It sets how many times the L-system's X production rule is expanded before the turtle draws the path. Enter a whole number from 0 to 6; each increase multiplies the segment count and traces a finer, denser approximation of the curve.

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.

Sierpinski Curve Generator - Space-Filling Square Curve Online

Overview

Introduction

The Sierpinski curve is a right-angle-turn space-filling curve, a single continuous path that, as it winds back and forth with 45-degree turns, progressively fills a square-shaped region. This tool renders its standard L-system construction as SVG line art.

Enter an iteration count and the tool expands the curve's X production rule that many times, then draws the resulting one-stroke path so you can see the fill pattern grow denser with each step.

What Is Generate a Sierpinski Curve?

A documented space-filling curve, sometimes named the Sierpinski square curve to distinguish it from the triangular Sierpinski arrowhead curve. It shares the Sierpinski name and a family resemblance in how it's generated, but it fills a square-ish area rather than tracing a triangle's outline.

It's built from an L-system with axiom "F+XF+F+XF" and a single rule for the letter X, interpreted by turtle graphics with 45-degree turns, the same general technique used for the Sierpinski arrowhead curve, Peano curve, and other L-system fractals in this category, just with a different axiom, rule, and angle.

How Generate a Sierpinski Curve Works

The generator starts from the axiom "F+XF+F+XF" and repeatedly replaces every X in the string with "XF-F+F-XF+F+XF-F+F-X" for as many iterations as you enter, where F draws a segment forward, + turns 45 degrees one way, and - turns 45 degrees the other way.

X itself is a non-drawing placeholder letter used only to control where the rule expands; only the F letters in the fully expanded string move the turtle forward and draw a segment.

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 unbroken path.

When To Use Generate a Sierpinski Curve

Use it whenever you want a right-angle, square-filling curve for generative art, plotter-friendly line art, or teaching how L-system rules and turn angles change the shape a curve traces.

It's a useful comparison piece next to the Sierpinski Arrowhead Curve Generator, both carry the Sierpinski name and use turtle-graphics L-systems, but one fills a square region with right-angle turns and the other traces a triangle with 60-degree turns.

If you want the filled, recursively-subdivided triangle instead, use the Sierpinski Triangle Generator.

Often used alongside Generate a Sierpinski Triangle and Generate a Sierpinski Arrowhead Curve.

Features

Advantages

A documented space-filling curve with a fixed, unambiguous axiom and rule, not an invented variant.
Renders as one continuous unbroken path, well suited to pen-plotter or single-stroke line-art use cases.
Downloadable as both SVG (for further vector editing) and PNG (for quick use anywhere).

Limitations

It approximates a square-filling curve rather than a triangular gasket, despite sharing the Sierpinski name with the triangle and arrowhead curve, which can be confusing on first glance.
At higher iteration counts near the maximum of 6, the dense back-and-forth path can make individual segments hard to distinguish without zooming into the downloaded SVG.

Examples

Low iteration count

Input

Iterations: 1

Output

A simple angular zigzag of a handful of segments, hinting at the right-angle turning pattern.

At iteration 1 the axiom has only been expanded once, so the individual 45-degree turns are easy to follow.

Default iteration count

Input

Iterations: 4

Output

A dense, woven path that visibly fills a square-shaped region of the canvas.

The default of 4 shows the space-filling behavior clearly while staying quick to render and download.

Best Practices & Notes

Best Practices

Start at iteration 1 or 2 to see the individual right-angle turns clearly before increasing toward the maximum of 6, where the fill pattern dominates.
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 "sierpinski-curve" preset in the shared L-system engine: axiom "F+XF+F+XF", rule X: "XF-F+F-XF+F+XF-F+F-X", 45-degree turn angle, default 4 iterations, capped at a maximum of 6. Like every other L-system tool in this category, expansion and turtle interpretation go through the shared generateFractalFromPreset/L-system engine before geometry normalization and SVG rendering.

Generate a Sierpinski Curve Use Cases

Teaching how axiom, rule, and turn angle together determine an L-system curve's shape
Pen-plotter or single-stroke generative art referencing a square-filling motif
Downloadable SVG line art for design projects wanting a woven, right-angle fractal pattern

Common Mistakes

Expecting a triangular shape because of the Sierpinski name, this curve fills a square-ish region with right-angle turns, for the triangular version see the Sierpinski Triangle Generator or Sierpinski Arrowhead Curve Generator.
Setting iterations to the maximum right away, past 4 or 5 the path becomes dense enough to make individual segments hard to distinguish visually.

Tips

Compare it directly against the Sierpinski Arrowhead Curve Generator to see how axiom, rule, and turn angle alone produce a square-filling curve versus a triangular one.
Zoom into the downloaded SVG in a vector editor to trace the individual right-angle turns at higher iteration counts.

Generate a Sierpinski Curve