How is the Moore curve different from the Hilbert curve?

Both are space-filling curves built from near-identical L-system rules, but the Hilbert curve is an open path with a distinct start and end corner, while the Moore curve is stitched into a closed loop, its endpoint connects back to its starting point, which is useful whenever a continuous cyclic sweep matters more than a fixed start and end.

What does "closed loop" mean for a fractal curve like this?

It means the traced path returns exactly to where it began, so you could keep tracing around it indefinitely without a break. That property makes the Moore curve popular in applications like disk-scheduling or texture-wrapping, where a fixed start and end point isn't wanted.

What does the Iterations input control?

It sets how many times the L-system's L/R production rules are applied. Enter a whole number from 0 to 6; each increase roughly quadruples the segment count and packs the loop 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.

Moore Curve Generator - Closed-Loop Hilbert Curve Variant

Overview

Introduction

The Moore curve is a well-known variant of the Hilbert space-filling curve, built to close into a loop rather than end at a separate corner. This tool renders its standard L-system construction as SVG line art.

Enter an iteration count and the tool expands the Moore curve's L/R production rules that many times, then draws the resulting closed path so you can see how the loop's fine structure grows with each step.

What Is Generate a Moore Curve?

A real, named space-filling curve closely related to the Hilbert curve, built from the same style of L-system alphabet (L and R symbols standing in for turtle sub-paths) but arranged so the finished curve starts and ends at the same point.

It's used in the same kinds of applications as the Hilbert curve, locality-preserving traversal of a 2D grid, but the closed-loop property makes it a better fit whenever a cyclic traversal is wanted instead of a fixed start and end.

How Generate a Moore Curve Works

The generator starts from the axiom "LFL+F+LFL" and repeatedly applies the rules L becomes "-RF+LFL+FR-" and R becomes "+LF-RFR-FL+" for as many iterations as you enter, where + and - are 90-degree turns.

These are the same style of production rules the Hilbert curve uses, but the specific axiom and rule pair are tuned so the resulting path closes into a loop instead of ending at a separate corner from where it started.

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 closed path.

When To Use Generate a Moore Curve

Use it whenever you specifically need a closed-loop space-filling curve, for locality-preserving traversal patterns, generative art, or teaching the difference between open and closed fractal curves.

It's also a good visual companion to a Hilbert curve tool, since the two share nearly identical construction logic but produce a visibly different overall shape.

If a fixed start and end point is fine (or preferred), use the classic open Hilbert curve or the Peano Curve Generator instead.

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

Features

Advantages

A well-documented, classic space-filling curve with a single, unambiguous closed-loop construction, not an invented variant.
Produces a visually balanced, symmetric loop at every iteration, since the curve always returns to its starting point.
Downloadable as both SVG (for further vector editing) and PNG (for quick use anywhere).

Limitations

Segment count grows roughly four-fold per iteration, so the range is capped at 6 to keep rendering and downloads responsive.
As a genuinely space-filling curve, high iteration counts produce very dense line art where individual segments become difficult to distinguish visually.

Examples

Low iteration count

Input

Iterations: 1

Output

A simple, blocky closed loop, the earliest recognizable Moore-curve shape.

At low iterations the closed-loop structure is easy to trace before the finer recursive detail fills in.

Default iteration count

Input

Iterations: 4

Output

A dense, evenly-filled closed loop that visibly sweeps through most of its square bounding area.

The default of 4 shows clear self-similar structure while staying quick to render and download.

Best Practices & Notes

Best Practices

Start at iteration 1 or 2 to see the basic closed-loop shape before increasing toward the denser, more grid-filling higher end of the range.
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 ("moore-curve" in l-system-presets.ts): axiom "LFL+F+LFL", rules L: "-RF+LFL+FR-" and R: "+LF-RFR-FL+", 90-degree turn angle, capped at 6 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 Moore Curve Use Cases

Teaching the difference between open and closed space-filling curves with a real, citable example
Generative art and print patterns built from a symmetric, densely looping curve
Downloadable SVG line art for design projects referencing a classic locality-preserving curve

Common Mistakes

Expecting the exact same visual layout as the Hilbert curve, the production rules are related but the axiom differs specifically to close the path into a loop, so the two curves look distinctly different.
Assuming higher iteration counts always look "better", past iteration 4 or 5 the loop becomes dense enough that the self-similar structure is hard to see without zooming into the downloaded SVG.

Tips

Compare against the Peano Curve Generator and Gosper Curve Generator to see how different space-filling constructions produce very different visual textures.
Zoom into the downloaded SVG in a vector editor to trace how the loop closes back on its own starting point at higher iteration counts.

Generate a Moore Curve