What is the Levy C curve?

It's a classic self-similar fractal curve first described by mathematician Paul Levy in 1938, built by repeatedly replacing a straight segment with a right-angle-ish detour of two shorter segments at 45 degrees. At higher iterations it curls into the dense, C-shaped, space-filling-looking pattern the curve is named for.

Why does this curve curl into a C shape rather than staying a simple zigzag?

The replacement rule, turn 45 degrees, forward, turn -90 degrees, forward, turn 45 degrees, is applied recursively to every new segment at every iteration, and because the same turn pattern compounds across many nested levels, the overall curve bends back on itself into the characteristic C-shaped curl rather than staying flat like a simple zigzag.

Is the Levy C curve open or closed?

It's open, starting from a single straight segment (the axiom) and never closing into a polygon, similar in that sense to the open Koch Triangle curve, though the two use very different rules and turn angles.

Why does the default sit at 11 iterations rather than lower?

The Levy C curve's self-similar curling pattern only becomes visually recognizable after a fair number of iterations, at very low iteration counts it just looks like a few connected line segments. 11 iterations, out of a 15 maximum, is where the dense, dragon-like curling structure the curve is known for becomes clearly visible.

Levy C Curve Generator - Classic Self-Similar Fractal Online

Overview

Introduction

First described by Paul Levy in 1938, the Levy C curve is a genuine landmark of fractal geometry, an open, self-similar curve built from a deceptively simple rule that curls into a dense, C-shaped pattern as it's iterated.

Set an iteration count and the tool renders the curve as an SVG line drawing, downloadable as SVG or PNG.

What Is Generate a Levy C Curve?

A generator for the classic Levy C curve, an open self-similar fractal built with a 45 degree turtle-graphics replacement rule, one of the well-documented curves in fractal geometry alongside the Koch curve and the dragon curve.

The output is pure line art, a single connected open path with no fill, drawn as a stroked SVG polyline.

How Generate a Levy C Curve Works

Construction starts from one straight segment. At each iteration, every existing segment is replaced with the rule turn 45 degrees, forward, turn -90 degrees, forward, turn 45 degrees, applied recursively so the compounding turns fold the curve back on itself across nested levels.

Because the starting shape is a single open segment rather than a closed polygon, the curve never wraps into a boundary, it stays a single continuous open path from a fixed start point to a fixed end point no matter how many iterations are applied.

The rendered path is normalized to the SVG viewbox and drawn as a single stroked polyline, with the same coordinate data backing both the SVG and PNG downloads.

When To Use Generate a Levy C Curve

Use it when you want a recognizable, well-documented fractal curve with a distinctive curling C shape, for teaching, illustration, or generative design work.

It's also a natural point of comparison against other turtle-graphics curves in this catalog, like the Koch curves, since it uses the same underlying construction method with a different rule and angle.

Often used alongside Generate a Koch Triangle, Generate a Triflake Curve and Generate a Vicsek Fractal.

Features

Advantages

A genuine, well-documented classical fractal, not an approximation or variant.
Renders instantly at any supported iteration count, with no server round trip.
Downloadable as clean vector SVG or a rasterized PNG.

Limitations

Capped at 15 iterations to keep segment counts and download sizes reasonable.
Stroke only, there is no fill option since the curve is open and doesn't enclose an area.

Examples

Iteration 0

Input

Iterations = 0

Output

A single straight line segment, the unmodified starting axiom.

With zero iterations the Levy rule hasn't been applied yet, so the output is just the base line.

Iteration 11 (default)

Input

Iterations = 11

Output

A dense, curling, dragon-like C-shaped fractal with thousands of tiny connected segments.

Eleven rounds of the 45 degree replacement rule have compounded into the curve's characteristic self-similar curl.

Best Practices & Notes

Best Practices

Start at a low iteration count, around 3 to 5, to see individual bends in the rule before jumping to the denser default of 11.
Download as SVG if you plan to recolor, scale, or place the curve into other vector artwork, use PNG for a quick raster embed.

Developer Notes

The curve comes from the shared L-system engine with axiom `F` and rule `F -> +F--F+` at a 45 degree turn angle. Segment count doubles per iteration, matched to the 0-15 iteration bound so even the deepest level stays well within a fast, in-browser render.

Generate a Levy C Curve Use Cases

Teaching a classic, well-documented self-similar fractal curve alongside the Koch and dragon curve families
Generating a distinctive curling line pattern for design or generative art work
Producing a downloadable SVG asset of a landmark fractal geometry curve

Common Mistakes

Expecting the curve to close into a boundary, it never does, it stays a single open path from start point to end point.
Judging the curve at very low iteration counts, the characteristic C-shaped curl only becomes visually clear after several rounds of the rule.

Tips

Compare this against Koch Triangle Generator, both are open curves built with the same turtle-graphics method, but very different rules and turn angles produce very different shapes.
Try mid-range iteration counts, around 6 to 8, to see the curl developing gradually before reaching the denser default.

Generate a Levy C Curve