How is this different from the standard Koch Snowflake?

The standard Koch Snowflake uses 60 degree turns and triangular bumps built from an equilateral-triangle base. The Quadratic Koch Island uses only 90 degree turns, so every notch is a right-angle step rather than a triangular spike, and it starts from a square base instead of a triangle. The result looks like a crenellated castle wall rather than a spiky star.

Where does this specific rule come from?

It's one of the classic quadratic Koch curve variants documented in Prusinkiewicz and Lindenmayer's "The Algorithmic Beauty of Plants," the standard reference for L-system-based curves. It's a real, well-established construction, not something invented for this tool.

Why does it stay capped at 4 iterations?

The rule replaces every segment with eight shorter ones, so segment count grows roughly 8-fold per iteration. By iteration 4 the outline already has tens of thousands of tiny segments, well past the point of adding visible detail, so the cap keeps the SVG output fast to render and download.

Does the shape stay closed at every iteration?

Yes. Because the rule is applied to all four sides of the starting square symmetrically, the outline always closes back on itself, forming a single continuous island boundary at any iteration from 0 up to the maximum.

Quadratic Koch Island Generator - Right-Angle Koch Fractal Online

Overview

Introduction

Most people picture the Koch curve with its familiar 60 degree triangular bumps, but the quadratic variants replace that with right-angle notches instead. Applied to a square base, the result is a closed, castle-wall-like island outline that's every bit as self-similar as the classic snowflake.

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

What Is Generate a Quadratic Koch Island?

A generator for the quadratic Koch island, a right-angle Koch curve variant applied to all four sides of a square, documented in Prusinkiewicz and Lindenmayer's L-system reference work on plant and fractal modeling.

The output is an unfilled, closed outline, rendered as a single stroked SVG path that traces the full boundary.

How Generate a Quadratic Koch Island Works

The starting axiom is a square, four straight sides connected by 90 degree turns. At each iteration, every segment is replaced by an eight-segment path that steps outward and back in a right-angle notch pattern, all turns are exactly 90 or -90 degrees, no diagonals.

Because the rule is applied uniformly to all four sides at once, the boundary always closes, deeper iterations just add finer notches along the same overall square silhouette.

The resulting path is normalized to the SVG viewbox and drawn as a single stroked polyline; the same coordinate data feeds both the SVG and PNG downloads.

When To Use Generate a Quadratic Koch Island

Use it when you want a Koch-family fractal outline with clean right angles instead of triangular spikes, for a more architectural or pixel-grid-adjacent look.

It's also a good companion to the triangular Koch curves in this catalog for showing how the same replace-the-middle-third idea plays out under a different turn angle.

Often used alongside Generate a Quadratic Cross Fractal, Generate a Cesaro Fractal and Generate a Koch Triangle.

Features

Advantages

Produces a closed, self-similar outline in a single render with no manual assembly of sides.
Right-angle geometry gives a distinctly different visual texture from the more common 60 degree Koch curves.
Downloadable as scalable SVG or a flattened PNG.

Limitations

Capped at 4 iterations since the 8-fold segment growth per step makes deeper levels visually redundant while inflating file size.
Outline only, there is no fill, since the tool renders the boundary as a stroked path rather than a filled polygon.

Examples

Iteration 0

Input

Iterations = 0

Output

A plain square outline, the unmodified starting axiom.

At zero iterations the quadratic rule hasn't been applied, so the shape is just the base square.

Iteration 3 (default)

Input

Iterations = 3

Output

A dense, castle-wall-like square boundary with right-angle notches at three levels of detail.

Every side of the square has had the eight-segment notch rule applied three times over.

Best Practices & Notes

Best Practices

View iteration 1 first to see a single pass of the notch rule clearly before moving to denser levels.
Use the SVG download if you plan to recolor or resize the outline in a vector editor, PNG for a quick raster embed.

Developer Notes

The curve comes from the shared L-system engine with axiom `F+F+F+F` and rule `F -> F+F-F-FF+F+F-F` at a 90 degree turn angle. Segment count grows as 8^n from the four-segment square base, which sets the practical 0-4 iteration bound.

Generate a Quadratic Koch Island Use Cases

Illustrating a right-angle alternative to the classic 60 degree Koch snowflake
Generating a castle-wall or circuit-board-like fractal border for design work
Producing a downloadable SVG asset of a documented L-system curve

Common Mistakes

Assuming this uses the same 60 degree turn as the Koch Snowflake, it doesn't, every turn here is a right angle.
Expecting a fillable shape, the tool renders the boundary as an unfilled stroked path, not a filled polygon.

Tips

Compare this against Quadratic Cross Fractal at the same iteration to see how a different quadratic rule string changes the silhouette despite the shared 90 degree turn angle.
Zoom into the downloaded SVG in a vector editor to inspect the right-angle notch pattern at the segment level.

Generate a Quadratic Koch Island