Why does this look like a cross or plus sign instead of a square outline?

It comes from the specific replacement rule. Like Quadratic Koch Island, this curve uses only 90 degree turns starting from a square base, but its rule string steps the boundary in and out in a pattern that carves deep notches toward the center on each side. At a few iterations that in-and-out stepping reads visually as a cross or plus-sign silhouette rather than a smoothly notched square.

How is this different from the Quadratic Koch Island tool?

Both start from a square and use only 90 degree turns, so they're in the same quadratic Koch family. The difference is the replacement rule string itself, Quadratic Koch Island's rule produces a castle-wall-like boundary that still reads as a square, while this rule's specific sequence of forward and turn steps produces the cross-like silhouette instead.

Why is the maximum iteration count only 3?

The rule replaces each segment with a long, multi-part path, over 30 new segments per original one, so segment count grows extremely fast. By iteration 3 the shape already has tens of thousands of segments and the cross silhouette is fully legible, going further would mainly inflate the SVG file size without adding visible detail.

Does the shape stay closed at every iteration?

Yes, the rule is applied uniformly to all four sides of the starting square, so the boundary always closes back on itself, at any iteration from 0 up to the maximum of 3.

Quadratic Cross Fractal Generator - Right-Angle Koch Cross Online

Overview

Introduction

Not every quadratic Koch rule produces a smooth castle-wall silhouette. This particular right-angle replacement rule, applied to a square base, steps deep enough toward the center of each side that the resulting boundary reads as a cross or plus-sign shape rather than a notched square.

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

What Is Generate a Quadratic Cross Fractal?

A generator for a quadratic (90 degree turn) Koch-family curve applied to a square base, using a longer, more elaborate replacement rule than the classic quadratic Koch island, one whose in-and-out stepping pattern produces a cross-shaped silhouette.

The output is an unfilled, closed outline rendered as a single stroked SVG path.

How Generate a Quadratic Cross Fractal Works

The starting axiom is a square, four sides joined by 90 degree corners. At each iteration, every segment is replaced by a long sequence of shorter segments and turns, all at exact 90 degree angles, that steps the boundary inward and outward multiple times rather than tracing a single simple notch.

Because the rule steps far enough inward along each side, the four sides' notches meet up visually near the center, and the overall silhouette resolves into a cross or plus-sign shape once the rule has been applied even once or twice.

The rule is applied uniformly to all four sides of the square, so the boundary always closes, and the final path is normalized to the SVG viewbox for both the SVG and PNG downloads.

When To Use Generate a Quadratic Cross Fractal

Use it when you want a Koch-family fractal boundary that reads clearly as a cross or plus-sign silhouette rather than a smoother notched square.

It's a useful companion to Quadratic Koch Island for showing how much the specific replacement rule string, not just the turn angle, shapes the final silhouette.

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

Features

Advantages

Produces a clearly recognizable cross silhouette from a genuine quadratic Koch-family rule, no manual shaping required.
Closed outline at every iteration, usable as a self-contained shape.
Downloadable as scalable SVG or a flattened PNG.

Limitations

Capped at 3 iterations, since the rule's roughly 30-fold segment growth per step makes deeper levels expensive to render for little added visible detail.
Outline only, there is no fill option since the shape is rendered as a stroked path.

Examples

Iteration 0

Input

Iterations = 0

Output

A plain square outline, the unmodified starting axiom.

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

Iteration 2 (default)

Input

Iterations = 2

Output

A clearly cross-shaped or plus-sign-like boundary, with deep right-angle notches carved into all four sides.

Two rounds of the long replacement rule have carved each side inward enough for the cross silhouette to fully resolve.

Best Practices & Notes

Best Practices

Start at iteration 1 to see the first pass of the rule before moving to the denser default of 2.
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+FF-FF-F-F+F+FF-F-F+F+FF+FF-F` at a 90 degree turn angle, considerably longer than the Quadratic Koch Island preset's rule string despite sharing the same axiom and angle. Iteration is capped at 3 given the roughly 30x per-step segment growth.

Generate a Quadratic Cross Fractal Use Cases

Generating a cross or plus-sign fractal silhouette for design or iconography work
Comparing rule-string variety within the same quadratic (90 degree) Koch family
Producing a downloadable SVG asset of a right-angle fractal boundary

Common Mistakes

Expecting the same castle-wall silhouette as Quadratic Koch Island, the rule string here is different and specifically produces a cross shape instead.
Setting a high iteration count expecting more visible change, the cross silhouette is already fully formed by iteration 2, deeper levels mainly add fine detail.

Tips

Compare this directly against Quadratic Koch Island at the same iteration count to see how much the replacement rule string alone can change a fractal's silhouette.
Try Vicsek Fractal for another cross-shaped construction in this catalog, built from an entirely different method, recursive grid subdivision rather than a turtle-graphics rule.

Generate a Quadratic Cross Fractal