Quadratic Koch Island Generator - Generate Quadratic Koch Fractals Online

What Is a Quadratic Koch Island?

The quadratic Koch island is a fractal curve that modifies the classic Koch snowflake construction by using square bumps instead of equilateral triangular bumps. Starting with a square as the base shape, each edge is divided into thirds and the middle third is replaced with a square protrusion that extends perpendicular to the original edge. This produces a fractal with a distinctly different character from the traditional Koch snowflake — angular, maze-like, and with four-fold rotational symmetry.

How Is a Quadratic Koch Island Constructed?

The construction follows a recursive algorithm similar to the Koch curve but with a different replacement rule:

Start with a square (iteration 0).
Divide each edge into three equal parts.
Replace the middle third with a square bump — two perpendicular segments extending outward and one segment parallel to the original, creating three new segments in place of one.
Repeat the process for all resulting edges.

Each iteration multiplies the number of segments by 5 (each segment becomes 5 sub-segments). After n iterations, the curve has 4 × 5ⁿ line segments.

Type 1 vs Type 2 Variants

This generator supports two variants of the quadratic Koch island:

Type 1 (All Outward): Every square bump extends outward from the center of the island. This produces a consistently expanding, bulky fractal that grows outward at each iteration.
Type 2 (Alternating): Square bumps alternate direction at each recursion level — outward at one level, inward at the next. This creates an intricate, maze-like pattern with much finer detail and a more complex boundary.

Mathematical Properties

Fractal Dimension: The quadratic Koch curve has a fractal dimension of log(5)/log(3) ≈ 1.4649, which is higher than the standard Koch curve (≈ 1.2619), meaning it fills more of the plane.
Self-Similarity: Like all Koch variants, the quadratic Koch island is exactly self-similar at every scale.
Infinite Perimeter: The perimeter grows by a factor of 5/3 at each iteration, diverging to infinity.
Finite Area: Despite the infinite perimeter, the enclosed area converges to a finite value.

Using This Generator

Choose between Type 1 and Type 2 variants, set the recursion depth, and customize colors and effects. Watch the fractal being constructed with the animation feature and export your creation as PNG, SVG, or JSON.

Frequently Asked Questions

What is the difference between a quadratic Koch island and a regular Koch snowflake?

The regular Koch snowflake uses equilateral triangle bumps on a triangular base, producing a curve with three-fold symmetry. The quadratic Koch island uses square bumps on a square base, producing four-fold symmetry and a higher fractal dimension (≈ 1.4649 vs ≈ 1.2619). The quadratic variant fills more of the plane and has a more angular, maze-like appearance.

What is the difference between Type 1 and Type 2?

Type 1 pushes all square bumps outward from the center at every recursion level. Type 2 alternates the direction of bumps — outward at one level, inward at the next. This alternation creates a much more intricate, interlocking pattern that resembles a maze or a Celtic knot. Both types share the same fractal dimension but look dramatically different visually.

Why does the quadratic Koch island have a higher fractal dimension than the standard Koch curve?

The fractal dimension depends on the self-similarity ratio. The standard Koch curve replaces each segment with 4 copies at 1/3 scale (dimension = log 4/log 3 ≈ 1.26). The quadratic Koch curve replaces each segment with 5 copies at 1/3 scale (dimension = log 5/log 3 ≈ 1.46). More copies per subdivision means the curve fills more of the plane.

How many iterations can I generate?

This tool supports up to 5 iterations. At iteration 5, the fractal has 4 × 5^5 = 12,500 line segments. The quadratic Koch island grows faster than the standard Koch curve (5^n vs 4^n per side), so higher iterations become computationally intensive more quickly. For most visual purposes, iterations 2 to 4 produce excellent results.

Can I use the generated quadratic Koch island for commercial purposes?

Yes. The fractals generated by this tool are created entirely in your browser and can be downloaded as PNG images, SVG vectors, or JSON coordinate data. You are free to use the exported files for any purpose, including commercial projects, educational materials, and artistic works.