Quadratic Cross Fractal Generator

What is the Quadratic Cross Fractal (Minkowski Sausage)?

The Quadratic Cross Fractal, also widely known as the Minkowski Sausage or Minkowski Curve, is a classic space-filling fractal curve formed exclusively with $90^\circ$ right angles. Unlike the traditional Koch curve which uses $60^\circ$ triangular bumps, the Minkowski Sausage replaces each line segment recursively with a square protrusion generator consisting of 8 smaller segments.

In its closed form, it is referred to as the Minkowski Island. The fractal starts from a simple square shape, and by recursively applying the generator rule to each of the 4 outer edges, it creates a beautifully intricate boundary that resembles a hyper-complex labyrinth or puzzle board.

Mathematical Formulation

The mathematical magic of the Quadratic Cross Fractal lies in its recursive scale division. At each iteration $n$:

Every individual segment is divided into $N = 8$ smaller sub-segments.
The length of each new sub-segment is scaled down by a factor of $S = \frac{1}{4}$.

To compute the Hausdorff fractal dimension $D$, we use the relation: $$D = \frac{\ln(N)}{\ln(1/S)} = \frac{\ln(8)}{\ln(4)} = 1.5$$

With a dimension of exactly $1.5$, the Minkowski Sausage sits perfectly halfway between a $1$-dimensional line and a $2$-dimensional plane, demonstrating a high degree of space-filling complexity while retaining its exact self-similarity at all zoom levels.

Features of This Generator

Our interactive generator is designed to let you explore the geometry of the Minkowski Sausage with full customizability:

Sausage & Island Variants: Toggle between the open Minkowski Curve (Sausage) and the closed Minkowski Island. Support for both standard Type 1 (outward) and Type 2 (alternating inward-outward) replacement rules.
Color Customization: Render with modern palettes including Neon Cyberpunk, Rainbow Spectrums, two-color gradients, or classic solid colors.
Live Drawing Animation: Control the drawing step-by-step to visualize the recursive turtle-graphics paths in real-time.
High-Quality Asset Export: Export your creations directly as high-resolution PNGs, vector SVGs for styling, or raw coordinate JSON data.

Frequently Asked Questions

What is the difference between Minkowski Type 1 and Type 2 curves?

Type 1 (Outward) replacement rules push all square protrusions consistently in one direction relative to the segment vector. Type 2 (Alternating) alternates the direction of protrusions, pushing some inward and others outward, resulting in a symmetrical, maze-like geometry that grows tightly packed.

Why is the recursion depth limited to 4?

Because each iteration multiplies the number of segments by 8, the total segments grow exponentially as $8^n$. For a closed Minkowski island:

Order 0 has 4 segments.
Order 1 has 32 segments.
Order 2 has 256 segments.
Order 3 has 2,048 segments.
Order 4 has 16,384 segments.

Limiting it to Order 4 ensures optimal rendering performance directly in your browser without lag.

What is the fractal dimension of a T-Square vs Minkowski curve?

The Minkowski curve has a Hausdorff dimension of $1.5$, whereas a T-Square fractal has a dimension of $\frac{\ln(4)}{\ln(2)} = 2.0$. This means the T-Square completely fills the 2D plane in the limit, whereas the Minkowski curve remains a highly complex boundary.