T-Square Curve Generator - T-Square Fractal Visualizer

What is the T-Square Fractal?

The T-Square Fractal is a beautiful, mathematically rich two-dimensional geometric fractal generated by recursively placing smaller squares at the corner vertices of existing squares. Starting from a single central square, the boundaries of the shape expand outward recursively, eventually forming a highly complex and self-similar tiling structure.

The name "T-Square" originates from the drafting tool of the same name, reflecting the clean, linear, and perpendicular intersections of the recursive square edges at $90^\circ$ angles.

Mathematical Formulation

The construction of the T-Square Fractal follows a simple recursive rule at each step $n$:

For every newly drawn square of side length $S$, we draw 4 smaller squares centered at its 4 corners.
The side length of each child square is exactly scaled by a factor of $R = \frac{1}{2}$.
To prevent drawing overlapping squares back towards the parent center, we only recurse on the $N = 3$ newly exposed "free" corners of each child square.

The Hausdorff fractal dimension $D$ of the T-Square Fractal is computed as: $$D = \frac{\ln(N)}{\ln(1/R)} = \frac{\ln(4)}{\ln(2)} = 2.0$$

Having a fractal dimension of exactly $2.0$ indicates that, in the limit as $n \to \infty$, the T-Square Fractal completely fills a bounded two-dimensional area. In fact, it is proved that the area of the T-Square fractal asymptotically approaches the entire bounding square box, yet the perimeter of the shape becomes infinitely long.

Interactive Visualization Features

Our T-Square Generator is built to give you maximum control over the fractal's layout and style:

Recursion Depth: Adjust the depth from $0$ up to $6$ to watch the fractal evolve from a single block into a highly intricate woven screen.
Filled vs Outline Modes: Toggle between filled blocks (which highlights the space-filling properties) and outline strokes (which reveals the delicate lattice-like nested boundaries).
Cyberpunk Styling: Style the squares with vibrant neon cyan and emerald color schemes, complete with custom background and neon glow toggles.
Step-by-Step Drawing: Enable the live draw animation to watch each recursive layer unfold corner-by-corner.
Asset Export: Download high-resolution PNGs, scale-free vector SVGs, or copy the computed raw vertex coordinate arrays as JSON.

Frequently Asked Questions

How does the area of the T-Square fractal grow with iterations?

If the starting square has an area of $1$, each recursion layer adds non-overlapping square subdivisions. Specifically, at iteration $1$, we add 4 squares of side $1/2$, adding an area of $4 \times (1/4) = 1$. However, because some squares overlap in later iterations, the total area grows asymptotically, eventually filling exactly $\frac{2}{3}$ of the bounding box without overlap, and up to $100\%$ with overlaps taken into account in the limit.

Why is it important to prevent recursing on the parent corner?

Preventing recursion on the parent corner is mathematically critical. If the algorithm recursed on all 4 corners of every child square, it would draw smaller squares directly back on top of the parent square's center, resulting in infinite redundancy and complete visual overcrowding. Restricting recursion to the 3 "free" corners ensures the fractal grows cleanly outward.

What is the fractal dimension of a T-Square compared to the Sierpinski Carpet?

The T-Square has a Hausdorff dimension of $2.0$, meaning it is a plane-filling fractal. In contrast, the Sierpinski Carpet has a dimension of $\frac{\ln(8)}{\ln(3)} \approx 1.8928$, meaning it contains "holes" at all scales and never completely fills the 2D plane.