Draw T Square Fractal

The T-square fractal is a 2D self-similar set built by repeatedly adding smaller squares at the corners of previously placed squares. Start with one square. At each iteration, every existing square gets four child squares of half its side, each child centered exactly at one corner of the parent. The new squares extend outside the parent’s outline, so the figure grows like a stylized cross. After infinitely many iterations the limit is a connected, space-filling region (similarity dimension 2).

(4 to the N+1 minus 1) divided by 3 – a geometric series, because the recursion at each non-leaf level pushes the parent itself plus 4 recursive children. Iter 1 = 5; iter 2 = 21; iter 3 = 85; iter 5 = 1,365; iter 8 = 87,381. An older version of this FAQ claimed 4 to the N (1,024 at iter 5; 65,536 at iter 8) – that count would only be the leaves, but this tool draws every level, so the real count is larger. The stats line shows the actual number.

The similarity dimension is log(4) divided by log(2), which is exactly 2.0. Each parent square is replaced (over the next iteration) by 4 self-similar children at scale 1/2, and that gives dim = log(N) / log(1/r) = log(4) / log(2) = 2. A dimension of 2 means the fractal is space-filling: in the infinite limit it covers a region of nonzero area with no “thickness deficit”. At any finite iteration the rendered figure is still made of overlapping squares – the value is the asymptote.

That is the classical T-square definition. Earlier code in this tool centered the children at the parent’s quadrant midpoints (offset = parent_side / 4) – that produces a quadtree-style 2×2 subdivision, not the T-square cross pattern. The current implementation uses offset = parent_side / 2 (child center exactly at parent corner), which matches the canonical figure shown in fractal-geometry references.

Each square’s recursion level (0 for the root, 1 for its 4 children, etc.) modulo 2 picks between the primary and secondary color. Even levels take primary; odd levels take secondary. This makes the recursion hierarchy easy to read – you can count generations by counting color swaps.

Because each parent’s 4 children stick outward from its corners, the silhouette grows in 4 directions. After many iterations the silhouette stabilizes into the famous T-square “cross-shaped” attractor, which when filled in is essentially the closed convex region the fractal approximates. Stop at iter 3 or 4 to see the cross shape clearly; at iter 8 the interior is fully colored.

Each child square has area 1/4 of its parent, so the total area added per level is 4 × (parent area / 4) = parent area. The series sums to a finite geometric total (the squares overlap heavily – the union of all squares is bounded). So even though the count of squares grows like 4ᴺ, the union of their pixels has finite area; that is precisely the space-filling cross.

No. The tool loads three static files (HTML, CSS, JS) and then runs entirely in your browser – recursion, drawing, PNG export, clipboard copy. You can disconnect from the internet after the page loads. No analytics, no tracking, no cookies.

Yes – free, unlimited, no signup, no watermark. Use the PNGs in lectures, papers, blog posts, or art freely. Attribution to is appreciated but not required.