Tool

T-square playground

Each square places four half-size squares at its corners. After several iterations the figure fills space, but with characteristic holes that never close.

Iteration depth: 6 (squares drawn: 5,461)

Construction rule

step 0: filled square of side s, centered at origin
step n: for each square of side s', center (cx, cy):
          place 4 half-size squares of side s'/2, centered at
            (cx − s'/2, cy − s'/2), (cx + s'/2, cy − s'/2),
            (cx − s'/2, cy + s'/2), (cx + s'/2, cy + s'/2)
        recurse

IFS form (4 maps at scale 1/2):
  f_k(x, y) = (x/2, y/2) + offset_k       for k = 1..4
  offset_k anchors the half-square at one of the four corners

Hausdorff dim: log 4 / log 2 = 2  (positive area, non-simply-connected)

Background at /fractals/t-square.

FAQ