Sierpinski Carpet

Recursive subdivision: remove center ninth at each depth — fractal dimension log8/log3 ≈ 1.893
3
0
—
Holes Removed
—
Area Remaining
1.893
Fractal Dim
The Sierpinski carpet is constructed by dividing a square into a 3×3 grid and removing the center square, then recursively applying the same rule to each of the 8 remaining squares. At depth d: 8d filled squares remain, each of size (1/3)d. Area remaining = (8/9)d → 0. Fractal dimension = log(8)/log(3) ≈ 1.893 — strictly between 1 and 2.