Iterated function systems
T-square fractal
The T-square is a Hausdorff-dimension-2 fractal whose early iterations resemble nested T-shapes built from squares. It is the attractor of a four-map IFS that takes the unit square to half-size squares anchored at each of its four corners.
At a glance
| Hausdorff dimension | 2 (limit has positive area) |
|---|---|
| Self-similar copies | 4 copies at scale 1/2 |
| Not simply connected | Riddled with infinitely many holes |
Construction
- Start with the unit square.
- Place a square of half side-length centered on each of its four corners.
- Recurse on each new square.
Properties
- Full dimension: 4 copies at ratio 1/2 give dim = log 4 / log 2 = 2; the limit set has positive Lebesgue measure.
- Non-simply-connected: at every finite step the union has many holes which never quite close.
- Self-similar boundary: the boundary of the limit region is itself fractal.
References
- Falconer, K., Fractal Geometry, Wiley, 2014.
- IFS Playground
Try it
Run an interactive playground at /tools/t-square.
Quick quiz
Test yourself on t-square
5 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.The T-square fractal is built by recursively placing smaller squares at:
Q2.Hausdorff dimension of the T-square is:
Q3.Despite dimension 2, the T-square has interesting structure because:
Q4.The 'T' in T-square refers to:
Q5.The IFS for the T-square uses four equal-probability affine maps with ratio: