Iterated function systems
Vicsek fractal
Hungarian physicist Tamás Vicsek studied this fractal in the 1980s in the context of percolation, aggregation, and gel formation. Take a 3 × 3 grid; keep the center and the four edge-center sub-squares (forming a plus sign), or the center and four corners (the saltire variant). Recurse.
At a glance
| Designer | Tamás Vicsek, 1980s |
|---|---|
| Hausdorff dimension | log3(5) ≈ 1.465 |
| Self-similar copies | 5 copies at scale 1/3 |
| Variants | Plus-sign (4 edge-centers + center) / saltire (4 corners + center) |
Construction
- Subdivide the unit square into a 3 × 3 grid.
- Keep the center sub-square and either the four edge-centers (plus) or four corners (saltire).
- Recurse on each of the five surviving sub-squares.
Applications
- Percolation theory: model of a self-similar percolating cluster.
- Aggregation models: diffusion-limited and ballistic aggregation give Vicsek-like clusters in some regimes.
- Antenna design: plus-shaped fractal patches for multi-band antennas.
References
- Vicsek, T., Fractal Growth Phenomena, World Scientific, 1989.
- IFS Playground
Try it
Run an interactive playground at /tools/sierpinski-carpet.