Geometric
Sierpinski carpet
Wacław Sierpiński introduced the carpet in 1916 as a 2D analogue of the Cantor set: subdivide a unit square into a 3 × 3 grid, remove the central sub-square, and recurse on the other eight. The carpet is a universal space for one-dimensional planar continua.
At a glance
| Designer | Wacław Sierpiński, 1916 |
|---|---|
| Hausdorff dimension | log3(8) ≈ 1.8928 |
| Topological dimension | 1 |
| Lebesgue measure | Zero |
| Self-similar copies | 8 copies at scale 1/3 |
Construction
- Start with the unit square
[0, 1]². - Subdivide into a 3 × 3 grid of nine sub-squares.
- Remove the central sub-square.
- Recurse on the remaining eight.
Universal curve
Whyburn proved that every planar continuum of topological dimension 1 with no interior embeds homeomorphically into the Sierpinski carpet. In 3D the same role is played by the Menger sponge.
Where it appears
- Antenna design: Sierpinski-carpet patches are wideband.
- 3D-printed metamaterials: carpet-patterned stiffness profiles.
- Mathematical universality: planar continuum embedding theorem.
References
- Sierpiński, W., “Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée,” Comptes Rendus, 1916.
- Whyburn, G., Analytic Topology, AMS, 1942.
- Sierpinski triangle · Menger sponge
Try it
Run an interactive playground at /tools/sierpinski-carpet.