Geometric
Sierpinski triangle
Constructed by Wacław Sierpiński in 1915, the Sierpinski triangle is the canonical example of an exactly self-similar fractal. Start with a filled equilateral triangle; remove the central inverted triangle; repeat forever inside each remaining triangle. Three copies of itself, each at half scale.
At a glance
| Designer | Wacław Sierpiński, 1915 |
|---|---|
| Hausdorff dimension | log₂(3) ≈ 1.585 |
| Area | Zero (Lebesgue measure) |
| Self-similar copies | 3 copies at scale 1/2 |
| Topological dimension | 1 |
Constructions (all equivalent)
- Subdivision: start with a filled triangle, remove the central inverted triangle, recurse on the three corner triangles.
- Chaos game: pick three vertices in the plane, start at any point. Repeatedly: pick a random vertex, move halfway to it, plot the new point. Skip the first few. The plotted points fill the Sierpinski triangle.
- Iterated function system: the attractor of the three affine maps fi(x) = (x + vi) / 2 where viare the three corner positions.
- Pascal’s triangle mod 2: shade in the odd binomial coefficients. The Sierpinski triangle emerges.
Dimension derivation
Three self-similar copies at scale 1/2: for a self-similar set with N copies at scale 1/r, the similarity dimension is log(N) / log(r). So log(3) / log(2) ≈ 1.585.
Cousins
- Sierpinski carpet: 2D square analogue, 8 copies at scale 1/3, dimension log3(8) ≈ 1.893.
- Sierpinski tetrahedron: 3D, 4 copies at scale 1/2, dimension exactly 2.
- Menger sponge: 3D analogue of the carpet.
References
- Sierpiński, W., “Sur une courbe dont tout point est un point de ramification,” Comptes Rendus, 1915.
- Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
- Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2014.
- Barnsley, M., Fractals Everywhere, Academic Press, 1993.
Try it
Run an interactive playground at /tools/sierpinski.