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.
Quick quiz
Test yourself on sierpinski
10 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.The Sierpinski triangle is constructed by removing what from each triangle?
Q2.The Hausdorff dimension of the Sierpinski triangle is:
Q3.How many contractive maps make up the Sierpinski triangle IFS?
Q4.Pascal's triangle mod 2 reveals which fractal?
Q5.The chaos game on three vertices with step factor 1/2 yields:
Q6.Topological dimension of the Sierpinski triangle is:
Q7.Sierpinski's 3D analogue, the Menger sponge, is built from how many copies per side?
Q8.Sierpinski first introduced the triangle in:
Q9.Total area of the Sierpinski triangle after infinitely many iterations is:
Q10.An L-system axiom F-G-G with rule F → F-G+F+G-F, G → GG, angle 120° draws: