Sierpiński triangle and carpet
Sierpiński constructions repeatedly keep selected subshapes and remove the rest. Their simple grids make scaling, dimension and connectivity unusually visible.
Triangle by subtraction or chaos game
Divide a triangle into four congruent triangles and remove the center, then repeat on the remaining three. The same limit appears through the chaos game: repeatedly move halfway from the current point toward a randomly chosen vertex.
The triangle keeps three copies at half scale in every stage, giving similarity dimension log 3/log 2. The chaos game reaches the same set by a surprisingly different route: choose a vertex at random and move halfway from the current point toward it. After a transient, the points lie almost entirely on the Sierpiński triangle despite the random choices. The contractions determine geometry; randomness only distributes samples.
Carpet and dimension
The carpet divides a square into nine parts and removes the center, retaining eight copies at one third scale. Its similarity dimension is log 8 / log 3, while its area tends to zero.
The carpet keeps eight of nine squares at one-third scale, so its similarity dimension is log 8/log 3. Its area is multiplied by 8/9 at each stage and vanishes in the limit. Yet the set remains connected and contains paths across many scales. The two quantities describe different aspects of the construction: area measures occupied proportion, while dimension measures the growth of detail coverage.
Structure survives the missing area
The triangle remains connected through points; the carpet contains paths around holes of every construction scale. These examples show that area, connectivity and dimensional complexity describe different aspects of a set.
The central hole is not an isolated defect; it repeats inside every remaining copy. Its pattern survives even as total area tends to zero. In a finite image, the effect depends strongly on stage depth: too few iterations show only coarse gaps, while too many create moiré and swallowed lines. A successful rendering matches construction depth to the pixel grid.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Sierpiński SieveWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
The Cantor set
Remove middle thirds forever and discover an uncountable set with zero length, self-similarity and dimension log 2/log 3.
The Menger sponge
Explore the three-dimensional relative of the Sierpiński carpet, with zero limiting volume and unbounded surface complexity.
Fractal dimension
What fractal dimension measures, how box counting works and why one object can have more than one meaningful dimension.