Sierpinski triangle

The Sierpinski triangle is a fractal formed by recursively removing the central triangle from an equilateral triangle. Named after Wacław Sierpiński, who described it in 1915, it is one of the most visually recognizable fractals — a triangular lattice of holes that repeats at every scale. It has Hausdorff dimension log(3)/log(2) ≈ 1.585, occupying a middle ground between a one-dimensional line and a two-dimensional plane.

Construction

Begin with a solid equilateral triangle. Connect the midpoints of its sides to form four smaller equilateral triangles, and remove the central one. Repeat this operation on each remaining solid triangle. After infinite iterations, the limit set is the Sierpinski triangle. At stage n, 3^n solid triangles remain, each of side length (1/2)^n. The total area vanishes as (3/4)^n → 0, yet the set contains uncountably many points.

Self-Similarity and Dimension

The Sierpinski triangle consists of three copies of itself, each scaled by one-half. This exact self-similarity gives its Hausdorff dimension immediately: log(3)/log(2). The set is connected but contains no interior: every point lies on the boundary of the removed triangles. It is a universal object in topology, appearing as the attractor of a simple iterated function system.

The triangle also arises in unexpected places. In cellular automata, particularly Rule 90 and Rule 60, the pattern of active cells traces the Sierpinski triangle. In the study of Pascal's triangle modulo 2, the pattern of odd numbers forms the same shape. These connections suggest the Sierpinski triangle is not merely a geometric construction but a structural attractor in discrete dynamical systems.

The Sierpinski triangle is the geometric fingerprint of binary choice. Every point that survives the infinite removal is a record of a sequence of left-or-right decisions. The triangle is not a shape; it is a decision tree made visible.