Sierpinski triangle

The Sierpinski triangle is constructed by repeatedly deleting the open central triangle from every triangle that remains. Begin with an equilateral triangle, join the midpoints of its three sides, and remove the inverted middle triangle; then apply the same operation to each of the three corner triangles. The process produces successive approximations rather than a single finite drawing.¹

At stage n, there are 3ⁿ retained triangles, each with side length 2⁻ⁿ times the original. If the initial area is 1, the retained area is (3/4)ⁿ, which tends to zero, while the number of boundary segments continues to grow. The limiting set is therefore uncountable and geometrically elaborate despite occupying no planar area.²

Its defining mathematical feature is exact self-similarity under three contractions of ratio one-half. The whole set is the union of three smaller copies placed at the vertices of the original triangle, making it a basic example of an iterated function system. This recursive description also gives the similarity, or Hausdorff, dimension: d satisfies 3(1/2)^d = 1, so d = log 3/log 2, approximately 1.585.²

The dimension lies between that of a line and a filled planar region. It measures scaling complexity rather than ordinary length or area: the figure has no area, yet its structure is too rich to behave like a one-dimensional curve. Its three-fold branching and two-fold scaling make calculations unusually transparent.

The figure is named for the Polish mathematician Wacław Sierpiński, who described it in the early twentieth century as part of his work on point sets and pathological geometric examples.³ The modern term belongs to the broader development of fractal geometry, although the construction itself predates the word “fractal.”⁴

A striking discrete analogue appears in Pascal's triangle: color the entries that are odd and enlarge the resulting pattern. The visible arrangement approaches the Sierpinski triangle because binomial coefficients modulo 2 obey a digit-by-digit rule in base two.¹ This links a continuous geometric object with combinatorics, number theory, and cellular patterns.

The Sierpinski triangle can be generated without explicitly deleting triangles. In the chaos game, a point repeatedly moves halfway toward a randomly selected vertex; after initial transients, the plotted points accumulate on the same fractal.¹ This provides an accessible demonstration of how a simple local rule can create a highly structured global pattern.

Its graph-theoretic shadow is also distinctive: replacing each triangular cell by three corner cells yields a branching structure related to the Sierpinski graph, used in studies of routing and recursively defined networks. Variants arise from changing the contraction maps, deleting different subregions, or working in higher dimensions; the three-dimensional analogue is commonly called the Sierpinski tetrahedron. The construction should not be confused with arbitrary triangular patterns that merely resemble it visually.

Fractal

A geometric or mathematical object exhibiting structure across multiple scales, often through recursive rules.

Iterated function system

A finite collection of contraction mappings whose repeated application defines a self-similar set.

Hausdorff dimension

A dimension defined by scaling behavior that can take non-integer values.

Chaos game

A random iterative procedure that can generate points distributed on a self-similar attractor.