The Sierpinski Triangle: Chaos, Recursion & Fractal Dimension

Two roads to the same triangle

The Sierpiński triangle (Wacław Sierpiński, 1915) can be built by two completely different processes that converge on exactly the same shape, and seeing both is the fastest way to understand what a fractal actually is. The recursive way: take a filled triangle, connect the midpoints of its three sides to carve out the central inverted triangle, and repeat the same operation on each of the three remaining corner triangles, forever.

sierpinski(triangle, depth):
  if depth == 0: draw(triangle); return
  mid = midpoints of triangle's three sides
  sierpinski(cornerTriangle(triangle.A, mid.AB, mid.CA), depth-1)
  sierpinski(cornerTriangle(mid.AB, triangle.B, mid.BC), depth-1)
  sierpinski(cornerTriangle(mid.CA, mid.BC, triangle.C), depth-1)
  // the middle triangle is never recursed into - it stays empty

The chaos game: pure randomness, structured result

The second construction looks nothing like the first and has no recursion at all. Fix the triangle's three vertices. Start at any point. Repeatedly pick one of the three vertices at random, jump halfway from the current point to that vertex, and plot the new point. Do this a few thousand times and, remarkably, the exact same Sierpiński triangle emerges from what looks like pure noise.

point = randomPointInsideTriangle()
repeat many thousands of times:
  vertex = triangle.vertices[randomInt(0, 2)]
  point  = midpoint(point, vertex)     // jump halfway toward the chosen vertex
  plot(point)                          // after the first few iterations,
                                        // every plotted point lies on the fractal

This is called the chaos game (Barnsley, 1988), and it works because the halfway-jump rule is exactly the same iterated function system that generates the triangle recursively, viewed from the other direction: recursion subdivides space top-down into smaller and smaller copies, while the chaos game is a Markov process whose long-run distribution of visited points is that same self-similar set — two views of the same underlying contraction mapping.

Hausdorff dimension: not a line, not a plane

A Sierpiński triangle is neither a one-dimensional curve nor a two-dimensional filled region — it has zero area (each subdivision step removes exactly 1/4 of the remaining filled area, and repeating that infinitely leaves none) yet it is not a simple curve either. Its Hausdorff dimension captures this precisely: scale the triangle up by a factor of 2 and it decomposes into exactly 3 copies of itself (not 4, because the middle piece is missing), and the dimension D solves 2^D = 3:

D = log(3) / log(2)  ~  1.585
// compare: a solid square scaled by 2 makes 4 copies => log(4)/log(2) = 2 (ordinary 2D)
//          a line segment scaled by 2 makes 2 copies => log(2)/log(2) = 1 (ordinary 1D)

A dimension of roughly 1.585 — strictly between 1 and 2 — is a precise, non-metaphorical statement that the triangle fills space more thoroughly than a curve but less thoroughly than a solid area, and it is the same self-similarity ratio (log(copies)/log(scale factor)) used to compute the dimension of every other exactly self-similar fractal, coastlines included, just with different copy counts and scale factors.

A surprising appearance: Pascal's triangle mod 2

Take Pascal's triangle of binomial coefficients, and colour a cell black if its value is odd and white if it is even (equivalently, compute each entry modulo 2). The resulting pattern, extended to enough rows, is the Sierpiński triangle — a purely number-theoretic object (which entries of a combinatorial table are odd) reproducing exactly the same self-similar shape as the geometric recursion and the random chaos game, a coincidence that turns out not to be a coincidence: all three are consequences of the same underlying binary self-similarity.

Frequently asked questions

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