The Sierpinski Triangle: Infinite Holes From One Rule

One rule, repeated forever

Start with a filled equilateral triangle and call it stage 0. Connect the midpoints of its three sides; this cuts it into four smaller congruent triangles, three pointing the same way as the original and one pointing down in the center. Delete the central, downward triangle. Three triangles remain, each a half-scale copy of the original. Now apply the very same operation to each of those three — remove their middles — to reach stage 2, then stage 3, and so on without end.

The bookkeeping is clean. At stage n there are 3ⁿ surviving triangles, each with side length (1/2)ⁿ of the original. The holes accumulate: at stage 1 you remove one triangle, at stage 2 you remove three more, at stage 3 nine more, and in general 3ⁿ⁻¹ new holes at stage n. Summed over all stages that is a countably infinite collection of open triangular holes — the “infinite holes” of the name.

The Sierpinski triangle itself is the limit, the set of points that are never removed at any stage. Formally, if S₀ is the solid triangle and S_n+1 is what you get after cutting every triangle in S_n, then the shapes are nested, S₀ ⊇ S₁ ⊇ S₂ ⊇ …, and the gasket is the intersection S = ⋂_n≥0 S_n. As an intersection of nested nonempty compact sets it is nonempty and compact — and it inherits the key property that will not go away: at every scale, it looks like three shrunken copies of itself.

Zero area, but an infinite edge

Track the area. Each cut deletes exactly one of four equal sub-triangles, so it keeps three-quarters of what was there. If A is the area of the original triangle, then stage n has area (3/4)ⁿ · A. Because the gasket sits inside every stage, its area is at most (3/4)ⁿ · A for all n, and (3/4)ⁿ → 0. So the Sierpinski triangle has area exactly zero — it is a measure-zero set in the plane, taking up no more space than a curve.

Now track the edges. At stage n there are 3ⁿ triangles, each with perimeter 3 · (1/2)ⁿ (taking the original side as 1). The total length of all those boundaries is 3ⁿ · 3 · (1/2)ⁿ = 3 · (3/2)ⁿ, which grows by a factor of 3/2 at every stage and therefore diverges to infinity. The limiting set's own one-dimensional length (its Hausdorff H¹ measure) is infinite for the same reason.

There is no paradox once you accept that “length” and “area” are the wrong rulers. A finite line segment has zero area and finite length; a solid disk has finite area. The gasket falls between: it is too sparse to have positive area, yet too rough and branchy to have finite length. That in-between-ness is precisely what a fractional dimension measures — and it is why the object needs a number like 1.585 rather than a clean 1 or 2.

The dimension log 3 / log 2 ≈ 1.585

Dimension can be defined by how detail multiplies as you zoom. For an ordinary shape, halving the ruler multiplies the number of pieces you need by a fixed factor: a line segment splits into 2 half-length copies (2 = 2¹), a filled square into 4 (4 = 2²), a cube into 8 (8 = 2³). The exponent is the dimension. Write it as d = log N / log (1/r), where the object is made of N copies each scaled by ratio r.

The Sierpinski triangle is built from N = 3 copies, each scaled by r = 1/2. Its similarity dimension is therefore

• d = log 3 / log 2 ≈ 1.58496…

You reach the same value by box-counting: cover the set with boxes of side (1/2)ⁿ and you need about 3ⁿ of them, so the box-counting dimension is lim (log 3ⁿ) / (log 2ⁿ) = log 3 / log 2. Because the three sub-triangles meet only at single points (the midpoints of the big triangle's sides) — they satisfy the open set condition — Moran's and Hutchinson's theorems guarantee that the box-counting and Hausdorff dimensions agree, and that the gasket carries positive, finite Hausdorff measure in dimension 1.585.

This is exactly Mandelbrot's working definition of a fractal: an object whose Hausdorff dimension strictly exceeds its topological dimension. The Sierpinski triangle is a connected curve-like continuum, so its topological dimension is 1, yet its Hausdorff dimension is 1.585. The gap between the two numbers is the roughness the eye reads as “fractal.”

A unique attractor: the fixed point of three maps

The construction hides a cleaner description. Place the triangle's vertices at points v₁, v₂, v₃ and define three shrink-toward-a-corner maps on the plane:

• f_i(p) = (p + v_i) / 2 — halve the distance from p to vertex v_i.

Each f_i is a contraction with ratio 1/2, and each sends the whole triangle onto one of its three corner sub-triangles. This trio is an iterated function system (IFS). Collect them into the Hutchinson operator W(X) = f₁(X) ∪ f₂(X) ∪ f₃(X), which takes a set and returns the union of its three half-size copies — one application of the removal rule.

Hutchinson (1981) showed that W is itself a contraction, not on points but on the space of nonempty compact subsets of the plane, measured by the Hausdorff metric (roughly, two sets are close if each lies within a small band of the other). That space is complete, so Banach's fixed-point theorem applies: W has exactly one fixed point, and iterating W from any starting compact set converges to it. The Sierpinski triangle is that unique attractor — the one set S with S = W(S). Start from a square, a circle, or a single blob; hit it with W a dozen times and the gasket emerges, because the shape is encoded entirely in the three rules, not in where you began.

The same shape in disguise

Because the gasket is the fingerprint of those three maps, it surfaces wherever the maps are hiding.

The chaos game. Mark the three vertices. Drop a point anywhere, then repeat: pick one vertex at random and step halfway toward it, plotting each landing spot. This is just applying a randomly chosen f_i over and over. After discarding a few transient points, the dots trace out the Sierpinski triangle. The reason is that the random IFS has a unique invariant (Hutchinson) measure supported on the attractor, and the orbit's points equidistribute onto it — order emerging from pure coin-flips, which is why Barnsley called it a “chaos game.”

Pascal's triangle mod 2. Print Pascal's triangle and color a cell black when the binomial coefficient is odd, white when even. The odd cells form the Sierpinski triangle. By Lucas' theorem, C(n, k) is odd exactly when, writing n and k in binary, every 1-bit of k also appears in n (equivalently k AND n = k; Kummer's theorem says this is the “no carries when adding” condition). That submask pattern is self-similar under doubling, so the first 2^k rows are three copies of the previous block — a gasket.

Rule 90. In this elementary cellular automaton each cell becomes the XOR of its two neighbors. Start from a single black cell on a white line and let time run downward; the space-time diagram is again the Sierpinski triangle, because Rule 90 computes exactly the binomial coefficients modulo 2.

Tower of Hanoi, history, and where it shows up

The pattern even governs a puzzle. Consider the Tower of Hanoi with n disks on 3 pegs, and draw its state graph: one vertex per legal arrangement, one edge per legal single-disk move. This graph has 3ⁿ vertices and is the Sierpiński graph S₃ⁿ — a finite gasket. The three corners are the “all disks on one peg” states; the shortest path between two corners has length 2ⁿ − 1, the famous minimum number of moves, running straight along one edge of the triangle. As n grows, the rescaled graph converges to the Sierpinski gasket itself.

Historically, Wacław Sierpiński introduced the set in 1915 as an example of a curve every point of which is a branch point — a deliberately pathological object at a time when such “monsters” unsettled analysts. Sixty years later Benoit Mandelbrot recast these monsters as fractals (1975), turning them from curiosities into a language for roughness in nature.

The gasket is not only decorative. Its set of points is uncountable (each point has an infinite “address” of vertex choices) yet nowhere dense and of measure zero, a vivid separation of “how many” from “how much.” Its self-similar branching makes it a real engineering tool: fractal antennas shaped like the gasket resonate on many frequencies at once thanks to their repeating structure, and the shape serves as a testbed for diffusion, resistance networks, and spectra on fractal domains — the mathematics of one endlessly repeated cut.

Classic self-similar fractals: how many shrunken copies, at what scale, and the dimension and size that result.

Fractal	Rule (copies × scale)	Hausdorff dimension	Size of the limit set
Cantor set	2 copies × 1/3	log 2 / log 3 ≈ 0.631	Length 0; uncountably many points
Koch curve	4 copies × 1/3	log 4 / log 3 ≈ 1.262	Infinite length; snowflake encloses finite area
Sierpinski triangle	3 copies × 1/2	log 3 / log 2 ≈ 1.585	Area 0; infinite boundary length
Sierpinski carpet	8 copies × 1/3	log 8 / log 3 ≈ 1.893	Area 0; universal plane curve
Menger sponge	20 copies × 1/3	log 20 / log 3 ≈ 2.727	Volume 0; infinite surface area