Topic · Geometry

The Sierpiński Triangle

One fractal. Three roads in.

Recursion, arithmetic, and a random walk all converge on the same triangular gasket. Each route is a different mathematics — and each one draws the same picture in the limit.

Discover the story → Open the Explorer

First encounter

One triangle inside itself, forever.

The big idea

A triangle, but every time you look closer you see three smaller copies of it tucked into the corners — and inside each of those, three smaller copies again, infinitely down. The astonishing part is that three completely different recipes — folding paper, counting Pascal's triangle, rolling a die — all draw the same shape.

A concrete example

Start with one solid triangle. Find the midpoints of its three sides; that splits it into four little triangles. Throw away the middle one. Now do the same to the three survivors. After just five steps you can already see the gasket — by step ten it's unmistakable.

Why it matters

The same shape arrives from (1) recursive subdivision, (2) Pascal's triangle with even cells erased, and (3) the random Chaos Game where you keep jumping halfway towards one of three corners. Three unrelated processes, one inevitable answer: the structure is built into the geometry itself.

Slide the depth knob below, then watch all three routes meet in the gasket.

Section 01 · The classical construction

Remove the centre, then recurse

Take an equilateral triangle. Connect the midpoints of its three sides — this slices it into four congruent smaller triangles, three pointing up and one pointing down. Erase the down-pointing one. Now you have three little triangles in the corners. Apply the same rule to each of them: connect midpoints, erase the central inverted triangle, keep the corners. Repeat. Each iteration multiplies the number of pieces by three and shrinks each piece to half its previous edge length. In the limit the leftover set has Lebesgue area zero, and yet it stays connected, self-similar at every scale, an uncountable perfect set with no isolated points. That limit set is the Sierpiński triangle.

Interactive · the recursive picture

Slide the depth — watch the gasket emerge

Each step replaces every solid triangle with three smaller copies — corner, corner, corner — and removes the central inverted triangle. The count of surviving triangles grows as 3ⁿ; the total area shrinks as (3/4)ⁿ. By depth 5 the gasket is clearly visible; by depth 7 the individual pieces are too fine to resolve.

Interactive · subdivision visualiser

Depth4

Triangles81

At depth n there are 3ⁿ filled triangles. The total area is (3/4)ⁿ of the original, tending to zero, yet the limit set, the gasket, keeps empty interior and stays self-similar at every scale.

Section 02 · The fractal dimension

log 3 / log 2 ≈ 1.585

How many copies of yourself fit inside you, at what scale? Three copies, each at half the linear size. The similarity dimension solves N = (1/r)^d, giving d = log 3 / log 2 ≈ 1.585. The Sierpiński triangle is strictly more than a curve and strictly less than a region: a true intermediate object. Its Hausdorff dimension and box-counting dimension coincide at this value, which is why the gasket is one of the cleanest concrete examples in a first lecture on fractal geometry.

Hausdorff dimension

log 3 / log 2 ≈ 1.585

Strictly between a curve (dim 1) and a region (dim 2). The same value is the box-counting dimension, so the gasket is dimensionally well-behaved.

Section 03 · Self-similarity at every scale

Three copies of itself, each at half size

The gasket S satisfies a single set-equation: S = f₁(S) ∪ f₂(S) ∪ f₃(S), where f₁, f₂, f₃ are the contraction maps that send the whole triangle into each of its three corners with ratio 1/2. Any point in S is the image of some point in S under one of these three maps — and that point, in turn, is the image of another point in S, forever. The triangle isn't merely approximately self-similar; it is the unique non-empty compact fixed set of this three-map iterated function system. Hutchinson's 1981 theorem guarantees such a set exists and is unique.

The three contractions

f₁(x, y) = (x/2, y/2) towards the bottom-left vertex

f₂(x, y) = ((x+1)/2, y/2) towards the bottom-right vertex

f₃(x, y) = ((x+1/2)/2, (y+√3/2)/2) towards the top vertex

S = f₁(S) ∪ f₂(S) ∪ f₃(S), the unique non-empty compact fixed set.

Section 04 · Three roads to the same gasket

Recursion · arithmetic · randomness

Route one is the deterministic subdivision above. Route two is arithmetic: write out Pascal's triangle and colour every odd entry, leaving the even ones blank. By Lucas's theorem, C(n, k) is odd exactly when the binary digits of k are a subset of the binary digits of n — and that combinatorial condition, drawn as a triangle, is the gasket. Route three is the random Chaos Game: drop a point anywhere, then repeatedly pick one of the three corners at random and jump halfway towards it; after a few hundred jumps the dots condense into the same shape. Recursion, divisibility, randomness — visit /pascalmod and /chaosgame to play the last two by hand.

Pascal mod 2

→ See the arithmetic route in full

Chaos Game

→ Play the random walk yourself

Subdivision

→ Recursive construction, deep

Interactive · three routes, one shape

Recursion, arithmetic, randomness — same gasket

Three completely different procedures, side by side. The subdivision route follows the geometric recipe. The Pascal route colours every odd binomial coefficient C(n, k). The Chaos Game route picks a triangle vertex at random and jumps halfway — a dot at a time. Watch the random walk fill in the same triangular silhouette as the deterministic two.

Interactive · three routes comparator

Subdivision · depth 6

Geometric recursion: keep the three corner triangles, erase the central one. Three copies at half size — the IFS in pictures.

Pascal mod 2 · 64 rows

Lucas's theorem in colour: C(n, k) is odd iff the bits of k fit inside the bits of n. The pattern of odd cells is the gasket.

Chaos game · live

Pick a vertex at random, jump halfway towards it, mark the spot. A few hundred dots are already enough for the silhouette.

Three unrelated recipes — geometric, combinatorial, stochastic — settle on the same compact set. That set is the unique fixed point of the three-map IFS.

Section 05 · Connections

Carpet · sponge · dust

Run the same recipe with a square divided into nine, keeping eight and erasing the centre: you get the Sierpiński carpet, dimension log 8 / log 3 ≈ 1.893. Run it again in three dimensions with a cube split into twenty-seven, keeping twenty and erasing the centre of each face plus the cube's central cell: Menger's sponge, dimension log 20 / log 3 ≈ 2.727. Run it in one dimension with a line split into three, keeping the outer two and erasing the middle third: Cantor dust. The whole family — dust, gasket, carpet, sponge — sits inside the same construction, just at different ambient dimensions.

The family

name	ambient	copies / scale	dimension
Cantor dust	1D	2 copies at 1/3	log 2 / log 3 ≈ 0.631
Sierpiński triangle	2D	3 copies at 1/2	log 3 / log 2 ≈ 1.585
Sierpiński carpet	2D	8 copies at 1/3	log 8 / log 3 ≈ 1.893
Menger sponge	3D	20 copies at 1/3	log 20 / log 3 ≈ 2.727

Section 06 · Mathematical history

Sierpiński 1915 and the rise of Polish set theory

Wacław Sierpiński published the triangle in 1915, working in Lwów (now Lviv) against the backdrop of an emerging Polish school of set theory and point-set topology that he, Janiszewski and Mazurkiewicz would soon institutionalise around the Warsaw journal Fundamenta Mathematicae. Sierpiński himself contributed the carpet in 1916; Karl Menger generalised the construction to three dimensions in 1926. (The famous Lwów school around Banach, Steinhaus, Mazur and Ulam at the Scottish Café came later, through the 1920s and 1930s.) The pattern itself, however, is far older: thirteenth-century Cosmatesque marble floors in Roman churches already carry recognisable Sierpiński gaskets, cut three or four iterations deep. The mathematics formalised what the masons had been laying out for seven centuries.

Take it further

Open the Explorer.

The Explorer lets you push subdivision deeper, run the Chaos Game on the triangle, switch the colour and step through all four constructions, subdivision, chaos game, Pascal mod 2 and Rule 90, one at a time. Everything you just read is one click away.

→ Open the Explorer

Where you meet it

How and where this technique lives in the world today.

Fractal antennas

Sierpiński gasket antennas are commercial products — cell phones, Wi-Fi routers and GPS devices use multi-band fractal patches that resonate at many frequencies in a small footprint.

Heat exchangers

Sierpiński-style branching channels appear in printed cooling plates used for high-power LEDs and chip cooling, maximising surface area.

Compression & graphics

Iterated Function Systems (Sierpiński-style IFS) underlie fractal image compression algorithms — still used in niche aerial-imagery encoders.

Networking

Hierarchical IP routing and tree topologies inherit Sierpiński-style fractal scaling properties for load balancing.