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Sierpiński Triangle — Definition, Formula & Examples

The Sierpiński Triangle is a fractal created by repeatedly removing the central triangle from an equilateral triangle. Each iteration leaves three smaller copies of the original shape, producing an infinitely detailed, self-similar pattern.

The Sierpiński Triangle is the compact set obtained by starting with a filled equilateral triangle and iteratively removing the open central triangle (formed by connecting the midpoints of each side) from every remaining filled triangle, carried out infinitely many times. Equivalently, it is the unique non-empty compact subset of R2\mathbb{R}^2 that is the attractor of three contraction mappings, each scaling by 12\frac{1}{2} toward a vertex of the original triangle.

Key Formula

d=ln⁡3ln⁡2≈1.585d = \frac{\ln 3}{\ln 2} \approx 1.585
Where:
  • dd = Fractal (Hausdorff) dimension of the Sierpiński Triangle
  • 33 = Number of self-similar copies at each iteration
  • 22 = Scaling factor (each copy is reduced by a factor of 2)

How It Works

Start with a solid equilateral triangle (Stage 0). Mark the midpoints of all three sides, then remove the upside-down triangle they form — this leaves three smaller solid triangles (Stage 1). Repeat the same removal process on each remaining solid triangle to get Stage 2, and continue indefinitely. At each stage, the number of filled triangles triples while each one shrinks to half its previous side length. The true Sierpiński Triangle is the limiting shape after infinitely many stages.

Worked Example

Problem: Starting with an equilateral triangle of side length 8 cm, how many filled triangles exist at Stage 3, and what is the side length of each?
Stage 0: You begin with 1 filled triangle of side length 8 cm.
Stage 1: Remove the central triangle. This leaves 3 filled triangles, each with half the side length.
Side length=82=4 cm\text{Side length} = \frac{8}{2} = 4 \text{ cm}
Stage 2: Repeat for each of the 3 triangles. The count triples and the side length halves again.
32=9 triangles, each 84=2 cm3^2 = 9 \text{ triangles, each } \frac{8}{4} = 2 \text{ cm}
Stage 3: Repeat once more.
33=27 triangles, each 88=1 cm3^3 = 27 \text{ triangles, each } \frac{8}{8} = 1 \text{ cm}
Answer: At Stage 3 there are 27 filled triangles, each with a side length of 1 cm.

Why It Matters

The Sierpiński Triangle is one of the first fractals students encounter, making it a gateway to understanding self-similarity, fractal dimension, and iterative processes. It appears in Pascal's triangle (shade the odd entries and the pattern emerges), connecting number theory to geometry. Antenna engineers have even used its shape to design compact, multi-frequency antennas.

Common Mistakes

Mistake: Assuming the fractal dimension must be a whole number like 1 or 2.
Correction: Fractals often have non-integer dimensions. The Sierpiński Triangle has dimension ln⁡3ln⁡2≈1.585\frac{\ln 3}{\ln 2} \approx 1.585, reflecting that it is more than a line but less than a filled region.