What is the fractal dimension of the Sierpinski Triangle?

Its Hausdorff (fractal) dimension is $\frac{\ln 3}{\ln 2} \approx 1.585$. This falls between 1 (a line) and 2 (a filled plane region), reflecting the fact that the Sierpinski Triangle is more than a curve but less than a solid shape. The value comes from the self-similarity ratio: 3 copies scaled by a factor of $\frac{1}{2}$ reproduce the whole.

Sierpinski Triangle (Sieve) — Definition, Formula & Examples

The Sierpinski Triangle (also called the Sierpinski Sieve) is a fractal created by starting with an equilateral triangle and repeatedly removing the middle triangle from every remaining filled triangle. This process continues infinitely, producing a shape with zero area but infinite perimeter.

The Sierpinski Triangle is a self-similar fractal subset of the Euclidean plane. Beginning with a solid equilateral triangle $T_0$ , each iteration $T_n$ is formed by subdividing every filled equilateral triangle into four congruent sub-triangles and removing the interior of the central one. The Sierpinski Triangle $S$ is the limit $S = \bigcap_{n=0}^{\infty} T_n$ , and its Hausdorff dimension is $\dfrac{\ln 3}{\ln 2} \approx 1.585$ .

Key Formula

A_n = A_0 \cdot \left(\frac{3}{4}\right)^n

Where:

$A_n$ = Total shaded area after n iterations
$A_0$ = Area of the original equilateral triangle
$n$ = Number of iterations (removal steps)

How It Works

To build a Sierpinski Triangle, start with a filled equilateral triangle. Find the midpoints of all three sides and connect them, forming four smaller congruent triangles. Remove the central upside-down triangle. Now repeat this process for each of the three remaining filled triangles. At each stage, the number of filled triangles triples while each one shrinks to one-quarter of its previous area, so the total shaded area is multiplied by

\tfrac{3}{4}

. After infinitely many iterations, the remaining area approaches zero, yet the boundary length grows without bound.

Worked Example

Problem: An equilateral triangle has side length 8 cm. After 3 iterations of the Sierpinski process, how many filled triangles remain and what is the total shaded area?

Step 1: Find the area of the original triangle using the equilateral triangle formula.

A_0 = \frac{\sqrt{3}}{4} \cdot 8^2 = 16\sqrt{3} \approx 27.71 \text{ cm}^2

Step 2: Count the filled triangles. Each iteration triples the count: after n iterations there are

3^n

filled triangles.

3^3 = 27 \text{ filled triangles}

Step 3: Apply the area formula. Each iteration keeps three-quarters of the remaining area.

A_3 = 16\sqrt{3} \cdot \left(\frac{3}{4}\right)^3 = 16\sqrt{3} \cdot \frac{27}{64} = \frac{27\sqrt{3}}{4} \approx 11.69 \text{ cm}^2

Answer: After 3 iterations, there are 27 filled triangles with a combined area of

\dfrac{27\sqrt{3}}{4} \approx 11.69

cm².

Another Example

Problem: What fraction of the original area remains after 5 iterations of the Sierpinski process?

Step 1: Use the ratio formula with n = 5.

\frac{A_5}{A_0} = \left(\frac{3}{4}\right)^5

Step 2: Compute the power.

\left(\frac{3}{4}\right)^5 = \frac{243}{1024} \approx 0.2373

Answer: About 23.7% of the original area remains after 5 iterations — roughly

\dfrac{243}{1024}

of the starting triangle.

Visualization

Why It Matters

The Sierpinski Triangle appears in precalculus and discrete math courses when studying sequences, geometric series, and limits. It also shows up in Pascal's Triangle — if you shade only the odd entries, the pattern converges to a Sierpinski Triangle. Computer scientists use it to teach recursive algorithms, and engineers encounter fractal antenna designs based on this shape.

Common Mistakes

Mistake: Thinking the area shrinks to zero after a finite number of steps.

Correction: Each iteration multiplies the area by 3/4, which never reaches zero. The area only reaches zero in the limit as n approaches infinity. After any finite number of steps, some positive area remains.

Mistake: Multiplying the number of triangles by 4 instead of 3 at each stage.

Correction: Each filled triangle splits into 4 sub-triangles, but you remove the central one. So only 3 filled triangles survive per parent, giving

3^n

total after n iterations.

Related Terms

Fractal — General category the Sierpinski Triangle belongs to
Geometric Sequence — Area at each stage forms a geometric sequence
Pascal's Triangle — Odd entries in Pascal's Triangle form this pattern
Self-Similarity — Key property where parts resemble the whole
Recursion — Construction method used to build each iteration
Infinite Series — Removed area sums to original area as a series