Sierpinski Triangle

The Sierpinski triangle (also called the Sierpinski gasket or Sierpinski sieve) is a fractal with the shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles.

Construction

1. Start with a solid equilateral triangle $S_0$
2. Remove the middle triangle whose vertices are the midpoints of the sides of $S_0$, leaving three smaller triangles
3. Repeat step 2 for each remaining triangle

The Sierpinski triangle is the limit:

$$ S_\infty = \bigcap_{n=0}^{\infty} S_n $$

Area

If the side length of $S_0$ is $s = 1$, then:

$$ \text{Area}(S_n) = \left(\frac{3}{4}\right)^n \cdot \frac{\sqrt{3}}{4} $$

As $n \to \infty$:

$$ \text{Area}(S_\infty) = \lim_{n \to \infty} \left(\frac{3}{4}\right)^n \cdot \frac{\sqrt{3}}{4} = 0 $$

Fractal Dimension

The Sierpinski triangle has Hausdorff dimension:

$$ d = \frac{\log 3}{\log 2} \approx 1.585 $$

This reflects that it is “more than” a 1-dimensional curve but “less than” a 2-dimensional surface.

Properties

• The Sierpinski triangle is non-empty, containing the vertices of all removed triangles at each stage
• It is self-similar: each of the three corner pieces is a scaled copy of the whole
• It has zero area but infinite perimeter

Related

• Koch_snowflake
• Cantor_set
• fractal_dimension