The Sierpinski Triangle is a fractal discovered by the Polish mathematician Wacław Sierpiński in 1915. It is constructed starting with an equilateral triangle and following a recursive process:

  1. Divide the triangle into 4 sub-equilateral triangles (by connecting the midpoints of each side).
  2. Remove the central triangle.
  3. Repeat the same procedure for each of the 3 remaining sub-triangles.

At each step, the area decreases because the central portion is always removed, while the perimeter increases, as each side is subdivided into a "zigzag" pattern.

Area

For an equilateral triangle with side length s, its initial area is:

\[ A_{0} = \frac{\sqrt{3}}{4} \, s^2 \]

In the next iteration, only 3/4 of the previous area is retained (as the central part is removed). Thus, the area in iteration n is:

\[ A(n) = A_{0} \left(\frac{3}{4}\right)^n \]

As n → ∞, (3/4)^n approaches 0, so the total area tends to zero.

Perimeter

The perimeter of the initial equilateral triangle is P0 = 3s. In each iteration, each side is subdivided, doubling its total length. Therefore, in iteration n:

\[ P(n) = P_{0} \times 2^n \]

In this way, the perimeter grows exponentially and approaches infinity as n → ∞.

Fractal Dimension

The fractal dimension (Hausdorff–Besicovitch) of the Sierpinski Triangle is calculated as:

\[ D = \frac{\log(N)}{\log\left(\frac{1}{r}\right)} \]

Where N = 3 (number of copies) and r = 1/2 (scaling factor for each side). Thus:

\[ D = \frac{\log(3)}{\log(2)} \approx 1.585 \]

Therefore, the fractal "occupies" more than a 1D object (a line) but less than a fully 2D surface.

Results by Iteration

Iteration Area (px²) Perimeter (px)