Sierpiński Triangle
A fractal triangle constructed by repeatedly removing the central sub-triangle from each remaining triangle.
The Construction
Begin with a filled equilateral triangle. At each step, find the midpoints of every remaining triangle’s three sides and remove the central sub-triangle formed by connecting them.
The IFS equivalent uses three contractions, each scaling by ½ and translating to one corner:
f₁(x,y) = (x/2, y/2)
f₂(x,y) = (x/2 + ½, y/2)
f₃(x,y) = (x/2 + ¼, y/2 + √3/4)
The filled triangle before any removal. This is the “initiator” — the seed from which the fractal grows.
First Removals
After one iteration, the central triangle is gone. Three sub-triangles remain at the corners, each similar to the original and scaled by ½.
The self-similarity is already visible: the whole shape is composed of three copies of itself, each half the size. The removed central triangle is as large as each remaining piece.
Fractal Dimension
Each iteration multiplies the number of triangles by 3 and scales each by ½. The Hausdorff dimension is:
d = log(3) / log(2) ≈ 1.585
After four iterations, 81 triangles remain. The area of the filled region approaches zero — in the limit, the Sierpiński triangle has zero area but a fractal boundary of dimension 1.585.