Sierpiński triangle

What just happened?

You started with a solid equilateral triangle. In each iteration you split every filled triangle into 4 smaller ones (by joining the midpoints of the sides) and remove the middle one — 3 corner ones remain. After many steps the solid figure becomes an ever more “holey” lace (the dashed triangle is the starting figure, from which the middles disappear).

The area decreases geometrically to zero:

Since 3 of the 4 equal sub-triangles remain, the area is multiplied by $\tfrac{3}{4}$ per iteration. The ratio $\tfrac{3}{4} < 1$, so the sequence $P_k$ is convergent to zero — after infinitely many steps the filled area vanishes, even though the number of sub-triangles grows like $3^k$.

Self-similarity and fractal dimension:

Each of the 3 corners is a miniature copy of the whole — the figure is self-similar. Its fractal dimension is $\log 3/\log 2 \approx 1.585$ — it lies between a line (1) and a plane (2): the structure is “more than a line, less than a plane”.

Who discovered it?

The Polish mathematician Wacław Sierpiński (1882–1969) described this fractal in 1915. He was one of the founders of modern set theory and topology, author of over 700 papers. Striking is the connection of his triangle with Pascal’s triangle: marking the odd numbers in it yields exactly the pattern of the Sierpiński triangle. The same shape also appears in the chaotic “chaos game” (random jumps toward the vertices) and in fractal antennas.

Why this matters for your exams:

It is a model example of a geometric sequence convergent to zero ($P_k = P_0 q^k$, $q = \tfrac{3}{4} < 1$) alongside a divergent sequence ($N_k = 3^k \to \infty$). It combines limits of sequences, powers and fractal geometry. Wacław Sierpiński is one of the two Polish touches in this collection (alongside Banach and Tarski).

Remember: the area of the Sierpiński triangle decreases ×3/4 per iteration and tends to zero ($P_0(3/4)^k \to 0$), while the number of sub-triangles grows ×3 ($3^k$). Vanishing area, infinitely rich structure.