Koch snowflake — infinite length, finite area

What just happened?

You started with an equilateral triangle. In each iteration every segment splits into 3 parts, and a smaller equilateral triangle “grows” over the middle one — a single segment turns into four, each 1/3 the length of the original. The boundary becomes more and more “jagged”, yet the whole figure still fits inside the same finite region (the dashed triangle is the starting figure).

The boundary length grows geometrically:

Since each segment turns into 4 segments of length 1/3, the perimeter is multiplied by $4\cdot\tfrac{1}{3}=\tfrac{4}{3}$ per iteration. The ratio $\tfrac{4}{3} > 1$, so the sequence $L_k$ is divergent — after infinitely many steps the length tends to infinity.

Yet the area stays finite:

In each iteration we add ever smaller triangles, but their combined area forms a convergent geometric series (ratio 4/9 < 1). The sum tends to a constant value: the area of the whole snowflake is only $\tfrac{8}{5}$ of the starting triangle’s area. An infinitely long boundary encloses a finite area — the heart of this fractal.

Who discovered it?

The Swedish mathematician Helge von Koch described this curve in 1904 as an example of a curve that is continuous but nowhere differentiable — it has no tangent at any point, nothing but “corners”. The Koch snowflake is one of the first and simplest fractals: a self-similar figure — every fragment of the boundary, magnified, looks like the whole.

Why this matters for your exams:

It is a model example that the intuitions from the world of “smooth” figures (finite perimeter ⇒ finite area) fail in fractal geometry. It combines geometric sequences and series with limits: the divergent sequence $L_k$ alongside the convergent series of areas. Fractal geometry — later developed by Mandelbrot — describes the “jagged” shapes of nature: coastlines, snowflakes, blood vessels.

Remember: the perimeter of the Koch snowflake grows ×4/3 per iteration and tends to infinity, but the area stays bounded (→ 8/5·A₀). Infinite length, finite area.