Koch Snowflake

The Koch Snowflake, introduced by the Swedish mathematician Helge von Koch in 1904, starts from an equilateral triangle and repeatedly replaces the middle third of every straight edge with two sides of a smaller equilateral triangle pointing outward, turning each straight segment into four shorter ones at every step. Each replacement adds detail without moving the endpoints of the original edge, so the snowflake's overall size barely changes, even as its total perimeter grows by a factor of $\frac{4}{3}$ at every iteration. Repeated infinitely, this makes the Koch snowflake one of the classic examples of a curve with infinite length enclosing a strictly finite area — a genuinely counterintuitive result that helped motivate the modern study of fractals and fractional dimension. Its boundary has a Hausdorff dimension of $\log 4/\log 3\approx 1.26$, strictly between a 1-dimensional line and a 2-dimensional area.