Koch Snowflake

The Koch snowflake is a fractal curve constructed by recursively replacing the middle third of each line segment with two sides of an equilateral triangle. First described by Helge von Koch in 1904, it is one of the earliest fractals to be studied and has infinite perimeter enclosing finite area — a paradox that anticipates the measure-theoretic complexities of later fractal geometry. The Koch snowflake has a Hausdorff dimension of log(4)/log(3) ≈ 1.26 and serves as a canonical example in fractal dimension calculations. Its construction is a simple case of an iterated function system and demonstrates that continuous curves can be nowhere differentiable, challenging the classical intuition that curves must be smooth to be well-behaved. The Koch snowflake is a special case of the broader family of space-filling curves and nowhere differentiable functions that demolished the naive geometric assumptions of nineteenth-century analysis.