Koch Curve
The Koch Curve is a well-known example of a fractal, introduced by the Swedish mathematician Helge von Koch in 1904. It begins with a simple straight line segment. In each iteration, the middle third of every line segment is replaced with two segments that form an equilateral triangle, creating a snowflake-like shape.
With each iteration, the Koch Curve becomes more complex, but never enclosed—its perimeter becomes infinitely long while the area it encloses remains finite. This paradox highlights key features of fractals: self-similarity, infinite complexity, and non-differentiability.
Philosophically, the Koch Curve challenges traditional geometry and our understanding of dimensionality. It provides insights into the roughness of natural forms, like coastlines and mountain ranges, that cannot be captured by classical Euclidean shapes.
The Koch Curve exemplifies how simple rules can generate intricate structures, influencing chaos theory, art, architecture, and even environmental modeling.