The Koch Snowflake is one of the most famous fractals, proposed by the Swedish mathematician Helge von Koch in 1904. It is constructed starting from an equilateral triangle, and in each iteration, a small protrusion forming another equilateral triangle is added to each side, gradually increasing its perimeter.
- Construction: Start with an equilateral triangle. Each side is divided into three equal segments, and the middle segment is replaced by two sides of a new equilateral triangle, forming a “protrusion.” This process is repeated recursively on each new side that is generated.
- Fractal Dimension: The dimension of the Koch Snowflake is \(\displaystyle \frac{\ln(4)}{\ln(3)} \approx 1.2619\). This means it "occupies more than a line" (dimension 1), but less than a two-dimensional surface (dimension 2).
- Infinite Perimeter: Although the area remains finite, the perimeter grows infinitely with each iteration, making it a figure that encloses a limited area but with an infinite edge.
In the following application, you can control the number of iterations to see how the initial triangle “twists” into the shape of a snowflake. Try experimenting with the slider or press “Start Animation” to see it grow!