Geometric

Koch snowflake

Helge von Koch published the Koch curve in 1904 as one of the first continuous, everywhere-nowhere-differentiable curves expressible by a simple geometric rule. The Koch snowflake is the closed version, the iconic example of a curve with finite area and infinite perimeter.

At a glance

Construction

1. Start with an equilateral triangle (Koch snowflake) or a single line segment (Koch curve).
2. On each segment, replace the middle third with two sides of an equilateral triangle pointing outward (so the segment becomes four segments, each 1/3 the original length).
3. Repeat on every new segment, forever.

Why infinite perimeter, finite area

At each step, the number of segments is multiplied by 4 and each is 1/3 as long. So total length is multiplied by 4/3, which grows without bound. But the area added at each step is a finite geometric series that converges: a triangle of fixed initial area has only finite area, total perimeter goes to infinity but the enclosing region remains bounded.

Dimension derivation

Four self-similar copies at scale 1/3: log(4) / log(3) ≈ 1.2619. Strictly more than 1 (the topological dimension of any curve) and strictly less than 2.

Variants

• Koch anti-snowflake: bumps inward instead of outward.
• Koch quadratic: replace middle third with three sides of a square; dimension log₃(5) ≈ 1.4650.
• Cesàro fractal: variable angle generalization.
• Snowflake sweep: space-filling curve based on Koch building blocks.

References

• von Koch, H., “Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire,” Arkiv för Matematik, 1904.
• Mandelbrot, B., “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science, 1967.
• Falconer, K., Fractal Geometry, Wiley, 2014.
• Sierpinski triangle · Cantor set

Try it

Run an interactive playground at /tools/koch.