Koch Curve

Koch curve

Koch curve is a kind of fractal curve. It appeared in a 1904 paper titled ‘On a Continuous Curve Without Tangents, Constructible from Elementary Geometry’ by the Swedish mathematician Helge von Koch.

Suppose there is a case in which you have to proceed by distance ‘d’ in some direction. First, proceed to d / 3. Next, turn 60° to the left and proceed to d / 3. Then turn the direction 120° to the right and proceed to d / 3. Finally, turn 60° to the left and proceed to d / 3. This task reaches the same destination. But this has 4 / 3 times longer length than the original. If you do the same task for the four short angular segments that are made in this way, the traveling distance will be (4 / 3)2.

If you repeat these tasks infinitely, the way to go will increase infinitely, and the whole shape will be the same shape as part of the snowflakes.
The Koch curve has a similar shape (self-similarity) to a part of it. Even if you enlarge it in some extent, it shows the same shape as the original.