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 to the same destination. but this have 4 / 3 times longer length than the original. If you do the same task for the four short angular segments that are maked 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 snow flakes.

The Koch curve has a similar shape (self-similarity) to a part of it. Even if you enlarges it in some extent, it shows the same shape as the original.