Koch’s Snowflake is a fractal curve and the third iteration is the 13^th Crease Pattern Challenge, designed by Ushio Ikegami. This one gave me a lot of trouble, but I wasn’t sure why. There are some tricky points, but it’s pretty straightforward overall. It is simply too intricate for me to fold at the scale I was trying. Instead, I ended up folding just the crease pattern presented, which is only 1/6^th of the snowflake.

This one has me on the fence, because I like a more spectacular result with this much effort. On the other hand, fractals are cool, and his inner detailing of triangles makes this a really elegant model. Anyway, the other 5 sixths are all identical (and so have no additional details), so I’m counting this as done. I will talk about this fractal tho.

A fractal is something with a pattern that repeats as you zoom in. An example of a 3D fractal is a Menger sponge, where you divide a cube into 27 equal sub-cubes, remove the center cube and middle cube of each face, and repeat on all the remaining cubes. Taken to infinity, this has infinite surface area and no volume. Also, the first iteration looks the same as a void cube puzzle and was featured in a Pokemon game.

To construct the Koch’s curve, we take a line (of a triangle for the snowflake). For the first iteration, you divide the line into 1/3^rds, remove the middle 1/3^rd, and replace it with the other two sides of an equilateral triangle it would have been a part of (going out in the snowflake form). Every subsequent iteration takes each remaining straight line and repeats the process on all of them.

The whole snowflake starts as a triangle. I’m not animating it too tho.