Frozen fractals all around

Watching the movie Frozen with my kids the other day, I was happy to hear the term “frozen fractals” in the acclaimed song, “Let It Go.”  It made me look closer at the snowflakes and ice shapes depicted in the movie – but I didn’t notice much in the way of fractal structure there. 

The archetypical snowflake images tend to exhibit “branches on branches,” indicating fractal self-similarity, though the six-fold symmetry appears to be the defining characteristic of iconic snowflake shapes. 

I wondered about making my own fractal snowflakes using a simple iterative process…but I wanted something different than the Koch snowflake.

Starting with a random shape, I repeated the following process (in Adobe Illustrator, which made it easy):

Create 5 more copies, rotated in multiples of 60º about the lower left corner of the bounding box.

Here are the first four iterations:
  
After that my computer was slowing down, but you get the idea.  I tried the same idea again, but this time the copies were rotated about one vertex so they overlapped and ended up making a cool tile pattern: