The two of us were admiring our favorite shirts at Shirt.Woot (a fantastic site- if you've never been there, go) and we happened upon the "Fractal Tree" design. Now I was confused, because I didn't think this was a fractal. David disagreed, although I have a sneaking suspicion this was just for the sake of disagreement, not because he actually knew. However, it turns out he was right either way-- the tree is a fractal, because each of the branches is a smaller iteration of the tree as a whole. A fractal, by definition, is an infinitely detailed geometric shape with small sections similar to large ones.
What I was thinking of was a specific type of fractal, known as a Koch curve or a Koch snowflake. I was going to just copy and paste the description of one from Wikipedia, but it seemed a little more complicated than I'm going for, so I'll give it a go on my own. A Koch snowflake starts as an equilateral triangle. Then, imagine adding another triangle, 1/3 the size of the original, to each of the triangle's sides. This would give you a six-pointed star. Now add smaller triangles to each of the sides of this new shape. Repeat infinitely. I'll illustrate this description with the gif image Wikipedia had, since that makes it a little more clear.
