LogoLSystem Sunday: Does Dragon Curve trace over itself?

It's LSystem Sunday! I was curious about investigating an assertion that I'd made the other day: that the dragon curve loops back and traces over itself. Turns out, I was talking nonsense:

The above shows the dragon curve at it's 7th expansion, with an animation of how this is drawn on the right. As you can see, it has overlapping points, but not overlapping line segments. I was wrong.

Btw, yes, I know, but today is still logical Sunday for me.

Note: animated gif rendering supported by http://elliot.kroo.net/software/java/GifSequenceWriter/

Btw, doing conditional GET in Jersey using ETags is piss-easy. Combine this with apache server-side caching and you have a very nice persistent caching solution. I can now do some pretty inefficient LSystem drawing without worrying about my server melting ;-) For example: