Imagine a particle in the complex plane, starting at $c_0$, a Guassian integer, moving initially $\pm$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left $90^\circ$. For example, starting at $12 - 7 i$, moving initially $+x$, this closed circuit results:
     12 - 7i http://cs.smith.edu/%7Eorourke/MathOverflow/12_-7i.jpg
Instead, starting at $3+5 i$, again $+x$, this (pleasingly symmetric!) closed cycle results:
     3 + 5i http://cs.smith.edu/%7Eorourke/MathOverflow/3_5i.jpg
My question is

Q0.What's going on?

More specifically,

Q1. Does the spiral always form a cycle?

Q2. Have these spirals been investigated previously?

(I am about to step on a plane; apologies for not acknowledging responses!)