Imagine a particle in the complex plane, starting at $c_0$, moving initially $\pm$ in the horizontal or vertical directions. When it hits a [Gaussian prime][1], it turns left $90^\circ$. For example, starting at $12 - 7 i$, moving initially $+x$, this closed circuit results: <br /> ![12 - 7i][2]<br /> Instead, starting at $3+5 i$, again $+x$, this (pleasingly symmetric!) closed cycle results: <br /> ![3 + 5i][3]<br /> My question is ><b>Q0</b>.*What's going on?* More specifically, > <b>Q1</b>. Does the spiral always form a cycle? > <b>Q2</b>. Have these spirals been investigated previously? (I am about to step on a plane; apologies for not acknowledging responses!) [1]: http://en.wikipedia.org/wiki/Gaussian_prime#As_a_unique_factorization_domain [2]: http://cs.smith.edu/~orourke/MathOverflow/12_-7i.jpg [3]: http://cs.smith.edu/~orourke/MathOverflow/3_5i.jpg