Imagine a particle in the complex plane, starting at $c_0$, a [Gaussian integer][1], moving initially $\pm$ in the horizontal or vertical directions. When it hits a [Gaussian prime][2], it turns left $90^\circ$. For example, starting at $12 - 7 i$, moving initially $+x$, this closed circuit results: <br /> ![12 - 7i][3]<br /> Instead, starting at $3+5 i$, again $+x$, this (pleasingly symmetric!) closed cycle results: <br /> ![3 + 5i][4]<br /> Here's another (added later), starting at $5+23 i$: <br /> ![5 + 23i][5]<br /> (Gaussian primes $a+bi$ off-axes have $a^2+b^2$ prime; on axis, $\pm(4n+3)$ prime.)<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!) ..._Later:_ > <b>Q3</b>. Under what conditions is the spiral (assuming it closes) symmetric w.r.t. reflection in a horizontal (as is $12-7 i$), or reflection in a vertical (as is $5 + 23 i$), or reflection in both (as is $3 + 5i$)? [1]: http://en.wikipedia.org/wiki/Gaussian_integer [2]: http://en.wikipedia.org/wiki/Gaussian_prime#As_a_unique_factorization_domain [3]: https://i.sstatic.net/ntCnE.jpg [4]: https://i.sstatic.net/pKTgD.jpg [5]: https://i.sstatic.net/BvZNI.jpg