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Imagine a particle in the complex plane, starting at $c_0$, a Gaussian 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:
     
Instead, starting at $3+5 i$, again $+x$, this (pleasingly symmetric!) closed cycle results:
     
Here's another (added later), starting at $5+23 i$:
     
(Gaussian primes $a+bi$ off-axes have $a^2+b^2$ prime; on axis, $\pm(4n+3)$ prime.)
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!) ...Later:

Q3. 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 $3 + 5i$)?

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian 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:
     
Instead, starting at $3+5 i$, again $+x$, this (pleasingly symmetric!) closed cycle results:
     
Here's another (added later), starting at $5+23 i$:
     
(Gaussian primes $a+bi$ off-axes have $a^2+b^2$ prime; on axis, $\pm(4n+3)$ prime.)
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!) ...Later:

Q3. 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 $3 + 5i$)?

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian 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
Here's another (added later), starting at $5+23 i$:
     5 + 23i http://cs.smith.edu/%7Eorourke/MathOverflow/5_23i.jpg
(Gaussian primes $a+bi$ off-axes have $a^2+b^2$ prime; on axis, $\pm(4n+3)$ prime.)
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!) ...Later:

Q3. 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 $3 + 5i$)?

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian 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:
     
Instead, starting at $3+5 i$, again $+x$, this (pleasingly symmetric!) closed cycle results:
     
Here's another (added later), starting at $5+23 i$:
     
(Gaussian primes $a+bi$ off-axes have $a^2+b^2$ prime; on axis, $\pm(4n+3)$ prime.)
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!) ...Later:

Q3. 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 $3 + 5i$)?