When exploring the structure of points of the integer lattice
whose two coordinates are relatively prime
(call these *r-prime points*),^{1}
I looked at spirals analogous to "Gaussian prime spirals."^{2}
Start at an r-prime point $(a,b)$,
walk vertically (North) until you hit another r-prime point,
then walk West until another r-prime point is hit, then South, then East,
continuing to turn counterclockwise
$90^\circ$ at r-prime points until you re-encounter an earlier point,
approached from the same direction as last hit, and so fall
into a cycle.
(The start point is
considered approached from its left.)
Unlike the Gaussian prime spirals, these "relatively primes spirals"
are not generally visually interesting.
Many are just $4$-cycles, e.g.:
$$
(223, 2), (223, 3), (221, 3), (221, 2)
$$
Let me illustrate one more before asking a question.
Starting at $(495,2)$ leads to a cycle of length $44$:
$$
(495, 2), (495, 4), (493, 4), (493, 3), (494, 3), (494, 5), (493, 5),
(493, 4), (495, 4), (495, 7), (494, 7), (494, 5), (496, 5), (496, 7),
(495, 7), (495, 4), (497, 4), (497, 5), (496, 5), (496, 3), (497, 3),
(497, 4), (495, 4), (495, 2), (497, 2), (497, 3), (496, 3), (496, 1),
(497, 1), (497, 2), (495, 2), (495, 1), (496, 1), (496, 3), (494, 3),
(494, 1), (495, 1), (495, 2), (493, 2), (493, 1), (494, 1), (494, 3),
(493, 3), (493, 2)
$$
Here is an illustration of this cycle:

A natural question is:

Does any start point lead to an infinite path that never cycles?

A candidate infinite path starts at $(5,2)$: $$ (5, 2), (5, 3), (4, 3), (4, 1), (5, 1), (5, 2), (3, 2), (3, 1), (4, 1), (4, 3), (2,3), (2,1), \ldots $$ Here is its first $200$ turns:

And here is its first $1000$ turns:

And here is its first $10000$ turns:

I've tracked it out to $10^6$ turns (reaching out to $(87652,87655)$), and still no cycle. So, in addition to the general question above, a more specific question is whether $(5,2)$ ever cycles.

*Added animation*:

^{1}Arbitrarily long composite anti-diagonals?