# Help with a prime number spiral which turns 90 degrees at each prime

I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the community.

I will try to explain the puzzle as best I can then detail the questions I am interested in after.

Imagine squared paper. In one square write the number $$1.1.$$ Continue to write numbers from left to right (as normal) until you reach a prime. The next number after a prime should be written in the square located $$9090$$ degrees clockwise to the last. You then continue writing numbers in that direction. This procedure should be continued indefinitely.

Here is a sample of the grid:

$$78910114041612124254314134443342633273228313029\begin{array}{} 7&8&9&10&11&40&41 \\6&1&2&&12&&42\\5&4&3&14&13&44&43\\&&34&&26\\&&33&&27\\&&32&&28\\&&31&30&29\end{array}$$

Note that the square containing 3 also contains 15 (I couldn’t put it in without confusing the diagram. In fact some squares contain multiple entries.
I would have liked to see an expanded version of the diagram. I originally thought of shading squares that contain at least one number.

Questions
Does the square surrounded by $$2,3,9,10,11,12,13,142,3,9,10,11,12,13,14$$ ever get shaded?
If so, will the whole grid ever be shaded?
Is there a maximum number of times a square can be visited? I have got to 4 times but it is easy to make mistakes by hand.
Are there any repeated patterns in the gaps?
I have other ideas but this is enough for now as I have genuinely no idea how easy or difficult this problem is.

Please forgive me for not taking it any further as it is so easy to make mistakes.
I hope this is interesting for the community and look forwards to any results.
Thanks.

Any questions I’ll do my best to clarify.

Side note: I observed that initially at least the pattern likes to cling to itself but I suspect it doesn’t later on.

Just for visual amusement, here are more pictures. In all cases, initial point is a large red dot.

Primes up to $10^5$:

Primes up to $10^6$:

Primes up to $10^6$ starting gaps of length $>6$:

Primes up to $10^7$ starting gaps of length $>10$:

Primes up to $10^8$ starting gaps of length $>60$:

For anyone interested, all the images were generated using Sage and variations of the following code:

d = 1
p = 0
M = []
prim = prime_range(10^8)
diff = []
for i in range(len(prim)-1):
diff.append(prim[i+1]-prim[i])
for k in diff:
if k>60:
M.append(p)
d = -d*I
p = p+k*d
save(list_plot(M,aspect_ratio = 1,axes = false,pointsize = 1,figsize = 20)+point((0,0),color = 'red'),'8s.png')