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 $1.$  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 $90$ 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:

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

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 $2,3,9,10,11,12,13,14$ 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 $&gt;6$:


Primes up to $10^7$ starting gaps of length $&gt;10$:


Primes up to $10^8$ starting gaps of length $&gt;60$:


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&gt;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')
As @ajotatxe explained, the point between $2$ and $12$ can never be crossed, but as regards your grid, the first time the point between $33$ and $27$ is reached at $6\ 716\ 606$:

Position[AnglePath[-Pi/2 Boole[PrimeQ[Range[10^7]]]],{2,-3}][[1, 1]]




With[{a = AnglePath[-Pi/2 Boole@PrimeQ@Range@6716606]}, Graphics[{Black, PointSize@0.002, Point@Last@a, Blue, Line@a, Thick, Red, Line@Take[a, 45]}, ImageSize -&gt; 5000]]