Let the following polar function :
r(x)=

where x is in radians
I use this UBASIC program to draw this function :
5 cls:screen 23
7 input "Scale:";C
8 clr J
9 line (319,240)(321,240):line (320,239)(320,241)
10 for I=2 to 5000000
12 Prime=prmdiv(I)
13 if Prime=I inc J
15 if Prime=I then pset ((I*cos(I))/C+320,(I*sin(I))/C+240),1
17 if inkey="+" then cancel for:C=C/2:cls:goto 9
19 if inkey="" then cancel for:C=C*2:cls:goto 9
20 next
That's what we have at different scales (when we go away from the center) :
Graph 1
Graph 2
Graph 3
At the beginning (Graph 1), we can only see 2 rotating arms in the spiral. But gradually (Graph 2 and 3), we can see 2 big arms in which we have exactly 10 little arms.Why 10 ? I never used this number neither in the function nor in the program. Now, if we take x in degrees, we obtain a simple Archimede spiral less and less dense (due to the prime numbers rarefaction). So, would it be an hidden property of pi ?
Thanks to :
 Isaac Keslassy (11 Apr 1999), who has found that it wasn't a property of pi but a property of a range including pi (3.141 to 3.144) : curious, no ?
 Douglas C. Sloan (02 Oct 1999), for his philosophical explanations.
 Erick Wong (03 Oct 1999), for his excellent explanations of all the phenomenas. Here are extracts from the mail:
The prime spiral on your web page is easily explained in terms of rational approximations to 2*Pi (this is why numbers close to Pi also work, since they have the same approximants).
The best rational approximations for 2*Pi, given by the continued fraction 6+1/(3+1/(1+1/(1+1/(7+1/(2+...))))) are 6/1, 19/3, 25/4, 44/7, 333/53, 710/113, ...
Now, imagine plotting the numbers (6n+1) in radians. We get the spiral instead of a straight line because 6 is not exactly 2*Pi.
The spirals are counterclockwise because 6 < 2*Pi radians.
I think that 19/3 and 25/4 are not close enough to 2*Pi to be recognized, and 44/7 is much better. Well, Phi(44)=20 and that's why there are 20 arms (the two large "gaps" correspond to the spirals 44n+{10,11,12} and 44n+{32,33,34}, which are never prime. Now the spirals are clockwise because 44>2*Pi*7.
As we zoom out further, I expect to find spirals corresponding to the very good approximation 710/113. and ther should be exactly Phi(710)=280 arms.
 Alan Powell (05 Oct 1999), for his good explanations about the rotation of the graph and the number of arms.
 Serge Boisse (07 Jan 2004), who has independently discovered the "prime spiral". See http://membres.lycos.fr/boisse/maths/primes_spiral.html for more information.