artistic view of the prime numbers.

• Starting from the origin of the coordinates (at the center of the picture), follow a square spiral-like path and number each integer points encountered (1, 2, 3,...) :

                      5 --- 4 --- 3
   
                      |           |
                      |           |
                                        .
                      6     1 --- 2     .
                                        .
                      |                 |
                      |                 |
   
                      7 --- 8 --- 9 --- 10
  

  
  
  
• display the N-th point with the false color f(d(N)) where :

                      d(N) = number of divisors of N, thus
                      d(N) = 2 if N is a prime number (including N=1 for the sake of simplicity),
  

  and :

                      f(x) = an arbitrary function, for example,
                      f(x) = x
  

  Thus a first interesting picture P1 is obtained with many linear structures made of points of the same color. For example, when chosing :

                      f(2) = white
  

  and :

                      f(d) = black \-/ d#2
  

  the picture displays all the prime numbers till the number of points of the picture (in this case 512x512 = 262144). Then, it is known as the Ulam spiral.
  
  
  
• Then transform the picture P1 into the picture P2 using the following conformal mapping :

                             1
                      z --> ---
                             z
  

  the picture plane being the Complex Plane.
  
  
  
• Finally, a Fourier filtering of picture P2 gives the final result...

Related pictures :

• the Eratosthene sieve displaying 10x10 numbers (jpg=38 KB).
• the Eratosthene sieve displaying 100x100 numbers (jpg=89 KB).
• the generalized Ulam spiral displaying 100 numbers (jpg=31 KB).
• the generalized Ulam spiral displaying 1000 numbers (jpg=58 KB).
• the Archimedes spiral displaying 100 numbers (jpg=27 KB).
• the Archimedes spiral displaying 1000 numbers (jpg=44 KB).