Artistic view of the prime numbers [Vue artistique des nombres premiers].

• 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, when N=1 is not a prime number),
  

  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
The Eratosthene sieve displaying 100x100 numbers

The generalized Ulam spiral displaying 100 numbers
The generalized Ulam spiral displaying 1024 numbers

The Archimedes spiral displaying 100 numbers
The Archimedes spiral displaying 1000 numbers