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

On this WWW site, all displayed still and animated pictures are Scientific Visualization with a high scientific contents, except for the "Artistic Creation" chapter. In this last one, pictures are exhibited just because I like them. So, the artistic view of the prime numbers is something like a joke, computed as follows:

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

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

• Then one displays 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 one transforms the picture P1 (after a first Fourier filtering) 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...

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(CMAP28 WWW site: this page was created on 10/31/1991 and last updated on 04/26/2015 11:24:03 -CEST-)

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