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
chapter. In this last one, pictures are exhibited just because
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,...):
| | .
| | .
6 1----2 .
- 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),
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
f(2) = white
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 (after a first Fourier filtering) into the picture P2 using the
following conformal mapping:
z --> ---
the picture plane being the Complex Plane.
- Finally, a Fourier filtering of picture P2 gives the final result...
(CMAP28 WWW site: this page was created on 10/31/1991 and last updated on 01/28/2014
[Please visit the related ArtisticCreation picture gallery [Visitez la galerie d'images ArtisticCreation associée]]
[Please visit the related NumberTheory picture gallery [Visitez la galerie d'images NumberTheory associée]]
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Copyright (c) Jean-François Colonna, 1991-2014.
Copyright (c) France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 1991-2014.