A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'pi' (141592...)-base 10- with 50.000 time steps [Une pseudo-marche aléatoire tridimensionnelle -coordonnées sphériques- définie à l'aide des 100.000 premières décimales de 'pi' (141592...) -base 10- avec 50.000 pas de temps ]

A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'pi' (141592...) -base 10- with 50.000 time steps [Une pseudo-marche aléatoire tridimensionnelle -coordonnées sphériques- définie à l'aide des 100.000 premières décimales de 'pi' (141592...) -base 10- avec 50.000 pas de temps].




The first decimals of an arbitrary real number (0.12345678901234567890... for example) are giving birth to two sets 'THETA' and 'PHI' using the arbitrary following way: 

                                                                           -------> THETA = 1               3               5               7               9               1                ...
                                                                          |                  \             / \             / \             / \             / \             / \             /
                                                                          |                   \           /   \           /   \           /   \           /   \           /   \           /
                                                                          |                    \         /     \         /     \         /     \         /     \         /     \         /
                    0.12345678901234567890... --> 12345678901234567890... |                     \       /       \       /       \       /       \       /       \       /       \       /
                                                                          |                      \     /         \     /         \     /         \     /         \     /         \     /
                                                                          |                       \   /           \   /           \   /           \   /           \   /           \   /
                                                                          |                        \ /             \ /             \ /             \ /             \ /             \ /
                                                                           -------> PHI   =         2               4               6               8               0               2

Then 'THETA' and 'PHI' are renormalized inside [0,π] and [0,20π/11] respectively and used in order to define the successive steps {RHO.cos(PHI).sin(THETA),RHO.sin(PHI).sin(THETA),RHO.cos(THETA)} (RHO being an arbitrary constant) of an "absolute" tridimensional random walk -spherical coordinates-.



See some related pictures (including this one):


 A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of an arbitrary random number (190496...)-base 10- with 50.000 time steps
An arbitrary random number
(0.1904968414...) 		 A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of the square root of 2 (414213...)-base 10- with 50.000 time steps
The square root of 2
(1.4142135623...) 		 A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'phi' -the golden ratio- (618033...)-base 10- with 50.000 time steps
The golden ratio
(1.6180339887...) 		 A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'e' (718281...)-base 10- with 50.000 time steps
e
(2.7182818284...) 		 A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'pi' (141592...)-base 10- with 50.000 time steps
pi
(3.1415926535...)
 A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of the Champernowne number (1 2 3 4 5 6 7 8 9 10 11 12...)-base 10- with 50.000 time steps
The Champernowne number
(0.1 2 3 4 5 6 7 8 9 10 11 12...)



The arbitrary colors used are displaying the rank of each decimal (from white for the first ones to dark blue for the last ones.)


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