Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

with D=1.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

with D=1.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

with D=1.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

with D=1.

Here is the meaning of the three {X,Y,Z} display coordinates:

The Goldbach conjecturestates that each even integer number N greater or equal to 4 can be written as the sum of two prime numbers. For example:

The horizontal axis represents the even numbers N={6, 8, 10, 12,…} starting at 6 (for compatibility with the other related visualizations). The “altitude” of each point exhibits the number of decompositions of N as sums of two prime numbers.

The horizontal and vertical axes display respectively two whole numbers A and B. Each disk display a couple of coprime numbers A and B:

The number C is the sum of A and B:

The function Radical(N) gives the product of the prime factors (with an exponent equals to 1) of N. For example:

Then the following function is computed:

The ABC conjecture states that k(A,B,C) is less than a certain constant (unknown, but greater than 1 and hopefully lesser than 2…) whatever the values of A and B.

The surface and the luminance of each disk are proportional to k(A,B,C).

For this picture, the numbers A and B belong to [1,100] giving birth to the following values:

The Syracuse sequence is defined as follows:

• The Syracuse conjecture states that sooner or later the {[[4,] 2,] 1} sequence will appear whatever the starting number N (and then repeats itself obviously ad vitam eternam). For example with N=7:
  U(0) = 7 U(1) = 22 U(2) = 11 U(3) = 34 U(4) = 17 U(5) = 52 U(6) = 26 U(7) = 13 U(8) = 40 U(9) = 20 U(10) = 10 U(11) = 5 U(12) = 16 U(13) = 8 U(14) = 4 U(15) = 2 U(16) = 1

• This picture is a circular display of sixteen different sequences from U(0)=5 (lower left) to U(0)=20 (upper right). For each sequence U(n) the following “star” is generated:
  Rho(n) = U(n) (with a renormalization inside [0,1])Teta(n) = 2.pi.n/(nm+1)X(n) = Rho(n).cos(Teta(n)) Y(n) = Rho(n).sin(Teta(n))

  where ‘nm’ denotes the maximal value of ‘n’:

  U(nm) = 1

  The colors used are a function of ‘n’ (from Dark Blue [n=0] to White with an increasing luminance).

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,…).

Then one displays the N-th point with the false color f(PA(N,B)) where:

Let’s define PA(N,B) with an obvious example:

Then the following sequence is computed:

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,…).

Then one displays the N-th point with the false color f(PM(N,B)) where:

Let’s define PM(N,B) with an obvious example:

Then the following sequence is computed:

An elementary monodimensional binary automaton is a monodimensional set of cells. At time ‘t’, each cell (with coordinate ‘x’) has a value ‘CELL(x,t)’ that equals either 0 (Black) or 1 (White) and has two neighbours (one at its left ‘CELL(x-1,t)’ and one at its right ‘CELL(x+1,t)’). The points outside the picture (at left and at right) are assumed to be White. The time evolution of this set of cells is defined by means of rules.

This picture was computed using successively the three following elementary monodimensional binary cellular automata:

automaton 86 (for t E [401,574]) -white background-

automaton 90 (for t E [101,400]) -light orange background-

automaton 106 (for t E [0,100]) -dark orange background-

The vertical axis is the time axis and the initial conditions are displayed on the bottom line.

The third coordinate ‘Z’ is the time ‘T’, whereas the two space coordinates X’ and ‘Y’ are periodical

The bidimensional life game was initially defined by Conway. It uses an empty square mesh (all vertices are turned off). At time t=0 some vertices are occupied (they are turned on): this is the initial state. To go from the time t to the time t+1, it suffices to count for each vertex -or “Cell”- C(x,y) the number N of its neighbours (it is less than or equal to 3^2-1=8) and then to possibly change the state of M according to the following bidimensional automata rules:

The boundary conditions can be periodical or not.

This process can extended in a tridimensional space. The number N of neighbours of the vertex -or “Cell”- C(x,y,z) is computed (it is less than or equal to 3^3-1=26) and the preceding rules can be extended as follows:

The bidimensional and tridimensional processes can be extended one step further using two binary lists ‘LD’ and ‘LA’ (“Dead” -off- and “Alive” -on- respectively):

(“1” means “to change the state” and “0” means “the state is unchanged”).

For this picture, the parameters have the following values:

The white peaks display the coefficients that are divisible by 2, 3, 5 and 7 (when the dark blue ones displays the coefficients that are not divisible by 2, 3, 5 and 7). The other colors display the other cases… The height of each point is proportional to the luminance of its color.

Here is the used grid with each digit (from 1 to 9) displayed with its own color: