AVirtualMachineForExploringSpaceTimeAndBeyond : The Syracuse conjecture for U(0)={5,6,7,8,...,68} -tridimensional display- (La conjecture de Syracuse pour U(0)={5,6,7,8,...,68} -visualisation tridimensionnelle-).

The Syracuse conjecture for U(0)={5,6,7,8,...,68} -tridimensional display- [La conjecture de Syracuse pour U(0)={5,6,7,8,...,68} -visualisation tridimensionnelle-].

The Syracuse sequence is defined as follows:

                    U  = N (an integer number [un nombre entier]) > 0
                     0


                    if U  is even [si U  est pair] :
                        n              n


                                                U
                                                 n
                                        U    = ----
                                         n+1    2


                    else [sinon] :


                                        U    = 3*U  + 1
                                         n+1      n

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 aeternam). For example with U(0)=7:

Here are 256 different sequences starting from U(0)=1 to U(0)=256.

This picture is a tridimensional display of sixty-four different sequences from U(0)=5 (lower left) to U(0)=68 (upper right). For each sequence U(n) the following tridimensional set of segments is generated:
```
                    X coordinates = {U(0),U(1),U(2),...,U(n),...}
                    Y coordinates = {U(1),U(2),U(3),...,U(n+1),...}
                    Z coordinates = {U(2),U(3),U(4),...,U(n+2),...}
```
with a renormalization inside [0,1] for the X and Y coordinates when a renormalization inside [-1/2,+1/2] for the Z coordinates is used. The colors used are a function of 'n' (from Dark Blue [n=0] to White with an increasing luminance ).