Z    = f(Z )
                     n+1      n

• Mandelbrot Set M ('C' denotes the Current point) :

                      Z  = C
                       0
  
  
                              2
                      Z    = Z  + C
                       n+1    n
  

• Julia Sets J(A) ('A' and 'C' denote respectively an Argument point and the Current point) :

                      Z  = C
                       0
  
  
                              2
                      Z    = Z  + A
                       n+1    n
  

                    Z  = C
                     0


                          2        2
                    Z  = Z  + A = C  + A
                     1    0


                          2         2     2
                    Z  = Z  + A = (C  + A)  + A
                     2    1


                          2          2     2     2
                    Z  = Z  + A = ((C  + A)  + A)  + A
                     3    2


                          2           2     2     2     2
                    Z  = Z  + A = (((C  + A)  + A)  + A)  + A
                     4    3


                    etc...

Mandelbrot set

Julia set

the Riemann Zeta function

number n+1 of iterations as a third dimension

tridimensional cross-section inside the quatenionic Mandelbrot set

this zooming in

                    z +z  = {x ,y }+{x ,y } = {x +x ,y +y }
                     1  2     1  1    2  2      1  2  1  2

                    z *z  = {x ,y }*{x ,y } = {x *x  - y *y ,x *y  + x *y }
                     1  2     1  1    2  2      1  2    1  2  1  2    2  1

                    {R ,T }*{R ,T } = {R *R ,T +T }
                      1  1    2  2      1  2  1  2

                    x  = R *cos(T )
                     i    i      i


                    y  = R *sin(T )
                     i    i      i

                         2     2
                    (R,T)  = (R ,2*T)

                    {x ,y ,z }+{x ,y ,z } = {x +x ,y +y ,z +z }
                      1  1  1    2  2  2      1  2  1  2  1  2

                    {R ,T ,P }*{R ,T ,P } = {R *R ,T +T ,P +P }
                      1  1  1    2  2  2      1  2  1  2  1  2

                    x  = R *cos(P )*sin(T )
                     i    i      i       i


                    y  = R *sin(P )*sin(T )
                     i    i      i       i


                    z  = R *cos(T )
                     i    i      i

                           n     n
                    (R,T,P)  = (R ,n*T,n*P)

• All that I have done for quaternions and octonions can be used; it suffices to define a new multiplication and to add a boolean allowing to switch between the theoretical multiplication and the new one.
• Four- and octo-dimensional spaces include bi- and tridimensional spaces.
• At last, this opens doors to metamorphoses as can be seen later with a Julia set.

                    {R ,T ,P ,A }*{R ,T ,P ,A } = {fR(R ,R ),fT(T ,T ),fP(P ,P ),fA(A ,A )}
                      1  1  1  1    2  2  2  2         1  2      1  2      1  2      1  2

                    
                    x  = R *cos(P )*sin(T )*sin(A )
                     i    i      i       i       i


                    y  = R *sin(P )*sin(T )*sin(A )
                     i    i      i       i       i


                    z  = R *cos(T )*sin(A )
                     i    i      i       i


                    t  = R *cos(A )
                     i    i      i

                    fR(R ,R ) = R * R
                        1  2     1   2

                    fT(T ,T ) = T + T
                        1  2     1   2

                    fP(P ,P ) = P + P
                        1  2     1   2

                    fA(A ,A ) = A + A
                        1  2     1   2

• 'q' (=(x,y,z,t)=(R,T,P,A)) denotes a pseudo-quaternion,
• 'qA' and 'qC' are respectively a pseudo-quaternionic parameter and the current point,
• a polynomial 'P' of the first degree is written 'P(q) = 1*q + ...' in order to force a multiplication.

• : this Mandelbrot set is a tridimensional cross-section (using the t=0 hyperplane) and was computed with a polynomial 'P' of the first degree and the following four functions:

                      P(q) = 1*q + q
                                    C
  

                                         2
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 2*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 2*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = 2*(A +A )
                          1  2        1  2
  





• : this Mandelbrot set is a tridimensional cross-section (using the t=0.1 hyperplane) and was computed with a polynomial 'P' of the second degree and the following four functions:

                              2
                      P(q) = q  + q
                                   C
  

                      fR(R ,R ) = R *R
                          1  2     1  2
  

                      fT(T ,T ) = 2*(T +T ) + pi
                          1  2        1  2
  

                      fP(P ,P ) = 2*(P +P ) + pi
                          1  2        1  2
  

                      fA(A ,A ) = A +A
                          1  2     1  2
  

  Its bird's-eye view looks like a snow flake with 5 spikes. It was computed using a polynomial 'P' of the second degree. For polynomials of the third, fourth, fifth and sixth degree, the numbers of spikes are 13 (=5*2+3), 29 (=13*2+3), 61 (=29*2+3) and 125 (=61*2+3) respectively.




• : this Mandelbrot set is a tridimensional cross-section (using the t=0 hyperplane) and was computed with a polynomial 'P' of the eighth degree and the following four functions:

                              8
                      P(q) = q  + q
                                   C
  

                      fR(R ,R ) = R *R
                          1  2     1  2
  

                      fT(T ,T ) = 1*(T +T ) + pi
                          1  2        1  2
  

                      fP(P ,P ) = 2*(P +P ) + pi
                          1  2        1  2
  

                      fA(A ,A ) = A +A
                          1  2     1  2
  





• : this Mandelbrot set is a tridimensional cross-section (using the t=0 hyperplane) and was computed with a polynomial 'P' of the second degree and the following four functions:

                              2
                      P(q) = q  + q
                                   C
  

                      fR(R ,R ) = R *R
                          1  2     1  2
  

                      fT(T ,T ) = 1*(T +T ) + pi
                          1  2        1  2
  

                      fP(P ,P ) = 2*(P +P ) + pi
                          1  2        1  2
  

                      fA(A ,A ) = A +A
                          1  2     1  2
  





• : this Mandelbrot set is a tridimensional cross-section (using the t=0 hyperplane) and was computed with a polynomial 'P' of the first degree and the following four functions:

                      P(q) = 1*q + q
                                    C
  

                                         8
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 8*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 8*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = 8*(A +A )
                          1  2        1  2
  





• : this Mandelbrot set is a tridimensional cross-section (using the t=0.7 hyperplane) and was computed with a polynomial 'P' of the first degree and the following four functions:

                      P(q) = 1*q + q
                                    C
  

                                         8
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 8*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 8*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = 8*(A +A )
                          1  2        1  2
  





• : this Julia set is a tridimensional cross-section (using the t=0 hyperplane) and was computed with a polynomial 'P' of the second degree and the following four functions:

                              2
                      P(q) = q  + {-0.5815147625160462,+0.6358885017421603,0,0}
  

                      fR(R ,R ) = R *R
                          1  2     1  2
  

                      fT(T ,T ) = T +T
                          1  2     1  2
  

                      fP(P ,P ) = P +P
                          1  2     1  2
  

                      fA(A ,A ) = A +A
                          1  2     1  2
  





• : this Julia set is a tridimensional cross-section (using the t=0 hyperplane) and was computed with a polynomial 'P' of the first degree and the following four functions:

                      P(q) = 1*q + {-0.5815147625160462,+0.6358885017421603,0,0}
  

                                         3
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 3*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 3*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = A +A
                          1  2     1  2
  





• : this Julia set is a tridimensional cross-section and was computed with a polynomial 'P' of the first degree and the following four functions ('Fractal(x,y,z,t)' is a non deterministic fractal function with value inside [2,12] identical to the ones that are used to generate and animate mountains and clouds):

                      P(q) = 1*q + {0,1,0,0}
  

                                         Fractal(x,y,z,t)
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 5*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 8*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = 11*(A +A )
                          1  2         1  2
  

  It is an original mixing between deterministic and non deterministic fractals.




• : this Julia set is a tridimensional cross-section and was computed with a polynomial 'P' of the first degree and the following four functions ('Fractal(x,y,z,t)' is a non deterministic fractal function with value inside [2,12] identical to the ones that are used to generate and animate mountains and clouds):

                      P(q) = 1*q + {0,1,0,0}
  

                                         Fractal(x,y,z,t)
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = Fractal(x,y,z,t)*(T +T )
                          1  2                       1  2
  

                      fP(P ,P ) = Fractal(x,y,z,t)*(P +P )
                          1  2                       1  2
  

                      fA(A ,A ) = Fractal(x,y,z,t)*(A +A )
                          1  2                       1  2
  

  It is an original mixing between deterministic and non deterministic fractals.




• : This metamorphosis made of sixteen pictures is obtained using a rotation in the quadridimensional space of the tridimensional cross-section inside a Julia set computed with a polynomial 'P' of the first degree and the following four functions:

                      P(q) = 1*q + {-0.5815147625160462,+0.6358885017421603,0,0}
  

                                         2
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 2*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 2*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = 2*(A +A )
                          1  2        1  2
  





• : This metamorphosis made of sixteen pictures is obtained using a rotation in the quadridimensional space of the tridimensional cross-section inside a Julia set computed with a polynomial 'P' of the third degree and the following four functions:

                                    2    3
                      P(q) = 1*q - q  + q  + {-0.5815147625160462,+0.6358885017421603,0,0}
  

                                         2
                      fR(R ,R ) = (R *R )
                          1  2      1  2
  

                      fT(T ,T ) = 2*(T +T )
                          1  2        1  2
  

                      fP(P ,P ) = 2*(P +P )
                          1  2        1  2
  

                      fA(A ,A ) = 2*(A +A )
                          1  2        1  2
  

5-PICTURE GALLERY: