N-Dimensional Deterministic Fractal Sets
using Quaternions, Octonions and more
(MandelBulb, JuliaBulbs and beyond...)

Jean-François COLONNA

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS
91128 Palaiseau Cedex, France

http://www.lactamme.polytechnique.fr

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Abstract: Is it possible to extend the complexity of bidimensional fractal deterministic sets in tri-, four- and eight-dimensional spaces? What are "MandelBulb" and "JuliaBulb"s? Is it possible to "mix" deterministic and non deterministic fractal sets? Iterations are fundamental!

Keywords: Fractal Geometry, Deterministic Fractal Sets, MandelBulb, JuliaBulb.

Contents of this page:

1-BIDIMENSIONAL DETERMINISTIC FRACTAL SETS:
2-QUADRIDIMENSIONAL AND OCTODIMENSIONAL DETERMINISTIC FRACTAL SETS:
3-"TRUE" QUADRIDIMENSIONAL AND OCTODIMENSIONAL DETERMINISTIC FRACTAL SETS:
4-ITERATIONS ARE FUNDAMENTAL:
5-PICTURE GALLERY:

Preliminary remark: These fractal sets are dubbed deterministic for no random process is used mathematically speaking (contrary to non deterministic fractals). After all, their computations are non linear ones and then are subject to rounding-off errors that can produce random-like artifacts.

1-BIDIMENSIONAL DETERMINISTIC FRACTAL SETS:

complex plane

                    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

to iterate

                    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...

the Riemann Zeta function

thêta

number n+1 of iterations as a third dimension

2-QUADRIDIMENSIONAL AND OCTODIMENSIONAL DETERMINISTIC FRACTAL SETS:

Real numbers

labels

Complex numbers

labels

labeled

quaternions (quaternionic numbers)

octonions (octonionic numbers)

tridimensional cross-section inside the quatenionic Mandelbrot set

this zooming in

3-"TRUE" QUADRIDIMENSIONAL AND OCTODIMENSIONAL DETERMINISTIC FRACTAL SETS:

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

                    {x ,y }*{x ,y } = {x *x  - y *y ,x *y  + x *y }
                      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.

Nota

multiplication

                    {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

it will be exhibited later

                    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

pseudo-quaternions (pseudo-quaternionic numbers)

for more information -in french-

'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.

                    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

                                  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

4-ITERATIONS ARE FUNDAMENTAL:

iteration

cellular automata

random walk

5-PICTURE GALLERY:

5.1-MandelBulbs:

5.2-JuliaBulbs:

5.3-Miscellaneous Sets:

Copyright (c) Jean-François Colonna, 2009-2012.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2009-2012.

N-Dimensional Deterministic Fractal Sets using Quaternions, Octonions and more (MandelBulb, JuliaBulbs and beyond...)