The quaternionic Julia set -degree=2- computed -pseudo-addition and pseudo-multiplication in C- with A=(-0.58...,+0.63...,0,0) -tridimensional cross-section- [L'ensemble de Julia -degré=2- dans le corps des quaternions calculé -pseudo-addition et pseudo-multiplication dans C- pour A=(-0.58...,+0.63...,0,0) -section tridimensionnelle-].




The quaternionic numbers are defined by means of the complex numbers. Then the addition and the multiplication of quaternionic numbers are defined by means of the addition and the multiplication of complex numbers. Let's recall that the addition ('+') of two complex numbers z1={x1,y1} and z2={x2,y2} is defined as:
                    z +z  = {x ,y }+{x ,y } = {x +x ,y +y }
                     1  2     1  1    2  2      1  2  1  2
when the multiplication ('*') is a little more complicated:
                    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
Even if it is a mathematical non sense, one can try more complicated definitions (the so-called pseudo-addition and pseudo-multiplication):
                    z + z  = A(x ,y ,x ,y )
                     1   2      1  1  2  2
                    z * z  = M(x ,y ,x ,y )
                     1   2      1  1  2  2
('A' and 'M' being two arbitrary complex functions) and for example:
                    A(x ,y ,x ,y ) = {x +x + alpha*(x *y ),y +y + beta*(x *y )}
                       1  1  2  2      1  2          1  1   1  2         2  1
('alpha' and 'beta' being two arbitrary real numbers) and:
                    M(x ,y ,x ,y ) = {x *x  - y *y ,x *y  + x *y }
                       1  1  2  2      1  2    1  2  1  2    2  1
(the multiplication being unchanged in this example...).



See some related pictures (possibly including this one):


alpha=beta=0
(standard addition and standard multiplication)

alpha=beta=0.1
(pseudo-addition and standard multiplication)

alpha=beta=0.2
(pseudo-addition and standard multiplication)

alpha=beta=0.3
(pseudo-addition and standard multiplication)

alpha=beta=0.4
(pseudo-addition and standard multiplication)

alpha=beta=0.5
(pseudo-addition and standard multiplication)


But more "complex" definitions can be used. For example, each of the real and imaginary parts of 'A' and 'M' can be defined with something like:
                    ((((((((((((((((((((a    )*x ))+a    )))*y ))+(((((a    )*x ))+a    )))))*x ))+((((((((((a    )*x ))+a    )))*y ))+(((((a    )*x ))+a    )))))))*y ))+(((((((((((((((a    )*x ))+a    )))*y ))+(((((a    )*x ))+a    )))))*x ))+((((((((((a    )*x ))+a    )))*y ))+(((((a    )*x ))+a    ))))))))
                                         1111   1    1110     1         1101   1    1100       2              1011   1    1010     1         1001   1    1000         2                   0111   1    0110     1         0101   1    0100       2              0011   1    0010     1         0001   1    0000
('aNNNN' denoting arbitrary real numbers).


See some other related pictures (possibly including this one):




See three other pictures: the first one -left-hand side- is obtained by means of the standard addition and the standard multiplication. The next ones are computed with all real numbers 'aNNNN' being random (inside [-0.01,+0.01] and [-0.02,+0.02] respectively) except those defining the standard addition ({a0010=+1,a0001=+1} for the real part and {a1000=+1,a0100=+1} for the imaginary one) and the standard multiplication ({a1100=-1,a0011=+1} for the real part and {a1001=+1,a0110=+1} for the imaginary one):




(CMAP28 WWW site: this page was created on 01/05/2019 and last updated on 02/11/2019 12:19:27 -CET-)



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