The quaternionic Mandelbrot set -tridimensional cross-section- [L'ensemble de Mandelbrot dans le corps des quaternions -section tridimensionnelle-].
The quaternionic Mandelbrot set -tridimensional cross-section- [L'ensemble de Mandelbrot dans le corps des quaternions -section tridimensionnelle-].
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...).
 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) |
 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) |
((((((((((((((((((((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).
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