The quaternionic Julia set -degree=2- computed -pseudo-addition and pseudo-multiplication in C- with A=(-0.581514...,+0.635888...,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=(-0.581514...,+0.635888...,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 z

zwhen the multiplication ('*') is a little more complicated:_{1}+z_{2}= {x_{1},y_{1}}+{x_{2},y_{2}} = {x_{1}+x_{2},y_{1}+y_{2}}

zEven if it is a mathematical non sense, one can try more complicated definitions (the so-called_{1}*z_{2}= {x_{1},y_{1}}*{x_{2},y_{2}} = {x_{1}*x_{2}- y_{1}*y_{2},x_{1}*y_{2}+ x_{2}*y_{1}}

z_{1}+z_{2}= A(x_{1},y_{1},x_{2},y_{2})

z'A' and 'M' being two arbitrary complex functions, for example:_{1}*z_{2}= M(x_{1},y_{1},x_{2},y_{2})

A(x('alpha' and 'beta' being two arbitrary real numbers) and:_{1},y_{1},x_{2},y_{2}) = {x_{1}+x_{2}+(alpha*x_{1}*y_{1}),y_{1}+y_{2}+(beta*x_{2}*y_{1})}

M(x(the multiplication being unchanged in this example...)._{1},y_{1},x_{2},y_{2}) = {x_{1}*x_{2}- y_{1}*y_{2},x_{1}*y_{2}+ x_{2}*y_{1}}

See some related pictures of quaternionic Julia sets (possibly including this one):

See some related pictures of quaternionic Mandelbrot sets (possibly including this one):

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('a_{1111}*x_{1})+a_{1110})*y_{1})+((a_{1101}*x_{1})+a_{1100}))*x_{2})+((((a_{1011}*x_{1})+a_{1010})*y_{1})+((a_{1001}*x_{1})+a_{1000})))*y_{2})+((((((a_{0111}*x_{1})+a_{0110})*y_{1})+((a_{0101}*x_{1})+a_{0100}))*x_{2})+((((a_{0011}*x_{1})+a_{0010})*y_{0001})+((a_{1}*x_{1})+a_{0000})))

See some other related pictures of quaternionic Julia sets (possibly including this one):

See three other pictures of quaternionic Julia sets: the first one -left-hand side- is obtained by means of the standard addition and the standard multiplication. The two next ones are computed with all real numbers 'a

(CMAP28 WWW site: this page was created on 12/18/2018 and last updated on 10/11/2024 21:06:34 -CEST-)

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