Expanding the Mandelbrot Set into Higher Dimensions

(Proceedings pages 247–254)

When in 1980 Benoit Mandelbrot described the
*z*→*z*^{2}+*c* formula, many
mathematicians and programmers tried to expand the Mandelbrot Set
into the third dimension. But all of them where stopped by the
non-equivalence in 3D to the 2D complex product
(*a*+*bi*)⋅(*c*+*di*),
something that was well known since times of mathematician W. R.
Hamilton. Also, as the 80’s computers where not able to produce the
calculations needed to represent an image of that kind, all research
moved towards other fractal fields. It was in 2007 when the search
was recovered by means of a controversial algorithm using algebra
based on spherical coordinates triplets {ρ, φ, θ}
(module, longitude
and latitude). Although, from a strict mathematical point of view,
the process is not correct, the stunning images of the 3D set,
especially when raised to higher polynomials
*z*→*z*^{n}+*c*
soon became an
iconic fractal named *Mandelbulb*. The expansion of the Mandelbrot
Set in 4D by means of quaternions is also possible. Recent experiments
reveal that adequate projecting surfaces provide an infinite group
of projections into 3D.

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