Expanding the Mandelbrot Set into Higher Dimensions

Javier Barrallo
Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010)
Pages 247–254 Regular Papers


When in 1980 Benoit Mandelbrot described the zz2+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 zzn+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.