Searching For The Mandelbrot In Higher Dimensions by Jesus Najera in Cantor’s Paradise.
Mandelbulb: Three Dimensional Fractals
Previously, we covered the history & the basics of iterating complex numbers in hopes of generating fractals. Starting from scratch with Julia Sets, we worked our way through defining & plotting the eminent Mandelbrot Set — arriving at the ground-breaking heart of fractal geometry:
The Mandelbrot Set Is A Dictionary Of All Julia Sets
The Mandelbrot Set visually communicates how varying starting constants, iterated ad nauseam in the function Z² + C, converge to Julia Sets (beautiful, connected patterns) or “blow up” into Fatou Sets (dis-connected clusters). As seen below with the red arrows, selecting starting complex numbers inside the Mandelbrot Set, in the filled area, generates continuous Julia Sets; starting with complex numbers outside the set, in the rest of the white space, converges to disconnected Fatou Sets.
Our initial curiosity indeed satisfied, like many discoveries in math, the uncovered Mandelbrot Set only lead us to a new realm of questions; most importantly, perhaps, is that of the equivalent in different dimensions — do fractals exist in different number systems? And more specifically, what’s the 3D equivalent of the Mandelbrot Set?
It appears that one Rudy Rucker beat me to the punch by roughly ~33 years. A brilliant mathematician, computer scientist & science fiction author, as well as one of the founders of the cyberpunk cultural movement, Rudy stayed on the cutting-edge of the STEM world. As a result, he was keenly aware of the Mandelbrot Set almost immediately after Benoit’s original publication. A creative, Rucker appreciated the Mandelbrot Set but his fiction imagination propelled him to the next step in the journey: the existence of a mathematically-equivalent 3D structure to the Mandelbrot Set.
Unfortunately, Rucker understood the computing limits of hardware (in the 80s) & knew that the billions of calculations required were likely painstakingly impossible. Limited by the technology of his time, Rucker did what he did best: he wrote about it. Rightfully, the very first authored evidence of the search for the Holy Grail in 3D came from Rucker, in 1987, in the form of a short story titled “As Above, So Below;” in it, he imagines the discovery of the Mandelbrot Set in 3D, giving it the name: Mandebulb.
We finally arrived at arguably the greatest break-through in fractal geometry since Benoit Mandelbrot first published his set in 1980: The Mandelbulb. Generated in the 3rd-dimension with the updated formula (z⁸ + c), it indeed does hold many of the properties exhibited in the Mandelbrot Set & expected in its 3D equivalent. It’s incontrovertibly visually captivating & extremely detailed; as many videos show, it also contains infinite complexity as one zooms in, much like the Mandelbrot Set.