The Hypercomplex Mandelbrot Set

Hypercomplex fractals are a new and largely unexplored area.   They are based on an extension of complex numbers into a four-dimensional number system.   Attempts have been made to make a satisfactory three-dimensional number system for fractals but these have produced only poor results.

In addition to hypercomplex fractals, fractals based on a different 4-D system called quaternions have been made.   These are in my opinion less aesthetic than hypercomplex forms.   Quaternion fractals tend to look stretched and distorted in places, whereas hypercomplex ones look like shadows and ghosts of overlapping Mandelbrot images that aren't stretched.   My thought is that the human mind better appreciates forms that are symmetrical, circular, or at least equally proportioned, and quaternions produce images that aren't proportioned in this way.

Mathematical note:   The normal and hypercomplex Mandelbrots have circular circles and aesthetic proportions because the iterative mapping is conformal.   This means right angles are preserved by the mapping.

Here are a sampling of hypercomplex images from my explorations into this area.


Fractint PAR file


Triskala

This is a hypercomplex Julia set.


Fragments

Fragments of broken iridescent crystal spirals.


Image

The ghostly image of a double spiral.


Pillar of Fire


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