# 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