Ordinary Mandelbrot set:
Z --> Z^2 + C
X transformed Mandelbrot set:
Z --> (Zx)(x^(-1)Z) + C
XY transformed Mandelbrot set:
Z --> (Zx)(y^(-1)Z) + C
Where x^-1 is the inverse of x such that x*x^-1 = 1. Obviously, in the complexes
and in the quaternion the X product reduces to the ordinary Mandelbrot set due
to associativity:
(Zx)(y^(-1)Z) = Z (xx^-1) Z = Z*1*Z = Z^2
Only in the Octonions do they make a difference and reselt in completely
emergent behavior. The fractals which result may only be described with one
word: liquid fractals because they retain some of the crystalline structure of
the mandelbrot while also being fluid. The XY-fractals loose the crystalline
property and are merely fluid, looking mostly like cigarrette smoke, affine
transformations or strange attractors.
On my homepage you will find images and animations of these fascinating fractals
:
www.hsr.no/~onar/
I haven't yet had the time to write about the underlying theory, but Tony Smith
has made a rough outline of the mathematical connection between fractals and
octonions obtainable on his homepage:
www.gatech.edu/tsmith/3x3OctCnf.html
The interesting thing is that Palmer and I had a completely different motivation
for studying the octonions that does Smith and Dixon. We believe that the
Octonion may be a mathematical model of the structure of consciousness while
Smith and Dixon believe that the division algebras are the underlying structure
of the laws of physics. Deep down there is likely to be a connection between the
two, but so far we are just noting the fascinating similarity between the two
models. Maybe in the future these will converge into one theory. That would be
very aesthetic.
Onar.