attempt on triplex numbers over exponential form

[kram1032]

Here is the idea:

complex numbers have this exponential form:

or
where 1 is the positive unit 1-vector (which is trivial of course )
quaternions have this form:


where

and
a,b,c are forming a unit 3-vector.

or


so, here's the idea on how to do extensions:

basically, take a unit 2-vector, that is, cos(angle) and sin(angle) (as sin²x+cos²x=1)
this way, I propose the exponential notation to be:



or


note that this also is consistent with how spherical coordinates might be defined

(spherical to cartesian -> )

this would allow for relatively simple power rules (at *least* for integer powers)

power n:



and this is where I am right now.

This, if developed further, is one of two things: - either it's the best representation for triplex numbers, I personally could think of, or it's just an other way to produce boring Mandelbrot-rotations-around-real-axis....
(or actually, maybe it's both)

either way, this could be interesting....
can anyone out there help me further on this?

for example, if I did it right, then it's kind of clear now how transforms with powers, but I can't yet directly see, what does...

[kram1032]

I think I need to understand powers of quaternions better.
Particularily how the a,b,c values change seperately from the angle alpha in exponential form...
else I wont get to the actual Mandelbulb formula to be derived from this idea.

For now, and I guess that would be wrong, this would be the iteration formula, obviously being way too simple..

r -> r^n

phi -> n*phi

theta -> theta

though, was this actually tried yet? I doubt the result would be tooooo interesting but I'd like to see it anyway...

[bugman]

I posted an exponential triplex formula here:
http://www.fractalforums.com/theory/triplex-algebra/msg9554/#msg9554