Print Page - attempt on triplex numbers over exponential form

Here is the idea:

complex numbers have this exponential form:

$z = r*exp(1*\phi*i)$ or $z=r*(cos(\phi)+sin(\phi))$
where 1 is the positive unit 1-vector (which is trivial of course :) )
quaternions have this form:

$z = r*exp(\phi*v)$
where
$v=a*i+b*j+c*k$
and
a,b,c are forming a unit 3-vector.

or
$z=r*(cos(\phi)+a*i*sin(\phi)+b*j*sin(\phi)+c*k*sin(\phi))$

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:

$z=r*exp(\phi*(i*cos(\theta)+j*sin(\theta)))$

or
$<br />z=r*(cos(\phi)+i*sin(\phi)*cos(\theta)+j*sin(\phi)*sin(\theta))$

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

(spherical to cartesian -> $[r*cos(\phi),r*sin(\phi)*cos(\theta),r*sin(\phi)*sin(\theta)]$ )

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

power n:

$r^n*exp(n*\phi*(i*cos(\theta)+j*sin(\theta)))$

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 $\phi$ transforms with powers, but I can't yet directly see, what $\theta$ does...

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...

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Fractal Math, Chaos Theory & Research => Theory => Topic started by: kram1032 on December 09, 2009, 10:21:48 PM