Formula?

[David Makin]

For the geometrical space mandelbrot iteration

newx =x2 - y2 - z2

newy = 2xy

newz = 2xz

But all you have there is the cut-down quaternion which just produces the "lathed" Mandelbrot.

[jehovajah]

My my! How things have changed in a few short weeks!

As i promised i have a few suggestions about the so called handedness term

                         ijyz + jizy

To be plotable ij and ji have to take on values from the following set
                      
                        {1, -1,i, -i,j, -j}

Setting out the possibilities in table form to facilitate exploration:

if ij =	1	-1	i	-i	j	-j
ji may =	1	i	-i	j	-j	i
or	-j	-1	1	i	-i	j
or	-i	j	i	-j	-1	1
or	j	-i	-j	-i	1	-1
or	-1	-j	j	1	j	-i
or	i	1	-1	-1	i	-j

 

Or an alternative layout

if ij =	1  -1    i   -i    j   -j
ji may =	1   1    1   1   1   1
or	-1  -1  -1  -1  -1  -1
or	i     i     i     i    i    i
or	-i    -i   -i    -i   -i   -i
or	j     j     j     j    j    j
or	-j    -j    -j   -j   -j   -j

In both cases the diagonal represents commutativity  
                                        
                                 ij = ji

and in this case the handedness term becomes 2yz and is added to or subtracted from the appropriate axis (or axes in case of non commutativity).

If David or somebody is prepared to run these transforms i would be interested to see the results.


I have designed a transform for the triplex ie ³ which is inspired by the transform on² and the notion of i being a transform in a set of transforms which include + - * / () and all algorithms. The syntax above the set imbues i with an additional property: that of a signal. Cardano when he first discussed these transforms used them as such. i will post the right handed transform later. There is of course a left handed transform which if the right handed one is interesting  i will derive from the construction process.

[jehovajah]

For the geometrical space mandelbrot iteration

newx =x² - y² - z²

newy = 2xy

newz = 2xz

But all you have there is the cut-down quaternion which just produces the "lathed" Mandelbrot.

Hiya David. Thanks for replying. I hope you follow my subsequent post. I am interested in the experiments you discarded when coming up with your alternative formula and would like to see what the results were if possible.

Your reference to the quaternion is interesting because the triplex transform i have designed is power 4 and so not quaternionic, but all attempts so far have therefore been as you say some modification of the quaternion. The analysis above introduces a handedness term exactly because the transform signals a fourth term is needed. On the basis of that and pending your investigations of the above, i  am prepared to say no power 2 formula is ever going to be adequate.

[David Makin]

Will get back to using the unit vector multiplication method shortly, am just trying some alternatives closer to the Mandelbulb method.

[jehovajah]

My my! How things have changed in a few short weeks!

As i promised i have a few suggestions about the so called handedness term

                         ijyz + jizy

To be plotable ij and ji have to take on values from the following set
                      
                        {1, -1,i, -i,j, -j}

Setting out the possibilities in table form to facilitate exploration:

if ij =	1	-1	i	-i	j	-j
ji may =	1	i	-i	j	-j	i
or	-j	-1	1	i	-i	j
or	-i	j	i	-j	-1	1
or	j	-i	-j	-i	1	-1
or	-1	-j	j	1	j	-i
or	i	1	-1	-1	i	-j

 

Or an alternative layout

if ij =	1  -1    i   -i    j   -j
ji may =	1   1    1   1   1   1
or	-1  -1  -1  -1  -1  -1
or	i     i     i     i    i    i
or	-i    -i   -i    -i   -i   -i
or	j     j     j     j    j    j
or	-j    -j    -j   -j   -j   -j

In both cases the diagonal represents commutativity  
                                        
                                 ij = ji

and in this case the handedness term becomes 2yz and is added to or subtracted from the appropriate axis (or axes in case of non commutativity).

If David or somebody is prepared to run these transforms i would be interested to see the results.

I have tried all the commutative ones in quasz
http://www.fractalforums.com/mystic-fractal-programs-gallery/mandelbrotin-3d/.
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/60/
I am going to have a go at the non commutative ones.

[jehovajah]

Recently looked at Terry Gintz idynamasz as a more appropriate programme for this exploration.