Triplex algebra

[Tglad]

"I have extended this idea to give 48 different polar forms"
I think you could say there are an infinity of polar forms, and I think it is right to say there is one for each rotation, which is equivalent to rotating the point by this rotation each iteration.

[Softology]

Have a look at Soler's PDF.  It does show variations that do cover 48 possible variations.  Some of them do match the orginal +SIN -SIN COS etc variations, but the rest are new and do give unique bulb types.  I have included Phase, Theta and Phi scaling/shifting in all the existing bulb formulas and these 48 do give new/unique variations.

Anyway, I hope the next 3D fractal that gets as much publicity as the Mandelbulb comes from these forums.  Keep going guys.  The Mandelbulb and Kaleidoscopic IFS were great examples of someone thinking "what if" outside the scientific community.

Jason.

[Nahee_Enterprises]

    Bugman has defined a triplex polar form using a particular matrix product.
    I have extended this idea to give 48 different polar forms.  If you are
    interested then please have a look at:
        http://soler7.com/Fractals/Matrices%20to%20Triplex.pdf

    Here is an image from one of the 48 variations:
        //soler7.com/Fractals/3D17/Mandelbulb1585.JPG
    Many more are at:  http://soler7.com/Fractals/3D0.html
    All were made using Visins of Chaos.

Greetings, and Welcome to this particular Forum !!!   

Some nice work you have there.  Looking forward to your future contributions.
 

[KRAFTWERK]

Here is an image from one of the 48 variations:
<Quoted Image Removed>
Many more are at
http://soler7.com/Fractals/3D0.html
All were made using Visins of Chaos.

Wow, nice image! Very interesting!

[Softology]

Here are another 20 new variations.

http://softologyblog.wordpress.com/2011/07/21/new-mandelbulb-variations/

A few sample images...





Jason.

[Softology]

27 more new variations
http://softologyblog.wordpress.com/2011/07/27/more-new-mandelbulb-variations/







Jason.

[jehovajah]

Very nice! Paticularly like the plant analogue.

[Softology]

And yet another 16 new varieties.
http://softologyblog.wordpress.com/2011/08/01/even-more-new-mandelbulb-variations/

Jason.

[DarkBeam]

Note:
For usability sake I merged "Cosine power expansions", "Sine power expansions" and "Triplex algebra" posts and topics because it's incredibly useful to have it all in one topic. Thanks bugman

Also, the multiplication rule for Cosine formula should be like that;

Code:

a = (z1/r1 + z2/r2); or a = (z1/r2 + z2/r1) ???
r1 = sqrt (x1*x1 + y1*y1); r2 = sqrt (x2*x2 + y2*y2);
xsq = a * (x1*x2 - y1*y2);
ysq = a * (x1*y2+x2*y1);
zsq = z1 * r2 + z2 * r1; or zsq = z1 * r1 + z2 * r2 ???