5th Order ABS Mandelbrot Variations by Kosmic-Stardust

5th Order ABS Mandelbrot Variations by Kosmic-Stardust

5th Order Mandelbrot Variations. These formula share remarkable similarities to my previous 3rd order formulas. The 5th Order polynomials each have six terms. Because i^2 is in the real domain, every even term is real and every odd term is imaginary. When Zi and Zr are factored out, that leaves all forth power terms remaining. To optimize the code for computational efficiency, additional variables are created for (Zr^2)^2, (Zi^2)^2 and (Zi^2)*(Zr^2). Each of these terms appears twice in the equation, reducing the number of necessary bignum multiplication operations significantly. It may be possible to further optimize code in software but this is beyond my scope.

Mandelbrot 5th:
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Mandelbar 5th (Vertical):
        zi = -zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Mandelbar 5th (horizontal):
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = -zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Burning Ship 5th:
        zi = abs(zi) * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Buffalo 5th:
        zi = abs(zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr)) +JuliaI;
        zr = abs(zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr)) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Burning Ship 5th Partial:
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Burning Ship 5th Partial Mbar:
        zi = -zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Celtic 5th (Buffalo 5th Partial):
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr)) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Celtic 5th Mbar:
        zi = -zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr)) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Quazi Burning Ship 5th (BS/Buffalo Hybrid):
        zi = -abs(zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr)) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

quazi Perpendicular 5th:
        zi = -zi * abs(5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Quazi Heart 5th:
        zi = zi * abs(5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;

Special thanks to Kalles Fraktaler and Command Line Cowboy (Panzerboy) for adding my 2nd and 3rd order fractals to their software plugins. Feel free to use any of the above formulas in any software program, image, or video render. Credit is appreciated but not mandatory. Thanks for viewing...

See Also:

2nd Order ABS Formula:
stardust4ever.deviantart.com/a…

3rd Order ABS Formula:
stardust4ever.deviantart.com/a…

4th Order ABS Formula:
stardust4ever.deviantart.com/a…

5th Order ABS Formula:
stardust4ever.deviantart.com/a…

Kaleidogal Featured By Owner Jun 5, 2017

I love all these Mandelbrot variations, very informative. I wish I could figure out how to get them into XaoS, my software of choice.

Rychveldir Featured By Owner Feb 20, 2018 Hobbyist Digital Artist

Hey!

I know I'm a bit late to the party, but I recently helped someone to translate some of the 2nd order formulas for use in Xaos, the process is relatively straightforward. Let me know if you are still interested!

Rychveldir

Kaleidogal Featured By Owner Mar 8, 2018

Thank you for reaching out to me on this. As it turns out, I don't use Xaos much anymore, I have moved on to Ultra Fractal. But one formula I would love to use in Xaos is the Burning Ship, I think it is so cool. I once tried a formula I found but it didn't work. Xaos is very picky with formula structure. How would I put these variations into UF?

Rychveldir Featured By Owner Edited Mar 9, 2018 Hobbyist Digital Artist

In Xaos the Burning Ship is given by:

(RABS(RE(Z))+I*RABS(IM(Z)))^2+C

Replace the 2 with higher powers for higher degree Burning Ships. I'm not at home right now, so I can't check the exact syntax for these formulas for UF. I'll check back later or next week when I figured them out.

As for the Burning Ship in Ultra Fractal, some of the formulas I saw are terribly complicated, just using:

z=abs(z)^n+#pixel

does the job fine. Then there's also

z=conj(z)^n+#pixel ; mandelbar
z=flip(z)^n+#pixel ;forgot what this one is called

Kaleidogal Featured By Owner Mar 9, 2018

Thank you so much for this, looking forward to trying the formula in Xaos when I get home :D (Big Grin)

Kaleidogal Featured By Owner Mar 9, 2018

I got it in Xaos, yay! Wow, that's an accomplishment. I found a formula on YouTube a year or so ago for it but it didn't work. Yours is great, thanks so much.

Rychveldir Featured By Owner Mar 19, 2018 Hobbyist Digital Artist

You're welcome!

I have been looking at different types of escape time fractals together with another user who is in the Fractals Discord channel. We sometimes exchange formulas and images, if you're interested I can send you a link.