The 2D pictures seems very similar to formula I just a month agou uploaded to Ultra Fractal database. Zuzubrot im EM.upr.

Julia fractals are exactly similar to picture 2.

If the images are the same, the formula should me the same. shouldn't?

I downloaded the Ultra Fractal formulas and looked up ours. And yes it is the same, although the formula I use is more general. Because mine still was in the Fractint language I translated it in Ultrafractal. The juliaform is still seperated. (it is the first time I use UF language). Here they are:

quadraticforms {

;by Jos Hendriks, origianally in Fractint, about 1990

init:

z=#pixel

float zx=0

float zy=0

float zzx=0

float zzy=0

loop:

zx=real(z)

zy=imag(z)

zzx=p1*zx*zx+p2*zy*zy+p3*zx*zy

zzy=p4*zx*zx+p5*zy*zy+p6*zx*zy

z=zzx + 1i* zzy+#pixel

bailout:

|z|<16

switch:

type = "Juliaquadraticforms"

seed = #pixel

p1=p1

p2=p2

p3=p3

p4=p4

p5=p5

p6=p6

default:

title = "quadratic forms"

float param p1

caption = "coefx x^2"

default=1.0

endparam

float param p2

caption = "coefx y^2"

default=-1

endparam

float param p3

caption = "coefx xy"

endparam

float param p4

caption = "coefy x^2"

endparam

float param p5

caption = "coefy y^2"

endparam

float param p6

caption = "coefy xy"

default=-2

endparam

}

Juliaquadraticforms {

;by Jos Hendriks, origianally in Fractint,about 1990

init:

z=#pixel

float zx=0

float zy=0

float zzx=0

float zzy=0

loop:

zx=real(z)

zy=imag(z)

zzx=p1*zx*zx+p2*zy*zy+p3*zx*zy

zzy=p4*zx*zx+p5*zy*zy+p6*zx*zy

z=zzx + 1i* zzy+@seed

bailout:

|z|<16

switch:

type = "quadraticforms"

p1=p1

p2=p2

p3=p3

p4=p4

p5=p5

p6=p6

default:

title = "juliaquadratic forms"

float param p1

caption = "coefx x^2"

endparam

float param p2

caption = "coefx y^2"

endparam

float param p3

caption = "coefx xy"

endparam

float param p4

caption = "coefy x^2"

endparam

float param p5

caption = "coefy y^2"

endparam

float param p6

caption = "coefy xy"

endparam

}

Actually You give a example of focusing on complex functions, is not always wise. You arrived at you formula, taking a complicated route, but it is a very simple and logic formula. No need for complex function theory.

This is one example. If I have some time I will show some more.

Syntopia:

Thanks for the reply. Before reading your post, I just had figured out that Fragmentarium is a program, also for rendering 3D images. I also discovered that it is possible to use the graphic processor for calculations. Because I have a very modest graphic card I did not even try it. But, surprise,surprise, I downloaded Fragmentarium and it runs very smooth. So I discovered that you are the creator. You must be very pround!!!

But, sadly, I cannot follow your script. I can see two functions: sqr and DE. But I do not see something as a main loop. The variable a, I don't understand what it is doing there. i expected the script runs over x,y,z. The page on tricomplex numbers on Wikipedia has been deleted. Anyhow if some kind if tri complex numbers is involved it is impossible to see if the sqr function does the same as my first formula. I suppose the tricomplex numbers are used to define some multiplication in R

^{3}, reflecting multiplication in R

^{2 }in one of the axes planes. Implementation of the total quadratic form, will give all possibilities, everything only using reals.