Newtonian Triples

[eiffie]

Your formula is actually a neat trick to turn the normal extruded mandelbrot:
nx=x*x-y*y+cx
ny=2*x*y+cy
nz=z*z+cz
... into one that isn't axis aligned I couldn't have come up with that if I tried.
It is very easy to get a more twisted version (almost pick at random). I like this one:
nx=x*x-2*y*z+cx; //mix a little of y and z into x, etc..
ny=y*y-2*x*z+cy;
nz=-z*z+2*x*y+cz;
This has no mathematical basis but it makes a neat shape!

[jehovajah]

Based on your experiences guys I have modified the formula for use in the quaternion frame with zz being the coefficient of the j vector/imaginary magnitude.
N(x,æy,ßz) = xñ +æyč+ßz$ where exp(-0.5)<æ, ß <exp(1)
Try
zx^2-2*æ*ß*zy*zz + cx=zzx
2*zx*æ*zy-ß^2*zz^2 +æcy=zzy
æ^2*zy^2+2*zx*ß*zz +ßcz =zzz
In the quaternion bracket.
Please allow zz to have its full variability.

Playing with small values of æ,ß may bring out the form I am expecting, thanks guys.
Still working on cube.

[jehovajah]

Your formula is actually a neat trick to turn the normal extruded mandelbrot:
nx=x*x-y*y+cx
ny=2*x*y+cy
nz=z*z+cz
... into one that isn't axis aligned I couldn't have come up with that if I tried.
It is very easy to get a more twisted version (almost pick at random). I like this one:
nx=x*x-2*y*z+cx; //mix a little of y and z into x, etc..
ny=y*y-2*x*z+cy;
nz=-z*z+2*x*y+cz;
This has no mathematical basis but it makes a neat shape!

:woot:
Thanks eiffie, this is what I was talking about! Right there! Your my main man!

Correction! This is an offshoot and not my formula so :Unwoot: , but you are still my main man!

[jehovajah]

N³(x,y,z)
Newx= x³ - y³ +z³-6xyz +cx

Newy=3(x²y-y²z-z²x)+cy


Newz=3(x²z+y²x-z²y)+cz

Render in a quaternion frame if possible please using the quaternion coefficients as x,y,z, but do not use the underlying quaternion algebra.
Thanks in advance

[eiffie]

Your math is correct at least since this creates the z=z^3+c brot extruded at the same 45 degree angle to all axii (just like the original formula). Sorry I don't have a decent rendering of it but imagine the power 3 brot extruded.

[jehovajah]

Your math is correct at least since this creates the z=z^3+c brot extruded at the same 45 degree angle to all axii (just like the original formula). Sorry I don't have a decent rendering of it but imagine the power 3 brot extruded.

Thanks eiffie. Would love to see an un-decent image cos my imagination is not that good!

I did the analysis and found that the extrusion is endemic for this 3d form of modulo 6 roots of unity, because at y= -z you get the actual Mandelbrot relation between the axes. Thus (x,-z,z) are point references in a plane at pi/4 to (x,y,0) plane. Th x axis is the axis of rotation for this plane rotation, but the relations are not circular in the (x,0,z)plane, they break the circular relation when the coefficients start to make the quadratic or cubic terms very large. As a consequence the bailout limits cut the form into the extruded shape. I am interested to see the result particularly for ver small y, and z to confirm my analysis, but Asdam has shown a cut out that gave me the clue to what may be happening.

Your variation is very interesting, by the way, so I hope you do not feel I am dismissing it. It is simply that I have no way of verifying my formulae and the coefficient limits as yet, except by the kindness of you guys, so I do not want to discuss the important meaning of what you have demonstrated at such an early stage. Nevertheless what you demonstrated is in my opinion highly significant!

Thanks again everyone for the interest and the computer time.

[Alef]

Brot with abs function (allways positive z) was not yours but this one:
http://www.fractalforums.com/new-theories-and-research/imho-reason-behind-no-3d-mandelbrot/

I rendered your brot without abs or any other additional function. Engine use quaternion numbers for 3D render, but here it is 3 part vector without 4th part. I itereted 3 variables and left z only for colours, becouse in 3D z is quaternion number, but in 2D z is complex number. So in 2D it would lose 3rd variable and result would not be accurate correspondence. You can trust, these pics are actual result of your formula;)


Cube power looks like cube power mandelbrot extruded in 45 degrees both horizontally and vertically, but closer look shows that this fractal is slightly curved. In some parts there are small stretching, probably when you 'll get to power 8, you 'll see larger curvature and more stretched horns;) 

[Alef]

Here is more to help your imagination.

In 2D this is very small angular piece, but it perfectly corresponds to cutout in z=0. Colour lines here are z value isogradients of different iterations. Julia set with swirls is cutout of julia set somewhere z=0.1, without cutout it was a hudge mass.

Here is full code:

Code:

zAnotherMandelbrot3D (quaternion) {
// PLACE YOR M-BROT FORMULA HERE.
// WORKS IN BOTH 2D AND 3D.

parameter real bailout;
quaternion C;
real zx, zy, zz, zw, zzx, zzy, zzz, zzw, Cx, Cy, Cz, Cw, root2;
parameter bool testInverted;
parameter quaternion julia, perturb;
parameter int settype, power;
parameter real zheight;

// block for cutout.
parameter int iscuted;
parameter quaternion Ctaxis;
real autobailout, coordX, coordY, coordZ;

void init_once(void)
	{
// all callculated coefficients place here.
		root2=sqrt(2);
	}
	
void init(void)
 {
	
 	 if (testInverted==true && settype=="Julia Set")
	 {
	z = recip(pixel);
	C = julia;
	}
     else if (settype=="Julia Set")
    {	
	z = pixel;
	C = julia;
    }
    else if (testInverted==true && settype=="Mandelbrot Set")
	 {
	 z= perturb;
	 C= recip(pixel);
	 }
    else
    {
	z= perturb;
	C= pixel;
    } 
	
		Cx=real(C);
		Cy=imag(C);
		Cz=part_j(C);	
// for 4th dim use Cw, zw and zzw. Cw=part_k(C);
		
//block for cutout. An unelegant code.
autobailout=0;
coordX= real(Ctaxis);
coordY= imag(Ctaxis);
coordZ= part_j(Ctaxis);
if (iscuted == "Larger than") 
{
	if (coordX!=0 && real(pixel) > coordX )
	{
	autobailout= bailout +1;
	}
	if (coordY!=0 && imag(pixel) > coordY )
	{
	autobailout= bailout +1;
	}
	if (coordZ!=0 && part_j(pixel) > coordZ )
	{
	autobailout= bailout +1;
	}
}	
else if (iscuted == "Smaller than")
{
	if (coordX!=0 && real(pixel) < coordX )
	{
	autobailout= bailout +1;
	}
	if (coordY!=0 && imag(pixel) < coordY )
	{
	autobailout= bailout +1;
	}
	if (coordZ!=0 && part_j(pixel) < coordZ )
	{
	autobailout= bailout +1;
	}
}	
	 
		zx=real(z);
		zy=imag(z);
		zz=part_j(z)+ zheight;
		
}		
void loop(void)	
{

//PLACE YOUR M-BROT FORMULA HERE:
if (power=="Square")
{
zzx=zx*zx - 2*zy*zz + Cx;
zzy= - zz*zz + 2*zx*zy + Cy;
zzz= zy*zy + 2*zx*zz + Cz;
}
else if (power=="Cube")
{
zzx=zx^3 - zy^3+ zz^3 - 6*zx*zy*zz + Cx;
zzy= 3*(sqr(zx)*zy- sqr(zy)*zz-zx*sqr(zz)) + Cy;
zzz= 3*(sqr(zx)*zz+ zx*sqr(zy) -sqr(zy)*zz ) + Cz; //ERROR CORRECTED
}

	zx=zzx;
	zy=zzy;
	zz=zzz;
	
/// All three number parts are iterated seperetely, hence zzx and zx. 
		z= quaternion(zzx, zzy, zzz, 0);
		
// in 2D z is complex number so zz, zy, zz must be iterated or zz would be lost.
		
}
bool bailout(void)
	{
// cabs(z) is included for smooth bailout in 2D, abs(zzz) is included coz in 2D z is complex variable, autobailot is for cutout.
	return( cabs(z)+  abs(zzx)+ abs(zzy) + abs(zzz)+ autobailout < bailout );
	}
void description(void)
	{
this.title = "Newtonian Triplet";
this.maxiter=34;
this.helpfile="http://www.fractalforums.com/";

	bailout.caption = "Bailout Value";
	bailout.default = 12.0;
   
settype.caption = "Set type";
settype.enum = "Mandelbrot Set\nJulia Set";
settype.default = 0;

    testInverted.caption="Inverted set";
    testInverted.default=false;
	 

separator.label1.caption  = "4th value is unused.";
    julia.caption = "Julia Parameter";
    julia.default = (-0.2,0.3,0.45,0);
    //julia.visible = (settype=="Julia Set");

    perturb.caption = "Calculation Start";
    perturb.default = (0,0,0,0);
    perturb.visible = (settype=="Mandelbrot Set");
	 
    power.caption = "Power";
    power.enum = "Square\nCube";
    power.default = 0;	 
	 
	 
	zheight.caption = "Z height";
	zheight.default = 0.0;	 
	 
//block for a cutout.	 
separator.label2.caption  = "CUTS along NONZERO coordinates.";		
iscuted.caption="CUTTING";
iscuted.enum="None\nLarger than\nSmaller than";
iscuted.default= 0;
		
Ctaxis.caption ="Coordinates of cut";
Ctaxis.default= (0,0,0,0);
Ctaxis.visible = (iscuted=="Larger than" || iscuted=="Smaller than");
	}
}

[jehovajah]

Well, what can I say?
To start off with a million thanks Asdam. Fantastic results, and the care to analyse is very much appreciated.
I am convinced of your results and your care in application. What got me excited was your 2d cutout!. The extruded form was unexpected because of the inherent formulaic twist, but of course I had only a prayer that the Madelbrot would even turn up! Your cut out showed not only it was in their but the essential requirement for it to be in there.
The extruded form required explanation, and I think I found it from the data.

The cubic was an intuition. The swirls being there were my signal that the rotation was in there. This would encourage me to dig deeper, and play around with coefficients.

Now you have given me data to analyse, and some glorious results.
Thanks again so much

[jehovajah]

Here is more to help your imagination....
Here is full code:
.......

zzz= 3*(sqr(zx)*zz+ zx*sqr(zy) -sqr(zy)*zz ) + Cz
this.helpfile="http://www.fractalforums.com/";

Could I beg of you just one corrected run? The highlighted above ,should be sqr(zz)*zy.
It may make no difference at all, but could you check?

Also I have posted a formula for very small coefficients . I don't suppose you have time to test that out, but if you could ....

In any case, it goes without saying that I am very grateful for whatever you might be moved to do.

Thanks again.

[jehovajah]

Here is 2D, m-brot inside and cutout on z=-0.0001. All the same colout method. IMHO, it needs much more.

Now I understand your generating code better I understand the cut out better.the cut out actually seems to cut across a 3d form showing the internal colouring from a cross sectional view. In both this and the cubic a definite cross section line(change in facet line) is visible. This is why I am so interested in the small valued coefficient formula.

If you had not set up your cut out box as you have this would have been totally obscured by a plane view at pi/4 to ij axes. Your cut out box seems to cut out a cross section of a form, but with z being so small you would expect a thin slice wouldn't you? In fact you seem to have isolated a small solid mandelbulb form!

[Alef]

Here is correct version. Now this is perfectly linear and extruded.
Haven't changed coefficients, but testing on other 3D brots small coeficient change resulted only in change of dimensions but not overall shape, unles you change them much.

[Alef]

Julia set alsou is perfectly extruded;) 2D version is pretty interesting. Changing coeffitients would be much more4 easy in Ultra Fractal, becouse it would be wisual, but no 3D. Hm, maybe this could be 2D UF formula.

[jehovajah]

Many thanks Asdam. Some lovely renders and colouring. I am hoping to get my computer up and running soon and I will be able to explore these fractals using yours as a map. The slices show interesting surfaces buried in the extrusions, and rather like a diamond it will be interesting to explore what faceted cuts reveal buried in this gnarly extrusion!

[jehovajah]

Igor has supplied the body parts! and now my Frankenstein LIVES!

So Quasz rocks, as usual. The results show the twisted deformation i expected from the formula and encourages a further exploration of the coefficient proportions.

While Asdam's extruded renders are the clue to what to look for, the bailout condition promotes an extruded sculpture. Quasz uses a spherical modulus condition with a more pleasing sculpted result. The sculpture is tilted at π/4 in the yz plane.