New numbers set.

[DarkBeam]

No, because I can not modify bailout conditions in MB, as I said this makes your fractal impossible to be ported in Mandelbulb. (Actually)

It can be ported in Fragmentarium for example, but I don't have the time to do much investigation.

Syntopia can help you I think. Good luck

(As the image states, that was a Makin 'brot)

[FFDiaz]

I have changed the structure of RC3 to unify the cycle basis (g) and (h) in a single that would be the conventional real numbers, in such a way that the square of a number z=x₀+x₁i+x₂j which belong to RC3 would be:

z²=(x₀²-x₁²)+(2x₀x₁+x₂²)i+2x₂(x₀+x₁)j

and the module of z would be: |z|=sqrt(x₀²+x₁²+x₂²)

with these changes, I made a small program in visualbasic with which I have obtained a series of sections of the Mandelbrot3D set thus generated.
As you can see in the attached images in the plane XY(z=0) gets the conventional Mandelbrot set. As we move away from it, the symmetry with respect to the X axis in the direction of the axis Y disappears, although  there is a symmetry with respect to the XY-plane in the Z direction. I also include an image of the silhouette of the complete set.

The code that I have generated the set is this if it is useful:

Public Class mandel3D
    Dim cx, cy, cz As Double
    Dim zx, zy, zz, x, y, z As Double
    Dim iternum, iter As Short
    Dim bailout, modulo As Single
    Dim xp, yp As Single
    Dim col As New System.Drawing.Pen(System.Drawing.Color.ForestGreen, 1)
    Dim formGraphics As System.Drawing.Graphics

    Private Sub Comenzar_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Comenzar.Click
        bailout = 2
        iternum = 1000
       
        For cz = -2.2 To 2.2 Step 0.005
            For cy = -1.2 To 1.2 Step 0.005
                For cx = -2 To 1.2 Step 0.005
                    modulo = 0
                    zx = 0
                    zy = 0
                    zz = 0
                    iter = 0

                    While iter < iternum And modulo < bailout
                        x = zx * zx - zy * zy + cx
                        y = 2 * zy * zx + zz * zz + cy
                        z = 2 * zz * zx + 2 * zy * zz + cz
                        zx = x
                        zy = y
                        zz = z
                        modulo = Math.Sqrt(zx * zx + zy * zy + zz * zz)
                        iter = iter + 1
                    End While

                    If modulo < bailout Then plotset()

                Next cx
            Next cy
        Next cz

    End Sub


    Private Sub plotset()
        Dim centrox, centroy As Integer

        centrox = 200
        centroy = 200
           
        xp = centrox - (0.8660254038 * cx - 0.7071067812 * cy) * 150
        yp = centroy - (cz - 0.5 * cx - 0.7071067812 * cy) * 150
        formGraphics.DrawLine(col, xp, yp, xp + 1, yp + 1)
    End Sub

    Private Sub mandel3D_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load
        formGraphics = Panelplot.CreateGraphics()
    End Sub
End Class

From this new perspective I will also explore RC4 and represent only the complex part, now I'll tell.

Greetings Paco Fdez.

[Syntopia]

So these are standard complex numbers, with a third component, satisfying: j*j=i, and j*i=i*j=j, right? This will give images like the attached one.

Many people have tried 3-component Mandelbrot system with Cayley table like-multiplication (especially extensions of the complex numbers). I have also tested the most reasonable combinations myself (link).

If you are interested, I used Fragmentarium with the following script - you can see the 3-vector multiplication in the code below, if you want to check it:

Code:

#include "DE-Raytracer.frag"
#group Hypercomplex Thing

uniform int Iterations;  slider[0,16,100]
uniform float Threshold; slider[0,10,100]
uniform bool JuliaMode; checkbox[false]

//Julia constant
uniform vec3 C123; slider[(-1,-1,-1),(0.18,0.88,0.24),(1,1,1)]

// FFDiaz mul
vec3 mul(vec3  a, vec3 b) {
	return vec3(
	a.x*b.x- a.y*b.y,
      a.x*b.y+a.y*b.x+a.z*b.z,
     a.x*b.z+a.z*b.x+a.y*b.z+a.z*b.y);
}

float DE(vec3 pos) {
	vec3 p = pos;
	vec3 dp = vec3(1.0,0.0,0.0);
	for (int i = 0; i < Iterations; i++) {
		dp = 2.0* mul(p,dp);
		p = mul(p,p) + (JuliaMode ? C123 : pos);
		float p2 = dot(p,p);
		orbitTrap = min(orbitTrap, abs(vec4(p.xyz,p2)));
		if (p2 > Threshold) break;
	}
	float r = length(p);
	return  0.5 * r * log(r) / length(dp);
}

[Alef]

Coincidence. I tried this too. But using Chaos pro. It have pretty usefull compiler, some 3D engine, and I alredy have alike formula, so just needed to switch my equations with posted. Could be that there are some mistake in code or in formula as 3D looks at the time producing standart 2D mandelbrot. In 2D Chaos pro just cuts off unused quaternion parts, so write whatever in j and part, and it still will generate mandelbrot. So rewriten code to loop all variables and not just quaternionic z.

Bailout conditions included too.

In 2D formula 1 generates mixed inside and outside in real positive side. The colouring aren't the most beautifull, but it nicely accents features. Both formulas produced similar results, what means, that at least there is no accidental mistakes. Much the same as above. All including julia versions have the same streching.

Code:

FfDiaz (quaternion) {

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

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);
		
		zx=real(z);
		zy=imag(z);
		zz=part_j(z);
		zw=0;
}		
void loop(void)	
{




if (power=="formula 2")
		{
 zzx = zx * zx - zy * zy + Cx;
 zzy = 2 * zy * zx + zz * zz + Cy;
 zzz = 2 * zz * zx + 2 * zy * zz + Cz;
		}

		
if (power=="formula 1")
		{
                       
								
	zzx=zx*zx+zw*zw+Cx;
     zzw=2*zx*zw+zy*zy;
     zzy=2*zy*(zx - zw)+zz*zz+Cy;
     zzz=2*zz*(zx-zw+zy)+Cz;
	  

		}	

if (power=="FunnyMistake")
		{
    zw=imag(z);    
	 //replication of my mistake I made with zw;              
								
	zzx=zx*zx+zw*zw+Cx;
     zzw=2*zx*zw+zy*zy;
     zzy=2*zy*(zx - zw)+zz*zz+Cy;
     zzz=2*zz*(zx-zw+zy)+Cz;
		}	
			
// Chaos pro cuts off unused number parts, so can't loop z.
	  zw=zzw;
	  zx=zzx;
	  zy=zzy;
	  zz=zzz;

z= quaternion(zzx, zzy, zzz, 0);

}
bool bailout(void)
	{
	return(  abs(zzx-zzw)+ abs(zzy) + abs(zzz) < bailout );
	}
void description(void)
	{
		this.title = "FfDiaz";
		//this.maxiter=17;
		this.helpfile="http://www.fractalforums.com/index.php?topic=9842.0";

		bailout.caption = "Bailout Value";
		bailout.default = 3.0;
		bailout.min = 0.5;
		bailout.hint = "Larger number increases detalisation";

    settype.caption = "Set type";
    settype.enum = "Mandelbrot Set\nJulia Set";
    settype.default = 0;

    testInverted.caption="Inverted set";
    testInverted.default=false;
	 
    power.caption = "formula";
    power.enum = "formula 2\nformula 1\nFunnyMistake";
    power.default = 0;

    julia.caption = "Julia Parameter";
    julia.default = (0.2,-0.5,-0.31,0);
    julia.hint = "4th value is not used";
    julia.visible = (settype=="Julia Set");

    perturb.caption = "Calculation Start";
    perturb.default = (0,0,0,0);
    perturb.hint = "Starts calculation with this number. 4th value is unused";
    perturb.visible = (settype=="Mandelbrot Set");
    perturb.min = -1.5;
    perturb.max = 1.5;
	}
}

[Alef]

During implementation I made a mistake, and it generated very interesting 2D, requiring bailout value of 3. Very catholic, throught I would prefared something kosher and 6 sided;) Fractal made of main square set with antenna with crosses and smaller square minibrots around. Like angular mandelbrot. Looks promising, a bitt like kalisets. Bailout value 2 produces vicsek fractal on antenna (jerusalem cross).

Pictures are pretty basic, coz I didn't used any colouring. IMHO this shows more of fractal, even if don't looks so good. Inside are coloured by bof60. This thing needs good tuning, and you'll have new kaliset like angular pattern 2D formula.

In attachments I included Chaos Pro fromula file. Just rename it to FFDiazBrot.cfm
p.s.
This forum part are very deep, probably too deep.

Code:

		Cx=real(pixel);
		Cy=imag(pixel);
		Cz=part_j(pixel);
		
		zx=real(z);
		zy=imag(z);
		zz=part_j(z);
		zw=0;

if (power=="FunnyMistake")
		{
    zw=imag(z);    
	 //replication of my mistake I made with zw;              
								
	zzx=zx*zx+zw*zw+Cx;
     zzw=2*zx*zw+zy*zy;
     zzy=2*zy*(zx - zw)+zz*zz+Cy;
     zzz=2*zz*(zx-zw+zy)+Cz;
		}
	  zw=zzw;
	  zx=zzx;
	  zy=zzy;
	  zz=zzz;

z= quaternion(zzx, zzy, zzz, 0);

[Alef]

I meant, way how cyclic numbers are implemented could contain mistake so the streching along y=-x. Maybe conditionals if, then, else, or some other non ordinary aproach...

Don't want to steal the thread or distract from seeking what everyone here are seeking. Here I optimised and tuned formula error I made. Negative equation influences just bailout, and not final z.
Allways the squares and crosses. Post function abs create some teuton orden fashioned burning ship. Features revealed by Kali suggested method using low iterations and exp smoothing.


Maybe the same thing deffined with higher powers would generate 5, 6, 7 sided figures, just that equations of it definetely aren't obvious. It looks that critical point is some x=0.7, which generates the most mandelbrot like thing.


Here dgoes Chaos pro:

Code:

Zuzubrot {
// Made this by trying to implement version of true 3D mandelbrot set 
// and making a mistake. Based on work of Francisco de Asis Fernandes Diaz
// about cyclic numbers and little bit of mine.
// http://www.fractalforums.com/index.php?topic=9842.0

// Similar to kalisets, but of chekuer squares and crosses.
// As with kalisets, details is revealed by exponent
// smoothing or twin lamps with small (colour) iteration values.  

// by Edgar Malinovsky 12.02.2012.
// You may redistribute this algorithm, modify, derivate or use comercialy as you wish as long as you give proper credits.

parameter real bailout;
parameter int settype, function;
parameter bool testInverted;
parameter complex julia, perturb;
complex C;
real zx, zy, zzx, zzy, zzw, Cx, Cy;

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);
				
}		
void loop(void)	
{

		zx=real(z);
		zy=imag(z);		  
	 //replication of my mistake I made implementing in 3D;              
								
	zzx=zx*zx+zy*zy+Cx;
	zzw=12*zx*zy+12*zy*zy; // just for a half-newton bailout.
	zzy=2*zy*zx+Cy;

	  zx=zzx;
	  zy=zzy;
z= complex(zzx, zzy);

if (function=="Abs")
{
z=abs(z);
}
else if (function=="Conj")
{
z=conj(z);
}
else if (function=="Naked singularity")
{
z=z;
}

}
bool bailout(void)
	{
	return(  abs(zzx-zzw)+ abs(zzy)  <= bailout );
	}
void description(void)
	{
		this.title = "Zuzubrot";
		this.helpfile="http://www.fractalforums.com/index.php?topic=9842.0";

		bailout.caption = "Bailout Value";
		bailout.default = 12.0;
		bailout.min = 0.5;
		bailout.hint = "Larger number increases detalisation";

    settype.caption = "Set type";
    settype.enum = "Mandelbrot Set\nJulia Set";
    settype.default = 0;

    testInverted.caption="Inverted set";
    testInverted.default=false;
	 
	function.caption = "Post function";
	function.enum= "Naked singularity\nAbs\nConj";
	function.default=0;

    julia.caption = "Julia Parameter";
    julia.default = (-1.6,0);
    julia.hint = "4th value is not used";
    julia.visible = (settype=="Julia Set");

    perturb.caption = "Calculation Start";
    perturb.default = (0.5,0);
    perturb.hint = "Starts calculation with this number. 4th value is unused";
    perturb.visible = (settype=="Mandelbrot Set");
	}
}

Today expires my UF, but here is UF code, too.

Code:

Zuzubrot {
; Made this by trying to implement version of true 3D mandelbrot set 
; and making a mistake. Based on work of Francisco de Asis Fernandes Diaz
; about cyclic numbers and little bit of mine.
; http://www.fractalforums.com/index.php?topic=9842.0

; Similar to kalisets, but of chekuer squares and crosses.
; As with kalisets, details is revealed by exponent
; smoothing or twin lamps with small (colour) iteration values.  

; by Edgar Malinovsky 12.02.2012.
; You may redistribute this algorithm, modify, derivate or use comercialy as you wish as long as you give proper credits.

global:
init:
complex c=0
float zx=0 
float zy =0
float zzx =0
float zzy =0
float zzw =0
float cx =0
float cy=0

IF (@settype=="Mandelbrot")
IF (@inverted)
c=1/#pixel
ELSE
c=#pixel
ENDIF
z=@start

ELSEIF (@settype=="Julia")
IF (@inverted)
z= 1/#pixel
ELSE
z= #pixel
ENDIF
c=@julia
ENDIF
	cx=real(c)
	cy=imag(c)
loop:
	zx=real(z)
	zy=imag(z)

	zzx=zx*zx+zy*zy+cx
	zzw=12*zx*zy+12*zy*zy; just for a half-newton bailout.
	zzy=2*zy*zx+cy

	zx=zzx
	zy=zzy
	z= zzx + 1i* zzy

IF (@function=="Abs")
z=abs(z)
ELSEIF (@function=="Conj")
z=conj(z)
ELSEIF (@function=="Naked singularity")
z=z
ENDIF

bailout:
abs(zzx-zzw)+ abs(zzy) <= @bailout

default:
title = "zTemplarbrot"
magn = 1
center = (-0.5, 0)
periodicity = 0
  
param settype
caption="Set type"
default=0
enum="Mandelbrot" "Julia" 
endparam

  param switchsettype
    caption = "switch to"
    default = 1
    enum = "Mandelbrot" "Julia"
    visible = false
  endparam

  
  float param bailout
    caption = "Bailout Value"
    default = 12.0
    min = 0.5
    hint = "This parameter defines how soon an orbit bails out while \
            iterating."
  endparam

param inverted
caption ="Inverted Set"
default = FALSE
hint="The same as circle inverse but implemented in equation."
endparam
 
param function
caption="Post function"
default=0
enum="Naked singularity" "Abs" "Conj"
endparam

  complex param julia
    caption="Julia Seed"
    default=(-1.6, 0)
    visible = (@settype=="Julia")
  endparam

 complex  param start
    caption = "Starting Point"
    default = (0.5,0)
    visible = (@settype=="Mandelbrot")
    hint = "The starting point parameter can be used to distort the set."
  endparam

switch:
type="Zuzubrot"
julia=#pixel
function=function
bailout=bailout
inverted=inverted
settype= switchsettype
switchsettype=settype
start=#pixel
}

[DarkBeam]

Looks like Kali's Mandelbrot on whatever numbers...

[FFDiaz]

In keeping with the simplification that I already commented of the ciclo-bases (g) and (h), I have represented the RC3 and set some RC4 to some fixed value x and drawing only the complex part. I include images and code in basic with which they have generated.

Code:

Public Class mandelRC4
    Dim cx, cy, cz, ct As Double
    Dim zx, zy, zz, zt, x, y, z, t As Double
    Dim iternum, iter As Short
    Dim bailout, modulo As Single
    Dim parametro_color As Single
    Dim xp, yp As Single
    Dim col As New System.Drawing.Pen(System.Drawing.Color.Black, 2)
    Dim formGraphics As System.Drawing.Graphics

    Private Sub Form1_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load
        formGraphics = PanelRC4.CreateGraphics()
    End Sub

    Private Sub Button_Comenzar_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button_Comenzar.Click
        bailout = 2
        iternum = 200
        cx = -0.5
        cy = 0
        cz = 0
        ct = 0
        For ct = -1.5 To 1.5 Step 0.005
            For cz = -1.8 To 1.8 Step 0.005
                For cy = -2 To 2 Step 0.01
                    
                        modulo = 0
                        zx = 0
                        zy = 0
                        zz = 0
                        zt = 0
                        iter = 0

                        While iter < iternum And modulo < bailout
                            x = zx * zx - zy * zy + cx
                            y = 2 * zx * zy + zz * zz + cy
                            z = 2 * zz * (zx + zy) + zt * zt + cz
                            t = 2 * zt * (zx + zy) + zz * zt + ct
                            zx = x
                            zy = y
                            zz = z
                            zt = t
                            modulo = Math.Sqrt(zx * zx + zy * zy + zz * zz + zt * zt)
                            iter = iter + 1
                        End While

                        If modulo < bailout Then
                            parametro_color = (600 * modulo) Mod 255
                            plotset()
                        End If
                   Next cy
            Next cz
        Next ct
    End Sub

    Private Sub Button_Detener_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button_Detener.Click
        End
    End Sub

    Private Sub plotset()
        Dim centrox, centroy As Integer

        centrox = 250
        centroy = 250
        col.Color = Color.FromArgb(255, parametro_color, 2 * parametro_color / 3, 255)

        xp = centrox - (0.8660254038 * cy - 0.7071067812 * cz) * 150
        yp = centroy - (ct - 0.5 * cy - 0.7071067812 * cz) * 150
        formGraphics.DrawLine(col, xp, yp, xp + 1, yp + 1)

    End Sub
End Class

As you can see my ability as a programmer it is rather limited.

Best regards

[FFDiaz]

Other also interesting results.

[Alef]

Result looks somewhat stretched and irregular, indicating that it may have some problems..

Maybe using unused 4D (all 5 number parts) could generate more nice mandelbrot set. At least hypothesis seems to be pretty logical and completely fit with exponent function colouring observation.

[FFDiaz]

I have been exploring RC3 for the iteration function f (z) = z ^ 4 + c which shows the lack of associativity and the cyclical properties of j. I have also improved the subroutine of representation with which I get a stronger 3D effect images.

Greetings Paco Fdez.