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Author Topic: Quadray Sets  (Read 11271 times)
Description: Quadray representations of the mandelbrot set
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Aexion
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« on: April 26, 2011, 05:59:03 PM »

Hello,

Here are some formulas that I want to share with you because they appears to be extremely interesting!! smiley

Those formulas basically used the Quadray coordinate system and they looks completely symmetrical.
Two of them are 4D and the last one is 8D.
Here are them:

Octomandelbrot

This formula is created by the following procedure:
Initial Condition:
Code:
//for every x,y,z
float ctx = fabs(x   + y   + z);
float cty =fabs(-x -y + z);
float ctz =fabs(-x + y   -z);
float ctw =fabs(x   -y -z);
x=0;
y=0;
z=0;
w=0;

and the iteration loop:
Code:
for (ic=0;ic<iter;ic++){
const float wz=2*w*z;
const float xy=2*x*y;
const float X1=x*x-y*y+wz+ctx;
const float Y1=y*y-x*x+wz+cty;
const float Z1=z*z-w*w+xy+ctz;
const float W1=w*w-z*z+xy+ctw;
x=X1;
y=Y1;
z=Z1;
w=W1;
const float radius=x*x+y*y+z*z+w*w;
if(radius>64)break;
}



OctoMandelbrot 2

This fractal is eightdimensional, and it consists in a mirrored quadray coordinates.
Initial Conditions:
Code:
//for every x,y,z
const float ctx = (x   + y   + z);
const float cty =(-x -y + z);
const float ctz =(-x + y   -z);
const float ctw =(x   -y -z);
x=0;
y=0;
z=0;
w=0;
float x1=0;
float y1=0;
float z1=0;
float w1=0;

Iteration Loop
Code:
for (ic=0;ic<iter;ic++){
const float wz=2*w*z;
const float xy=2*x*y;
const float wz1=2*w1*z1;
const float xy1=2*x1*y1;
const float X1=x*x-y*y+wz1+ctx;
const float Y1=y*y-x*x+wz1+cty;
const float Z1=z*z-w*w+xy1+ctz;
const float W1=w*w-z*z+xy1+ctw;
const float X2=x1*x1-y1*y1+wz-ctx;
const float Y2=y1*y1-x1*x1+wz-cty;
const float Z2=z1*z1-w1*w1+xy-ctz;
const float W2=w1*w1-z1*z1+xy-ctw;
x=X1;
y=Y1;
z=Z1;
w=W1;
x1=X2;
y1=Y2;
z1=Z2;
w1=W2;
if((x*x+y*y+z*z+w*w+x1*x1+y1*y1+z1*z1+w1*w1)>64)break;
}



Mandelbrot Six

I have found this set while playing with the first set of the series.
It basically consists in six mandelbrot sets on where four of them lies on  the same plane and two into the poles.
A small value (I call it delta) is substracted from the quadray coordinates for generating them. You can play with those values if you wish.
These last two ones produces an incredible amount of structures..I saw many minibrots, 3d spirals and the like, and I strongly recomend you to explore them..

Initial Conditions:
Code:
//for every x,y,z
float ctx = fabs(x   + y   + z)-1.35;
float cty =fabs(-x -y + z)-1.35;
float ctz =fabs(-x + y   -z)-1.35;
float ctw =fabs(x   -y -z)-1.35;
x=0;
y=0;
z=0;
w=0;

And the iteration loop (its equal to the first example):
Code:
for (ic=0;ic<iter;ic++){
const float wz=2*w*z;
const float xy=2*x*y;
const float X1=x*x-y*y+wz+ctx;
const float Y1=y*y-x*x+wz+cty;
const float Z1=z*z-w*w+xy+ctz;
const float W1=w*w-z*z+xy+ctw;
x=X1;
y=Y1;
z=Z1;
w=W1;
const float radius=x*x+y*y+z*z+w*w;
if(radius>64)break;
}

For all sets, I have used a minimun of 20 iterations.

There are some renders on my DA gallery, if you want to see how these sets look:
http://aexion.deviantart.com/art/MiniBrot-206346586
http://aexion.deviantart.com/art/Hidden-Treasure-206276225
http://aexion.deviantart.com/art/Mandel-205892341
http://aexion.deviantart.com/art/OctoMand-2-205804254
http://aexion.deviantart.com/art/OctoMand-205781023
http://aexion.deviantart.com/art/Air-Neuron-205879803

Hope that you like those simple formulas and if anyone implement them, please let me know, since my renders are a little crude and I wish to explore them at least in realtime.

Thanks

Aexion


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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #1 on: April 26, 2011, 07:25:27 PM »

 undecided sad

Dear, I tried to implement this fractal but the result is not not good, don't know why ... When I zoom I get awful details!

Something wrong? Dunno...


* aaaaa iko.jpg (74.39 KB, 724x653 - viewed 504 times.)
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #2 on: April 26, 2011, 07:33:32 PM »

Okay, this is what I get starting from zero. It's better, now!  cheesy


Mandelbulb3Dv16{
N.....S....O/...w....2....kRY1qjmlsszOav3PzaOT/EnhIOSJyfpu9nNXEWQTbWzoSV2lrBtdsj
................................R6x80lNlO.2........A./..................y.2.....
................/ME//....6k6/...Z0....E2.....6aGlGpR3LoD/..........c./...w1....U
z.....kD..../..........wz........................................y1...sD...../..
.w1...sDQ2QrGRLl3x1..........oKQz.7mJ5bj7cCfSX3CdtHW7371QfgDzIhINc5G7vnDwcPlrBOC
dtH4ot8T6ZgDzea/wb.r8vnD......Y8......................sD.2kz0...................
..............................LRR4.0qpN.UOLb/UnRR4.orpN.cVLb/.ESR4..............
.....................wzzz1.U..6.P....M4...EB....W....61....F....8/...I1....UJl22
...U.iVFwxDE./ozPM2Tzz7lz16.mc..zXCc.El18XGQeGyDjvIRhrVAkz1............28.kFrA0.
FWb96aAIVzvh1se7Umvxz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................2CcN/UvNP6.eeWCNq0.yRii.EJJUk1f..XR
S1.mx3CcN/UvNP6.QsLsUa3.ibhV..bTV1OK.sSq40.ly3CcN/UvNP6.MwLsUa3.ibhV.kqTV1OK.sSq
40.kz3CcN/UvNP6...EsUa3.eeWCNq0.IJ36wk8.wyLsUa3.................................
E....6....E.....I....A....kLVF4N1/..............................................
...................wz0........zj........kz9.....................................
................................................................................
.....................2.....3.........2JRVFaQVZbEmx4R.65.........................
8.............................0E........kz9.....................................
................................................................................
................................}
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DarkBeam
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Posts: 2492


Fragments of the fractal -like the tip of it


« Reply #3 on: April 26, 2011, 07:43:30 PM »

Too bad, _Translate don't take care of w so it's useless for this formula fiery ... Anyway, that's a detail for

x'= ... - 1
y'= ...
z'= ... - 1
w'= ...


Mandelbulb3Dv16{
N.....S....O/...w....2....kHphfP5Ancz.6WZrJxX8.EAdnyiFmF3zXL9M7cclQyyYL92OYSxxwD
................................xymvWTMAn.2........A./..................y.2.....
................/ME/I....6Ut7...r1....E2.....g9xNuU5StmD/..........c./...w1.BnAH
y.EnAnQD..../..........wz........................................y1...sD...../..
.w1...sDeD7TowFM2qnsEr.1SsZ9zKEgTCdXUXZDX1R1AsVLiwHdSLYA6y9ovw9vpKdXUXZj22vbHu6s
Mt1kgLPZC0CKysDf/aU5StmD......ow/.....................sD.2kz0...................
..............................LRR4.0qpN.UOLb/UnRR4.orpN.cVLb/.ESR4..............
.....................wzzz1.U..6.P....M4...EB....W....61....F....8/...I1....UJl22
...U.iVFwxDE./ozPM2Tzz7lz16.mc..zXCc.El18XGQeGyDjvIRhrVAkz1............28.kFrA0.
FWb96aAIVzvh1se7Umvxz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................2CcN/UvNP6.eeWCNq0.yRii.EJJUk1f..XR
S1.mx3CcN/UvNP6.QsLsUa3.ibhV..bTV1OK.sSq40.ly3CcN/UvNP6.MwLsUa3.ibhV.kqTV1OK.sSq
40.kz3CcN/UvNP6...EsUa3.eeWCNq0.IJ36wk8.wyLsUa3.................................
E....A....E.....I....A....kLVF4N1/..............................................
...................wz0........zj........kz9.....................................
................................................................................
.....................2.....3.........2JRVFaQVZbEmx4R.65.........................
8.............................0E........kz9.....................................
................................................................................
................................/....E/...k.....TFZQVtqQg34RZ/..................
........................................kz9...................zj................
................................................................................
............................................}

too bad, I see still many bad details... Probably the "whipped cream" effect undecided - because, due to the "imperfections" of the quaternion numbers in some zones I see stretchings or weird deformations, even if I zoom...
« Last Edit: April 26, 2011, 07:49:33 PM by DarkBeam » Logged

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Aexion
Conqueror
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Posts: 116


The Fractal Hermit


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« Reply #4 on: April 26, 2011, 10:41:39 PM »

Too bad, _Translate don't take care of w so it's useless for this formula fiery ... Anyway, that's a detail for

x'= ... - 1
y'= ...
z'= ... - 1
w'= ...

too bad, I see still many bad details... Probably the "whipped cream" effect undecided - because, due to the "imperfections" of the quaternion numbers in some zones I see stretchings or weird deformations, even if I zoom...

Oh
At this rate I will become "Aexion, Mr. Formula Impossible"..  ugly

But let me try to help you..

First, the system is not quaternionic, hypercomplex or triplex..its just what you see..
Lets see the first formula
The init part is:
Code:
//for every x,y,z
//I mean, for every point in the x,y,z space..
//Calculate the C constant using Quadray
float ctx = fabs(x   + y   + z);//fabs all the way
float cty =fabs(-x -y + z);//
float ctz =fabs(-x + y   -z);//
float ctw =fabs(x   -y -z);//very important..dont remove it
//
x=0;//set them to 0
y=0;//ditto
z=0;//ditto
w=0;//also important!!

I think that on this section theres nothing special..

Now, let see the iteration loop:
Code:
//iteration loop
for (ic=0;ic<iteration;ic++){

     //multiply wz and xy by 2.0
const float wz=2.0*w*z;
const float xy=2.0*x*y;

//calculate X1,Y1,Z1,W1 using the formula and the constants ctx,cty,ctz,ctw
const float X1=x*x-y*y+wz+ctx;
const float Y1=y*y-x*x+wz+cty;
const float Z1=z*z-w*w+xy+ctz;
const float W1=w*w-z*z+xy+ctw;

//reassign to the old x,y,z,w
x=X1;
y=Y1;
z=Z1;
w=W1;//do not forget it!!..

//calculate the 4D radius..you must add the w*w here..not just x*x+ y*y + z*z..
const float radius=x*x+y*y+z*z+w*w;
if(radius>64)break;
}

Probably you get problems on the radius..

Another thing: in order to get detail, you must really use more than 20 iterations!..

Please, let me know if those explanations helps you..

I think that this formula is easy.. you don't want to see the others (higher dimensions)..

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Syntopia
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« Reply #5 on: April 26, 2011, 11:42:59 PM »

Thanks for sharing your systems, Aexion - I've always been a fan of your work!

For realtime you will need to find a suitable distance estimate.

So far this is the best I got (GLSL code):

Code:
float DE(vec3 p)
{
vec4 ct = abs(vec4(p.x+p.y+p.z,-p.x-p.y+p.z,-p.x+p.y-p.z,p.x-p.y-p.z));
vec4 w = vec4(0.0);
float dr = 1.0;
float r = 0.0;
for (int iter=0;iter<Iterations;iter++) {
w = w*w - (w*w).yxwz + 2.0*w.wwxx*w.zzyy + ct;
        r = length(w);
dr = dr*r + 1.0;
if (r > 8.0) break;
}
return r*log(r)/dr;
}

It seems to overshoot quite a lot (multiply by 0.03 or so), but you can get images like the one attached.


* Quadray.jpg (31.52 KB, 500x467 - viewed 1381 times.)
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Aexion
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« Reply #6 on: April 27, 2011, 12:25:25 AM »

Thanks for sharing your systems, Aexion - I've always been a fan of your work!

For realtime you will need to find a suitable distance estimate.

So far this is the best I got (GLSL code):

Code:
float DE(vec3 p)
{
vec4 ct = abs(vec4(p.x+p.y+p.z,-p.x-p.y+p.z,-p.x+p.y-p.z,p.x-p.y-p.z));
vec4 w = vec4(0.0);
float dr = 1.0;
float r = 0.0;
for (int iter=0;iter<Iterations;iter++) {
w = w*w - (w*w).yxwz + 2.0*w.wwxx*w.zzyy + ct;
        r = length(w);
dr = dr*r + 1.0;
if (r > 8.0) break;
}
return r*log(r)/dr;
}

It seems to overshoot quite a lot (multiply by 0.03 or so), but you can get images like the one attached.


Many Thanks Syntopia for the translation to GLSL!! smiley
I'm very outdated on the current escapetime methods and this really helps me a lot to learn about them (in particular to GLSL!)
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KRAFTWERK
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« Reply #7 on: April 27, 2011, 08:23:51 AM »

Wow, this looks beautifully interesting Aexion!
And DarkBeam... I hope you get this one right for MB3D some day...  A Beer Cup smiley
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msltoe
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« Reply #8 on: April 27, 2011, 02:34:04 PM »

Nice stuff, Aexion. I've been looking into some random four-dimensional transforms, lately, too. It's interesting how they bring about symmetries. Now if only we can generate chaotic symmetries - where after each iteration, the symmetry doubles.

-mike
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Jesse
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« Reply #9 on: April 27, 2011, 04:27:36 PM »

What nice formulas, the 6 brots one has now a formula for m3d:
http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944
 A Beer Cup


* aexion1.jpg (211.26 KB, 800x600 - viewed 521 times.)
« Last Edit: April 27, 2011, 05:41:45 PM by Jesse » Logged
Aexion
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« Reply #10 on: April 27, 2011, 05:13:35 PM »

What nice formulas, the 6 brots one has now a formula foe m3d:
http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944
 A Beer Cup

A foe??..  evil
Let me drop the Holy Fractal Grenade of Antioch  wink :
<a href="http://www.youtube.com/v/GR7tm-osFto&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/GR7tm-osFto&rel=1&fs=1&hd=1</a>

Complete with minibrots and spirals.. cheesy

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Jesse
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« Reply #11 on: April 27, 2011, 05:47:41 PM »

A foe??..  evil

i knew i better had fixed that typo  wink

Quote
Let me drop the Holy Fractal Grenade of Antioch  wink :
...
Complete with minibrots and spirals.. cheesy
...

Very nice, but where are the spirals?
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Aexion
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« Reply #12 on: April 27, 2011, 06:08:41 PM »

Quadray Box (please name it,since its not a box!)

Here is another quadray set, this time is a mandelbox based one.
For this set, theres a delta value that changes the overall aspect of the set.
Here is a delta=7, on where the box, breaks in six octahedrons and eight tetrahedrons.
The renders show here arent good, but I suppose that you can get the idea. The set is very beautiful and complex (tetrahedrons, triangles and octahedrons..but not boxes)
Init conditions:
Code:
const float delta=7;
const double ctx =fabs(x   + y   + z)-delta;
const double cty =fabs(-x -y + z)-delta;
const double ctz =fabs(-x + y   -z)-delta;
const double ctw =fabs(x   -y -z)-delta;
float w=0;
x=0;
y=0;
z=0;


Iteration Loop:
Code:

for (ic=0;ic<iter;ic++){
if(x>1)x=2-x;
else
if(x<-1)x=-2-x;
if(y>1)y=2-y;
else
if(y<-1)y=-2-y;
if(z>1)z=2-z;
else
if(z<-1)z=-2-z;
if(w>1)w=2-w;
else
if(w<-1)w=-2-w;
const float vm=sqrt(x*x+y*y+z*z+w*w);
if(vm<0.5){
x=x*4;
y=y*4;
z=z*4;
w=w*4;
} else
if(vm<1){
const float vsq=vm*vm;
x/=vsq;
y/=vsq;
z/=vsq;
w/=vsq;
}
x=x*2+ctx;
y=y*2+cty;
z=z*2+ctz;
w=w*2+ctw;
if ((x*x+y*y+z*z+w*w)>3600) break;
}
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Aexion
Conqueror
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Posts: 116


The Fractal Hermit


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« Reply #13 on: April 27, 2011, 11:02:23 PM »

A foe??..  evil

i knew i better had fixed that typo  wink

Quote
Let me drop the Holy Fractal Grenade of Antioch  wink :
...
Complete with minibrots and spirals.. cheesy
...

Very nice, but where are the spirals?

Oh.. no foe??  embarrass
My english is not very good, thats why I misunderstood.. smiley
Thanks for adding my formulas to M3D!!
As for the missing spirals, they are visible around the perimeter of the mandelbrot..
A good example is at: http://aexion.deviantart.com/art/MiniBrot-206346586
This is the largest minibrot located in the mandelbrot anntena..
I have zoomed on the elephant trunk valley and have seen them, but my main problem is the fact that they connections are very fine and they become visible thanks to the minibrot swarms.. Anyways, let me make a zoom into the easy ones.. smiley


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« Reply #14 on: April 27, 2011, 11:19:37 PM »

lol, people you found the "Balloon Formula" respects!
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