Logo by Fiery - Contribute your own Logo!

END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

it was a great time but no longer maintainable by c.Kleinhuis contact him for any data retrieval,
thanks and see you perhaps in 10 years again

this forum will stay online for reference
News: Visit us on facebook
 
*
Welcome, Guest. Please login or register. March 28, 2024, 10:45:36 AM


Login with username, password and session length


The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!


Pages: [1]   Go Down
  Print  
Share this topic on DiggShare this topic on FacebookShare this topic on GoogleShare this topic on RedditShare this topic on StumbleUponShare this topic on Twitter
Author Topic: New 3D Mandelbrot formula ( Orangeman )  (Read 10840 times)
Description: based on complex analogy
0 Members and 1 Guest are viewing this topic.
bkercso
Fractal Lover
**
Posts: 220



« on: May 14, 2011, 09:42:52 AM »

Hi All,

 I think I found a new way to extend Mandelbrot set to 3D. I read twinbee's article about the Real 3D Mandelbrot set. The idea is brilliant! But I think it contains a little mistake: when you square a complex number ((a,b) vector) its angle with x-axis doesn't change to 2 times bigger. The relation between original and new angle is: original tangent=b/a, new tangent=2ab/(a^2-b^2).
 In my formula I also used 2 following rotations and calculated angle in this way. I squared z=(x,y,u): z^2=Z=(A,B,C) ; where B/A=2xy/(x^2-y^2), C/B=2Bu/(B^2-u^2), and abs(Z)=abs(z)^2.

The iterative function is (yes, in pascal...  embarrass ):

null:=1E-10;
dim1:=x; dim2:=y; dim3:=u;
zabs:=0;
while zabs<4 do
begin
     if (abs(x)<null) and (abs(y)<null) then
     begin
      xtemp:=dim1;
      ytemp:=-sqr(u)+dim2;
      utemp:=dim3;
     end
     else begin
      sqrx:=sqr(x); sqry:=sqr(y); sqru:=sqr(u);
      abs1sq:=sqrx+sqry+sqru;
      xtemp1:=sqrx-sqry;
      ytemp1:=2*x*y;

      denom:=sqrx*sqry-sqru;
      if (0<=denom) and (denom<null) then denom:=null 
       else if (-null<denom) and (denom<0) then denom:=-null;
      utemp1:=2*sqrx*sqry*u/denom;

      abs2:=sqrt(sqr(xtemp1)+sqr(ytemp1)+sqr(utemp1));
      quotient:=abs1sq/abs2;
      xtemp:=xtemp1*quotient+dim1;
      ytemp:=ytemp1*quotient+dim2;
      utemp:=utemp1*quotient+dim3;
     end;

     x:=xtemp;
     y:=ytemp;
     u:=utemp;

     zabs:=sqr(x)+sqr(y)+sqr(u);

     iter:=iter+1;
end;

 The formation contains the 2D Mandelbrot set, and this 3D extension seems very logic to me: line at head changed to plane, and form has more and more smaller furrows like 2D Mandelbrot has more and more smaller circles. As the standard 2D Mandelbrot called Appleman I call this form Orangeman  cheesy

 I took some pics. Last 5 made with 7 iterations only. Unfortunately I am not a graphic artist yet, but I hope you will render some nice pics  wink
(For anaglyph pics put red-cyan glasses on!)


* 4DM_01_.png (225.06 KB, 527x393 - viewed 480 times.)

* 4DM_02_.png (191.3 KB, 647x616 - viewed 746 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #1 on: May 14, 2011, 09:48:08 AM »

.


* 4DM_03_Bottom.png (243.75 KB, 456x394 - viewed 879 times.)

* 4DM_04_zoom1.png (213.39 KB, 670x573 - viewed 605 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #2 on: May 14, 2011, 09:49:01 AM »

.


* 4DM_05_zoom2.png (208.97 KB, 492x434 - viewed 888 times.)

* 4DM_06_zoom2.png (211.44 KB, 468x430 - viewed 885 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #3 on: May 14, 2011, 09:49:49 AM »

.


* 4DM_07_7iter_botom.PNG (250.5 KB, 478x370 - viewed 2484 times.)

* 4DM_08_7iter_z-direction.PNG (222.24 KB, 434x336 - viewed 899 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #4 on: May 14, 2011, 09:50:42 AM »

.


* 4DM_09_7iter_y-direction.PNG (228.73 KB, 434x334 - viewed 853 times.)

* 4DM_10_7iter_top.PNG (209.07 KB, 434x336 - viewed 896 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #5 on: May 14, 2011, 09:51:13 AM »

.


* 4DM_11_7iter_z-dir,bottom.PNG (236.23 KB, 555x430 - viewed 520 times.)
Logged
Syntopia
Fractal Molossus
**
Posts: 681



syntopiadk
WWW
« Reply #6 on: May 14, 2011, 05:46:15 PM »

But I think it contains a little mistake: when you square a complex number ((a,b) vector) its angle with x-axis doesn't change to 2 times bigger.

To multiply two complex numbers in polar form, you multiply the moduli (lengths) and add the arguments (angle to x-axis). So the angle should double.
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #7 on: May 14, 2011, 10:48:03 PM »

oh, really! Sorry...

Then the difference between the two 3D Mandelbrot set as I see:
- twinbee doubled (original vectors xy projection; x-axis) angle and (original vector; z-axis) angle
-      I     doubled (original vectors xy projection; x-axis) angle and (NEW vectors yz projection; y-axis) angle.

Now I also tried double (original vectors xy projection; x-axis) angle and (ORIGINAL vectors yz projection; y-axis) angle, see picture.




* 4DM_v2_01.PNG (140.54 KB, 521x519 - viewed 689 times.)
Logged
Pages: [1]   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
3D Mandelbrot Formula based on the Hopf Map Theory « 1 2 » bugman 19 36760 Last post May 03, 2017, 02:27:40 PM
by faxingberlin
Sound wave generation from 1D Mandelbrot formula (new) Theories & Research Kali 7 2927 Last post February 07, 2011, 12:11:19 AM
by jehovajah
Changing Mandelbrot Formula Mandelbrot & Julia Set hoddie54 1 5312 Last post March 13, 2012, 04:55:01 PM
by lkmitch
Please try this... new 3d mandelbrot formula, animations included Mandelbulb Implementation « 1 2 » Fred Decker 18 11236 Last post January 26, 2013, 12:42:24 AM
by Fred Decker
Terms z0 and c in the Mandelbrot/Julia Iteration Formula General Discussion « 1 2 » aleph0 16 8073 Last post August 04, 2016, 05:04:54 PM
by valera_rozuvan

Powered by MySQL Powered by PHP Powered by SMF 1.1.21 | SMF © 2015, Simple Machines

Valid XHTML 1.0! Valid CSS! Dilber MC Theme by HarzeM
Page created in 0.19 seconds with 28 queries. (Pretty URLs adds 0.013s, 2q)