Jump to content

File:Mandelbrot set Components.jpg

Page contents not supported in other languages.
This is a file from the Wikimedia Commons
From Wikibooks, open books for an open world

Original file (1,000 × 1,000 pixels, file size: 69 KB, MIME type: image/jpeg)

File:Mandelbrot Components.svg is a vector version of this file. It should be used in place of this JPG file when not inferior.

File:Mandelbrot set Components.jpg → File:Mandelbrot Components.svg

For more information, see Help:SVG.

In other languages
Alemannisch  Bahasa Indonesia  Bahasa Melayu  British English  català  čeština  dansk  Deutsch  eesti  English  español  Esperanto  euskara  français  Frysk  galego  hrvatski  Ido  italiano  lietuvių  magyar  Nederlands  norsk bokmål  norsk nynorsk  occitan  Plattdüütsch  polski  português  português do Brasil  română  Scots  sicilianu  slovenčina  slovenščina  suomi  svenska  Tiếng Việt  Türkçe  vèneto  Ελληνικά  беларуская (тарашкевіца)  български  македонски  нохчийн  русский  српски / srpski  татарча/tatarça  українська  ქართული  հայերեն  বাংলা  தமிழ்  മലയാളം  ไทย  한국어  日本語  简体中文  繁體中文  עברית  العربية  فارسی  +/−
New SVG image

Summary

Description
English: Boundaries of 53 hyperbolic components of Mandelbrot set for periods 1-6 made with polynomial maps from the unit circle
Polski: Brzeg składowych zbioru Mandelbrot obliczony na podstawie równań brzegowych
Date
Source Own work by uploader in Maxima and Gnuplot with help of many people ( see references )
Author Adam majewski
 
This plot was created with Gnuplot.

Description with Maxima code

Boundaries of hyperbolic components of Mandelbrot sets are closed curves : cardioids[1] or circles.

Douady-Hubbard-Sullivan theorem (DHS) states that unit circle can be mapped to boundary of hyperbolic component. This relation id defined by boundary equations. Here these equations, are used to draw boundaries of hyperbolic components.

Douady-Hubbard-Sullivan theorem

Douady-Hubbard-Sullivan theorem (DHS) states that the multiplier map " of an attracting periodic orbit is a conformal isomorphism from a hyperbolic component of the Mandelbrot set onto the unit disk and it extends homeomorpically to the boundaries." [2]

Here it is important that it maps boundary of hyperbolic component to boundary of unit disk ( = unit circle ) :

and it's inverse function maps unit circle to boundary of hyperbolic components :

The algorithm

The algorithm consist of 2 big steps :

In datails there are more steps.

For given period do steps :

  • Decide how many points of closed curve you want to draw ( iMax ).
  • Compute
  • start with
  • while repeat :
    • compute point of the unit circle in the standard plane where is an internal angle,
    • map points onto the parameter plane (complex mapping ) using one of 2 methods :
      • using explicit function ( it is possible only for periods 1-3)
      • solving implicit equation with respect to ( it is posible for periods 1-8 using numerical methods)
    • compute new angle
  • draw set of points, which looks like curve [3]

Relations between hyperbolic components and unit circle

Definitions

Complex quadratic map :

f(z,c):=z*z+c;

Iterated function (map)  :

F(n, z, c) :=
   if n=1 then f(z,c)
   else f(F(n-1, z, c),c);

Multiplier of periodic orbit  :

_lambda(n):=diff(F(n,z,c),z,1);

Unit circle = boundary of unit disk

where coordinates of point of unit circle in exponential form are :

Boundary equations

Boundary equation

  • defines relations between hyperbolic components and unit circle for given period ,
  • allows computation of exact coordinates of hyperbolic componenets.

is boundary polynomial ( implicit function of 2 variables ).


Equations are in papers of Brown[4],John Stephenson[5], Wolf Jung[6]. Methods of finding boundary equations are also described in WikiBooks.

For boundary points :

so boundary equations can be in 4 equivalent forms :

period exponential trigonometric
1
2

For higher periods only P-form is used, because it is the shortest and usefull for computations.

for period 3 :

for period 4 :

for period 5 :

Solving boundary equations with respect to c

Boundary equations for periods:

  • 1-3 it can be solved with symbolical methods and give explicit solution :
    • 1-2 it is easy to solve [7]
    • 3 it can be solve using "elementary algebra" ( Stephenson )
  • >3 it can't be solved explicitly and must be solved numerically with respect to .
period 1
circle to cardioid conversion
circle to cardioid conversion

There is only one period 1 component. [8] Because boundary equation is simple :

so it is easy to get inverse multiplier map :

For each internal angle one computes :

  • point on unit circle ,
  • point

Result is a list of boundary points .

period 2

Because boundary equation is simple :

so it is easy to get inverse multiplier map  :

For each internal angle one computes :

  • point on unit circle ,
  • point

Result is a list of boundary points .

period 3
Period 3 hyperbolic components as a images of unit circle
Period 3 hyperbolic components as a images of unit circle

There are 3 period 3 components[9] Here solution of boundary equation gives 3 inverse multiplier maps .

It is possible in 3 ways :

  • Munafo method[10] (every functions maps one half of one component and one half of other component)
  • Giarrusso-Fisher method [11] ( one function for one component )
  • Walter Hannah method

I use functions by Robert Munafo.

(%i3) b3:c^3+2*c^2+(1-P)*c+(P-1)^2=0$
(%i4) solve(b3,c);
(%o4) [
c=(-(sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(((sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-  1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3,
c=((sqrt(3)*%i)/2-1/2)*
(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
((-(sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-1)*sqrt(27*P^2-22*P+23)) /(6*sqrt(3))-
(27*P^2-36*P+25)/54)^(1/3))-2/3,
c=(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(3*P+1)/(9*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))

For each internal angle one computes :

  • point on unit circle ,
  • points :

Result is a list of boundary points .

period 4

Boundary equation one can find in Mu-Ency. It can't be solved symbolicaly so it must be evaluated numerically [12].

It is 1 equation with 2 variables. To solve it one has to compute and put in . Now it is equation with 1 variable and it can be solved numerically.

For each internal angle one computes :

  • point on unit circle ,
  • Boundary polynomial
  • solve boundary equation with respect to . Result is 6 roots ( each for one of 6 period 4 components).

Result is a list of boundary points .

b4(w):=c^6 + 3*c^5 + (w/16+3)* c^4 + (w/16+3)* c^3  - (w/16+2)* (w/16-1)* c^2 - (w/16-1)^3;
l(t):=%e^(%i*t*2*%pi);
iMax:200; /* number of point */
dt:1/iMax;
/* point to point method of drawing */
t:0; /* angle in turns */
w:rectform(ev(l(t), numer)); /* "exponential form prevents allroots from working", code by Robert P. Munafo */
/* compute equation for given w */
per4:expand(b4(w));
/* compute 6 complex roots and save them to the list cc4 */
cc4:allroots(per4);
/*  create new lists and save coordinates  to draw it later */ 
xx4:makelist (realpart(rhs(cc4[1])), i, 1, 1); 
yy4:makelist (imagpart(rhs(cc4[1])), i, 1, 1);
for j:2 thru 6 step 1 do
 block
 (
  xx4:cons(realpart(rhs(cc4[j])),xx4),
  yy4:cons(imagpart(rhs(cc4[j])),yy4)
 );
for i:2 thru iMax step 1 do
block
( t:t+dt,
  w:rectform(ev(l(t), numer)), /* code by Robert P. Munafo  */
  per4:expand(m4(w)),
  cc4:allroots(per4),
  for j:1 thru 6 step 1 do
   block
   (
    xx4:cons(realpart(rhs(cc4[j])),xx4),
    yy4:cons(imagpart(rhs(cc4[j])),yy4)
   )
  );
period 5

one computes in the same way as for period 4, only implicit function is diffrent and there are 15 components.

period 6

one computes in the same way as for period 4, only implicit function is diffrent (see Stephenson paper II ) and there are 27 components.

period 7

one computes in the same way as for period 4, only implicit function is diffrent (degree in c is 63; see Stephenson paper III ) and there are 63 components.

period 8

Implicit equation can be computed but "is too large to exhibit" (see Stephenson paper III ). There are 120 components.

Higher periods

"Although extension of the arithmethic method to higher orders is possible in principle, the computations become to big in space and time " (Stephenson paper III )

Relations between boundary equation, multiplier map, inverse multiplier map and multiplier

period
1
2
3

Symbolic solution of boundary equation is possible only for periods 1-3 ( with respect to or ). Every function can be in 4 equivalent forms : P, w, exponential t, trigonometric t (see boundary equations for details).

Period 1

Solving with respect to gives 2 results. hoose attracting one.

Period 2

Solving is simple because these are degree 1 equations ( with respect to both and ).

Period 3

Solving with respect to is possible in 3 ways.

Solving with respect to gives 2 results. One have to choose attracting.


Maxima source code

/* 
batch file for Maxima
http://maxima.sourceforge.net/
wxMaxima 0.7.6 http://wxmaxima.sourceforge.net archive copy at the Wayback Machine
Maxima 5.16.1 http://maxima.sourceforge.net
Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
Distributed under the GNU Public License. 
based on :
http://www.mrob.com/pub/muency/brownmethod.html
*/
start:elapsed_run_time ();
iMax:200; /* number of points to draw */
dt:1/iMax;
/* 
unit circle D={w:abs(w)=1 } where w=l(t) 
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
*/
l(t):=%e^(%i*t*2*%pi);
/* 
conformal maps from unit circle 
to hyperbolic component of Mandelbrot set of period 1-4 
These functions ( maps ) are computed in other batch file 
*/
/* ---------------  inverse function of multiplier map : explicit function : c=gamma_p(P)  where P = w/(2^period) ---------------- */
gamma1(P):=P-P^2;
gamma2(P):=P - 1;
/* code of functions by Robert P. Munafo */
gamma3a(P):=(-(sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(((sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3;
gamma3b(P):=((sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
((-(sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-  1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3;
gamma3c(P):=(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+(3*P+1)/(9*(((P-1)*sqrt(27*P^2-22*P+23))/ 
(6*sqrt(3))-   (27*P^2-36*P+25) /54)^(1/3))-2/3;
/* ---------- boundary equation (implicit function)  b_p(P,c)=0 ------------------------------------------------------------------ */
b4(P):=c^6 + 3*c^5 + (P+3)* c^4 + (P+3)* c^3  - (P+2)*(P-1)*c^2 - (P-1)^3;
/* ------ period 5 ------------- */
b5(P):=c^15 + 
8*c^14 + 
28*c^13 + 
(P + 60)*c^12 + 
(7*P + 94)*c^11 + 
(3*(P)^2 + 20*P + 116)*c^10 + 
(11*P^2 + 33*P + 114)*c^9 +
(6*P^2 + 40*P + 94)*c^8 + 
(2*P^3 - 20*P^2 + 37*P + 69)*c^7 + 
(3*P - 11)*(3*P^2 - 3*P - 4)*c^6 + 
(P - 1)*(3*P^3 + 20*P^2 - 33*P - 26)*c^5 + 
( 3*P^2 + 27*P + 14)*((P - 1)^2)*c^4 - 
(6*P + 5)*((P - 1)^3 )*c^3 + 
(P + 2)*((P - 1)^4)*c^2 - 
c*(P - 1)^5  + 
(P - 1)^6 ;
/*-----period 6 ----------------------- */
b6(P):=
c^27+
13*c^26+
78*c^25+
(293 - P)*c^24+
(792 - 10*P)*c^23+
(1672 - 41*P)*c^22+
(2892 - 84*P - 4*P^2)*c^21+
(4219 - 60*P - 30*P^2)*c^20+
(5313 + 155*P - 80*P^2)*c^19+
(5892 + 642*P - 57*P^2 + 4*P^3)*c^18+
(5843 + 1347*P + 195*P^2 + 22*P^3)*c^17+
(5258 + 2036*P + 734*P^2 + 22*P^3)*c^16+
(4346 + 2455*P + 1441*P^2 - 112*P^3 + 6*P^4)*c^15 + 
(3310 + 2522*P + 1941*P^2 - 441*P^3 + 20*P^4)*c^14 + 
(2331 + 2272*P + 1881*P^2 - 853*P^3 - 15*P^4)*c^13 + 
(1525 + 1842*P + 1344*P^2 - 1157*P^3 - 124*P^4 - 6*P^5)*c^12 + 
(927 + 1385*P + 570*P^2 - 1143*P^3 - 189*P^4 - 14*P^5)*c^11 + 
(536 + 923*P - 126*P^2 - 774*P^3 - 186*P^4 + 11*P^5)*c^10 + 
(298 + 834*P + 367*P^2 + 45*P^3 - 4*P^4 + 4*P^5)*(1-P)*c^9 + 
(155 + 445*P - 148*P^2 - 109*P^3 + 103*P^4 + 2*P^5)*(1-P)*c^8 + 
2*(38 + 142*P - 37*P^2 - 62*P^3 + 17*P^4)*(1-P)^2*c^7 + 
(35 + 166*P + 18*P^2 - 75*P^3 - 4*P^4)*((1-P)^3)*c^6 + 
(17 + 94*P + 62*P^2 + 2*P^3)*((1-P)^4)*c^5 + 
(7 + 34*P + 8*P^2)*((1-P)^5)*c^4 + 
(3 + 10*P + P^2)*((1-P)^6)*c^3 + 
(1 + P)*((1-P)^7)*c^2 +
-c*((1-P)^8) + (1-P)^9;
/*-----------------------------------*/
/* point to point method of drawing */
t:0; /* angle in turns */ 
/* compute first point of curve, create list and save point to this list */
/* point of unit circle   w:l(t); */
w:rectform(ev(l(t), numer)); /* "exponential form prevents allroots from working", code by Robert P. Munafo */ 
/* ---- period 1 -------------------*/
P:w/2;
c1:gamma1(P);
xx1:makelist (realpart(c1), i, 1, 1); /* save coordinates  to draw it later */ 
yy1:makelist (imagpart(c1), i, 1, 1);
/* -----period 2 --------------*/
P:P/2;
c2:gamma2(P); 
xx2:makelist (realpart(c2), i, 1, 1); 
yy2:makelist (imagpart(c2), i, 1, 1); 
/* period 3 components */
P:P/2;
c3:gamma3a(P); 
xx3a:makelist (realpart(c3), i, 1, 1); 
yy3a:makelist (imagpart(c3), i, 1, 1); 
c3:gamma3b(w);
xx3b:makelist (realpart(c3), i, 1, 1); 
yy3b:makelist (imagpart(c3), i, 1, 1); 
c3:gamma3c(w);
xx3c:makelist (realpart(c3), i, 1, 1); 
yy3c:makelist (imagpart(c3), i, 1, 1);
/* period 4 */ 
P:P/2;
per4:expand(b4(P)); /* compute equation for given w ( t) */
cc4:allroots(per4); /* compute 6 complex roots and save them to the list cc4 */
/*  create new lists and save coordinates  to draw it later */ 
xx4:makelist (realpart(rhs(cc4[1])), i, 1, 1); 
yy4:makelist (imagpart(rhs(cc4[1])), i, 1, 1);
for j:2 thru 6 step 1 do
block
(
xx4:cons(realpart(rhs(cc4[j])),xx4),
yy4:cons(imagpart(rhs(cc4[j])),yy4)
);
/* period 5 */
P:P/2;
per5:expand(b5(P)); /* compute equation for given w ( t) */
cc5:allroots(per5); /* compute 15 complex roots and save them to the list cc5 */
/*  create new lists and save coordinates  to draw it later */ 
xx5:makelist (realpart(rhs(cc5[1])), i, 1, 1); 
yy5:makelist (imagpart(rhs(cc5[1])), i, 1, 1);
for j:2 thru 15 step 1 do
block
(
xx5:cons(realpart(rhs(cc5[j])),xx5),
yy5:cons(imagpart(rhs(cc5[j])),yy5)
);
/* period 6 */
P:P/2;
per6:expand(b6(P)); /* compute equation for given w ( t) */
cc6:allroots(per6); /* compute 15 complex roots and save them to the list cc5 */
/*  create new lists and save coordinates  to draw it later */ 
xx6:makelist (realpart(rhs(cc6[1])), i, 1, 1); 
yy6:makelist (imagpart(rhs(cc6[1])), i, 1, 1);
for j:2 thru 27 step 1 do
block
(
 xx6:cons(realpart(rhs(cc6[j])),xx6),
 yy6:cons(imagpart(rhs(cc6[j])),yy6)
) ;
/* ------------*/
for i:2 thru iMax step 1 do
block
( t:t+dt,
 w:rectform(ev(l(t), numer)), /* "exponential form prevents allroots from working", code by Robert P. Munafo */ 
 P:w/2,
 c1:gamma1(P),
 /* save values to draw it later */
 xx1:cons(realpart(c1),xx1),
 yy1:cons(imagpart(c1),yy1),
 P:P/2,
 c2:gamma2(P),
 xx2:cons(realpart(c2),xx2),
 yy2:cons(imagpart(c2),yy2),
 P:P/2,
 c3:gamma3a(P),
 xx3a:cons(realpart(c3),xx3a),
 yy3a:cons(imagpart(c3),yy3a),
 c3:gamma3b(P),
 xx3b:cons(realpart(c3),xx3b),
 yy3b:cons(imagpart(c3),yy3b),
 c3:gamma3c(P),
 xx3c:cons(realpart(c3),xx3c),
 yy3c:cons(imagpart(c3),yy3c),
 /* period 4 */
 P:P/2,
 per4:expand(b4(P)),
 cc4:allroots(per4),
 for j:1 thru 6 step 1 do
  block
   (
   xx4:cons(realpart(rhs(cc4[j])),xx4),
   yy4:cons(imagpart(rhs(cc4[j])),yy4)
   ),
 /* period 5 */
 P:P/2,
 per5:expand(b5(P)), /* compute equation for given w ( t) */
 cc5:allroots(per5), /* compute 15 complex roots and save them to the list cc5 */
 for j:1 thru 15 step 1 do
 block
  (
  xx5:cons(realpart(rhs(cc5[j])),xx5),
  yy5:cons(imagpart(rhs(cc5[j])),yy5)
  ),
 /* period 6 */
 P:P/2,
 per6:expand(b6(P)), /* compute equation for given w ( t) */
 cc6:allroots(per6), /* compute 27 complex roots and save them to the list cc6 */
 for j:1 thru 27 step 1 do
  block
  (
  xx6:cons(realpart(rhs(cc6[j])),xx6),
  yy6:cons(imagpart(rhs(cc6[j])),yy6)
  )    
 );
stop:elapsed_run_time ();
time:fix(stop-start); 
load(draw);
draw2d(
  file_name = "", /* file in directory  C:\Program Files\Maxima-5.16.1\wxMaxima */
  terminal  = 'screen, /* jpg when draw to file with jpg extension */
  pic_width  = 1000,
  pic_height = 1000,
  yrange = [-1.5,1.5],
  xrange = [-2,1],
  title= concat("Boundaries of 53 hyperbolic components of Mandelbrot set made in ",string(time),"sec"),
  xlabel     = "c.re ",
  ylabel     = "c.im",
  point_type    = dot,
  point_size    = 5,
  points_joined =true,
  user_preamble="set size square;set key out vert;set key bot center",
  key = "one period 1 component = {c:c=(2*w-w*w)/4} ",
  color         = red,
  points(xx1,yy1),
  key = "one period 2 component = {c:c=(w/4 -1)} ",
  color         = green,
  points(xx2,yy2),
  key = "",
  color         = blue,
  points_joined =false, /* there are 3 curves so we can't join points */
  points(xx3a,yy3a),
  points(xx3b,yy3b),
  key = "three period 3 components (blue)",
  points(xx3c,yy3c),
  key = "six period 4 components (magenta)",
  color         = magenta,
  points(xx4,yy4),
  key = "fifteen period 5 components (black)",
  color         = black,
  points(xx5,yy5),
  key = "27 period 6 components (black)",
  color         = black,
  points(xx6,yy6)
);

Questions

Other implementations

See also

References

  1. The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid. These curves are notr cardioids but higher degree curves
  2. Multipliers of periodic orbits of quadratic polynomials and the parameter plane by Genadi Levin
  3. Algebraic solution of Mandelbrot orbital boundaries by Donald D. Cross
  4. A. Brown, Equations for Periodic Solutions of a Logistic Difference Equation, J. Austral. Math. Soc (Series B) 23, 78–94 (1981).
  5. John Stephenson : "Formulae for cycles in the Mandelbrot set", Physica A 177, 416-420 (1991); "Formulae for cycles in the Mandelbrot set II", Physica A 190, 104-116 (1992); "Formulae for cycles in the Mandelbrot set III", Physica A 190, 117-129 (1992)
  6. Wolf Jung : "Some Explicit Formulas for the Iteration of Rational Functions" , unpublished manuscript of August 1997 containing Maple code
  7. Thayer Watkins : The Structure of the Mandelbrot Set
  8. Enumeration of Features by Robert P. Munafo
  9. M. Lutzky: Counting hyperbolic components of the Mandelbrot set. Physics Letters A Volume 177, Issues 4-5, 21 June 1993, Pages 338-340
  10. Brown Method by Robert P. Munafo
  11. A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set Dante Giarrusso; Yuval Fisher Proceedings of the American Mathematical Society, Vol. 123, No. 12. (Dec., 1995), pp. 3731-3737
  12. Exact Coordinates by Robert P. Munafo
  13. Internal Rays of the Mandelbrot Set by Walter Hannah
  14. Mark McClure "Bifurcation sets and critical curves" - Mathematica in Education and Research, Volume 11, issue 1 (2006). archive copy at the Wayback Machine

Acknowledgements

This program is not only my work but was done with help of many great people (see references). Warm thanks (:-))

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

Captions

Add a one-line explanation of what this file represents

Items portrayed in this file

depicts

17 August 2008

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current16:12, 31 August 2008Thumbnail for version as of 16:12, 31 August 20081,000 × 1,000 (69 KB)Soul windsurfer{{Information |Description= |Source= |Date= |Author= |Permission= |other_versions= }}
16:44, 28 August 2008Thumbnail for version as of 16:44, 28 August 20081,000 × 1,000 (64 KB)Soul windsurfer{{Information |Description= |Source= |Date= |Author= |Permission= |other_versions= }}
20:03, 26 August 2008Thumbnail for version as of 20:03, 26 August 20081,000 × 1,000 (60 KB)Soul windsurfer{{Information |Description= boundaries of hyperbolic components of Mandelbrot set |Source= |Date= |Author= |Permission= |other_versions= }}
19:53, 17 August 2008Thumbnail for version as of 19:53, 17 August 20081,000 × 1,000 (59 KB)Soul windsurfer{{Information |Description={{en|1=Boundaries of hyperbolic components of Mandelbrot set }} |Source=Own work by uploader |Author=Adam majewski |Date=17.08.2008 |Permission= |other_versions= }} <!--{{ImageUpload|full}}-->

Global file usage

Metadata