File:Jung50e.png

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Original file(1,000 × 500 pixels, file size: 23 KB, MIME type: image/png)

Summary[edit]

Description
English: External rays and equipotential lines of Mandelbrot set as an images of external rays and circles of unit circle under Jungreis function
Date 4.01.2009
Source Own work with help of : Richard J. Fateman and G. A. Edgar
Author Adam majewski

Compare with[edit]

Maxima source code[edit]

 /*  batch file for maxima 
 uses :
 - symmetry around horizontal ( 0X ) axis
 - Psi_M function to map conjugate plane to parameter plane
 - jungreis algorithm to 
 time :3818
 */ 

 start:elapsed_run_time ();
 jMax:50; /* precision  = proportional to details and time of computations  */
 iMax:200; /* number of points to draw */
 iMaxBig:400;

 /* computes b coefficient of Jungreis function*/
 betaF[n,m]:=block
 (
 [nnn:2^(n+1)-1],
 if m=0
 then 1.0
 else if ((n>0) and (m < nnn)) 
  then 0.0
  else (betaF[n+1,m]- sum(betaF[n,k]*betaF[n,m-k],k,nnn,m-nnn)-betaF[0,m-nnn])/2.0
 )$
 b[m]:=betaF[0,m+1]$

 /* -------------------------------*/
 /* Power of w to j */
 wn[w,j]:= if j=0 then 1 else w*wn[w,j-1]$

 /* ---------Jungreis function ;  c = Psi_M(w) ----------------------------- */
 Psi_M(w):=w + sum(b[j]/wn[w,j],j,0,jMax)$

 /* exponential for of complex number with angle in turns */
 GiveCirclePoint(t):=R*%e^(%i*t*2*%pi)$ /* gives point of unit circle for angle t in turns */

 GiveWRayPoint(R):=R*%e^(%i*tRay*2*%pi)$ /* gives point of external ray for radius R and angle tRay in turns */

 NmbrOfCurves:9;

 /* coordinate of w-plane  not c-plane */
 R_max:1.5;
 R_min:1;
 dR:(R_max-R_min)/NmbrOfCurves; /* for circles */
 dRR:(R_max-R_min)/iMax; /* for rays */ 

 /* --------------------------------------f_0 plane = w-plane -----------------------------------------*/

 /*-------------- unit circle ------------*/
 R:1;
 circle_angles:makelist(i/iMax,i,0,iMax/2)$
 CirclePoints:map(GiveCirclePoint,circle_angles)$ 

 /* ---------------external circles ---------*/
 circle_radii: makelist(R_min+i*dR,i,1,NmbrOfCurves)$ 
 circle_angles:makelist(i/iMaxBig,i,0,iMaxBig/2)$
 WCirclesPoints:[]$
 for R in circle_radii do 
 WCirclesPoints:append(WCirclesPoints,map(GiveCirclePoint,circle_angles))$

 /* -------------- external w rays -------------*/
 ray_radii:makelist(R_min+dRR*i,i,1,iMax);
 ray_angles:[0,1/3,1/7 , 2/7 ,3/7  ]; /* list of angles < 1/2 of root points */
 WRaysPoints:[];
 for tRay in ray_angles do 
 WRaysPoints:append(WRaysPoints,map(GiveWRayPoint,ray_radii));

 /* -------------------------parameter plane = c plane -----------------------------------*/
 MPoints:map(Psi_M,CirclePoints); /* Mandelbrot set points */

 CRaysPoints:map(Psi_M,WRaysPoints); /* external z rays */

 Equipotentials:map(Psi_M,WCirclesPoints);

 /* add points below horizontal axis */
 for w in CirclePoints do CirclePoints:cons(conjugate(w),CirclePoints);
 for w in WRaysPoints do WRaysPoints:cons(conjugate(w),WRaysPoints);
 for w in WCirclesPoints do WCirclesPoints:cons(conjugate(w),WCirclesPoints);
 for c in MPoints do MPoints:cons(conjugate(c),MPoints);
 for c in CRaysPoints do CRaysPoints:cons(conjugate(c),CRaysPoints);
 for c in Equipotentials do Equipotentials:cons(conjugate(c),Equipotentials);

 /* time */
 stop:elapsed_run_time ();
 time:fix(stop-start); 

 /* ---------------- draw *--------------------------------------------------------------------------*/

 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(file_name = "jung50es_2",
  pic_width=1000, 
  pic_height= 500,
  terminal  = 'png,
  columns  = 2,
  gr2d(title = " unit circle with external rays & circles ",
  point_type = filled_circle,
  points_joined =true,
  point_size    =  0.34,
  color         = red,
  points(map(realpart, CirclePoints),map(imagpart, CirclePoints)),
  points_joined =false,
  color         = green,
  points(map(realpart, WCirclesPoints),map(imagpart, WCirclesPoints)),
  color         = black,
 points(map(realpart, WRaysPoints),map(imagpart, WRaysPoints))
 ),
 gr2d(title      = "Parameter plane : Image under Psi_M(w) ",
 points_joined =true,
 point_type = filled_circle,
 point_size    =0.34,
 color         = blue,
 points(map(realpart, MPoints),map(imagpart, MPoints)),
 points_joined =false,
 color         = green,
 points(map(realpart, Equipotentials),map(imagpart, Equipotentials)),
 color         = black,
 points(map(realpart, CRaysPoints),map(imagpart, CRaysPoints))
 ) 
 );

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File history

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Date/TimeThumbnailDimensionsUserComment
current19:10, 8 January 2009Thumbnail for version as of 19:10, 8 January 20091,000 × 500 (23 KB)Adam majewski{{Information |Description=Better version made by Robert Dodier using Maxima/Clisp and jMax=100 |Source= |Date= |Author= |Permission= |other_versions= }}
13:42, 4 January 2009Thumbnail for version as of 13:42, 4 January 20091,000 × 500 (23 KB)Adam majewski{{Information |Description={{en|1=External rays and equipotential lines of Mandelbrot set}} |Source=Own work |Author=Adam majewski |Date=4.01.2009 |Permission= |other_versions= }} <!--{{ImageUpload|full}}-->

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