From Wikibooks, open books for an open world
Jump to navigation Jump to search

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


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

Maxima source code

 /*  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 */

 /* computes b coefficient of Jungreis function*/
 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

 /* -------------------------------*/
 /* 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 */


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

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

 /*-------------- unit circle ------------*/

 /* ---------------external circles ---------*/
 circle_radii: makelist(R_min+i*dR,i,1,NmbrOfCurves)$ 
 for R in circle_radii do 

 /* -------------- external w rays -------------*/
 ray_angles:[0,1/3,1/7 , 2/7 ,3/7  ]; /* list of angles < 1/2 of root points */
 for tRay in ray_angles do 

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

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


 /* 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 ();

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

 load(draw); /* Mario Rodríguez Riotorto */
 draw(file_name = "jung50es_2",
  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))


I, the copyright holder of this work, hereby publish it under the following licenses:
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.

GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
You may select the license of your choice.

File history

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

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}}-->

Global file usage

The following other wikis use this file: