# File:Jung50e.png

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

## Summary

 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

## 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 */
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_angles:makelist(i/iMaxBig,i,0,iMaxBig/2)\$
WCirclesPoints:[]\$
WCirclesPoints:append(WCirclesPoints,map(GiveCirclePoint,circle_angles))\$

/* -------------- external w rays -------------*/
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

/* -------------------------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))
)
);

```

## Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
 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 attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
 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.

## File history

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

Date/TimeThumbnailDimensionsUserComment
current19:10, 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 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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