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Description
English: Basilica Julia set with spine
Date
Source Own work
Author Adam majewski
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Maxima CAS src code

 /*  
batch file for Maxima CAS

maxima
batch("b.mac")



 */ 

start:elapsed_run_time ();

kill(all);
remvalue(all);

 /* --------------------------definitions of functions ------------------------------*/
 f(z,c):=z*z+c; /* Complex quadratic map */
 finverseplus(z,c):=sqrt(z-c)$
 finverseminus(z,c):=-sqrt(z-c)$ 

/* */
fn(p, z, c) :=
  if p=0 then z
  elseif p=1 then f(z,c)
  else f(fn(p-1, z, c),c)$

/*Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors */
F(p, z, c) := fn(p, z, c) - z $

/* Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors*/
G[p,z,c]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(F(p,z,c)),
for i in f do 
 t:t*G[i,z,c],
g: F(p,z,c)/t,
return(ratsimp(g))
)$

GiveRoots(g):=
 block(
 [cc],
 cc:bfallroots(expand(%i*g)=0),
 cc:map(rhs,cc),/* remove string "c=" */
 cc:map('float,cc),
 return(cc)
  )$ 




/* endcons the complex point to list in the format for draw package */ 
endconsD(point,list):=endcons([realpart(point),imagpart(point)],list)$
consD(point,list):=cons([realpart(point),imagpart(point)],list)$

GiveForwardOrbit(z0,c,iMax):=
   /* 
   computes (without escape test)
   forward orbit of point z0
   and saves it to the list for draw package */
block(
 [z,orbit,temp],
 z:z0, /* first point = critical point z:0+0*%i */
 orbit:[[realpart(z),imagpart(z)]], 
 for i:1 thru iMax step 1 do
        ( z:expand(f(z,c)),
          orbit:endcons([realpart(z),imagpart(z)],orbit)),
         
 return(orbit) 
)$



 /* Gives points of backward orbit of z=repellor       */
 GiveBackwardOrbit(c,repellor,zxMin,zxMax,zyMin,zyMax,iXmax,iYmax):=
  block(
   hit_limit:4, /* proportional to number of details and time of drawing */
   PixelWidth:(zxMax-zxMin)/iXmax,
   PixelHeight:(zyMax-zyMin)/iYmax,
   /* 2D array of hits pixels . Hit > 0 means that point was in orbit */
   array(Hits,fixnum,iXmax,iYmax), /* no hits for beginning */
  /* choose repeller z=repellor as a starting point */
  stack:[repellor], /*save repellor in stack */
  /* save first point to list of pixels  */ 
  x_y:[repellor], 
 /* reversed iteration of repellor */
  loop,
  /* pop = take one point from the stack */
  z:last(stack),
  stack:delete(z,stack),
  /*inverse iteration - first preimage (root) */
  z:finverseplus(z,c),
  /* translate from world to screen coordinate */
  iX:fix((realpart(z)-zxMin)/PixelWidth),
  iY:fix((imagpart(z)-zyMin)/PixelHeight),
  hit:Hits[iX,iY],
  if hit<hit_limit   
   then 
    (
    Hits[iX,iY]:hit+1,
    stack:endcons(z,stack), /* push = add z at the end of list stack */
    if hit=0 then x_y:endcons( z,x_y)
    ),
  /*inverse iteration - second preimage (root) */
  z:-z,
 /* translate from world to screen coordinate, coversion to integer */
  iX:fix((realpart(z)-zxMin)/PixelWidth),
  iY:fix((imagpart(z)-zyMin)/PixelHeight),
  hit:Hits[iX,iY],
  if hit<hit_limit   
   then 
    (
     Hits[iX,iY]:hit+1,
     stack:endcons(z,stack), /* push = add z at the end of list stack to continue iteration */
     if hit=0 then x_y:endcons( z,x_y)
    ),
   if is(not emptyp(stack)) then go(loop), 
 return(x_y) /* list of pixels in the form [z1,z2] */
 )$

 
 
 /*-----------------------------------*/ 
 Psi_n(r,t,z_last, Max_R):=
 /*   */
 block(
  [iMax:200,
  iMax2:0],
  /* -----  forward iteration of 2 points : z_last and w --------------*/
  array(forward,iMax-1), /* forward orbit of z_last for comparison */
  forward[0]:z_last,
  i:0,
  while cabs(forward[i])<Max_R  and  i< ( iMax-2) do
  (     
  /* forward iteration of z in fc plane & save it to forward array */
  forward[i+1]:forward[i]*forward[i] + c, /* z*z+c */
  /* forward iteration of w in f0 plane :  w(n+1):=wn^2 */
  r:r*2, /* square radius = R^2=2^(2*r) because R=2^r */
  t:mod(2*t,1),
  /* */
  iMax2:iMax2+1,
  i:i+1
  ),
  /* compute last w point ; it is equal to z-point */
  R:2^r,
  /* w:R*exp(2*%pi*%i*t),       z:w, */
  array(backward,iMax-1),
  backward[iMax2]:rectform(ev(R*exp(2*%pi*%i*t))), /* use last w as a starting point for backward iteration to new z */
  /* -----  backward iteration point  z=w in fc plane --------------*/
  for i:iMax2 step -1 thru 1 do
  (
  temp:float(rectform(sqrt(backward[i]-c))), /* sqrt(z-c) */
  scalar_product:realpart(temp)*realpart(forward[i-1])+imagpart(temp)*imagpart(forward[i-1]),
  if (0>scalar_product) then temp:-temp, /* choose preimage */
  backward[i-1]:temp
  ),
  return(backward[0])
 )$
 
 
 GiveRay(t,c):=
 block(
  [r],
  /* range for drawing  R=2^r ; as r tends to 0 R tends to 1 */
  rMin:1E-5, /* 1E-4;  rMin > 0  ; if rMin=0 then program has infinity loop !!!!! */
  rMax:2, 
  caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */
  /* upper limit for iteration */
  R_max:300,
  /* */
  zz:[], /* array for z points of ray in fc plane */
  /*  some w-points of external ray in f0 plane  */
  r:rMax,
  while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */
  R:2^r,
  w:rectform(ev(R*exp(2*%pi*%i*t))),
  z:w, /* near infinity z=w */
  zz:cons(z,zz),
  unless r<rMin do
  (     /* new smaller R */
  r:r*caution,  
  R:2^r,
  /* */
  w:rectform(ev(R*exp(2*%pi*%i*t))),
  /* */
  last_z:z,
  z:Psi_n(r,t,last_z,R_max), /* z=Psi_n(w) */
  zz:cons(z,zz)
  ),
  return(zz)
 )$





  


/* 
converts complex number z = x*y*%i 
to the list in a draw format:  
[x,y] 
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$

ToPoints(myList):= points(map(d,myList))$


/* give Draw List from one point*/
ToPoint(z):=points([d(z)])$


GiveSpine(zc, Alfa,c, iMax):=block(
  	[s, center, cut], 
  	
  	/* first center of component, only one !!! */
  	center: zc, 
  	s:[center],
  	
  	/* first pair of cut points  */
  	s:   cons(  Alfa,s),
  	cut : -Alfa, 
  	s:endcons( cut,s),
  	
  	for i: 1  thru iMax step 1 do (
  	
  	/* pair of component's centers  */
  	center: finverseminus(center,c), 
  	s: cons(center, s),
  	center : - center,
  	s: endcons (center, s),
  	
  	/* pair of cut points  */
  	cut : finverseminus(cut,c),  
  	s:   cons(  cut,s),
  	cut : - cut,
  	s:endcons( cut,s)
  	
  	),
  	
  	
  	
  	
  	
  	/* convert to draw format and return list */  
  	s: ToPoints(s) 
  
  
  )$
  
  
  





compile(all)$

 /* ----------------------- main ----------------------------------------------------*/


period:3$

  

 /* external angle in turns */
 /* resolution is proportional to number of details and time of drawing */
 iX_max:1000$
 iY_max:1000$
 /* define z-plane ( dynamical ) */
 ZxMin:-2.0$
 ZxMax:2.0$
 ZyMin:-2.0$
 ZyMax:2.0$

 

/* give c a value */
 c:-1$ /* center of period 2 component */

 

 /* compute fixed points */
 Beta:float(rectform((1+sqrt(1-4*c))/2))$ /* compute repelling fixed point beta */
 alfa:float(rectform((1-sqrt(1-4*c))/2))$ /* other fixed point */

 /* compute backward orbit of repelling fixed point */
 xy: GiveBackwardOrbit(c,Beta,ZxMin,ZxMax,ZyMin,ZyMax,iX_max,iY_max)$ 


  /* compute ray points & save to zz list */
 eRayZero:GiveRay(0/1,c)$
 
 eRay1o2:GiveRay(1/2, c)$
   

 spine: GiveSpine(0, alfa,c, 4)$

 /* time of computations */
 time:fix(elapsed_run_time ()-start)$

 /* draw it using draw package by */
 
 
 
 load(draw)$ 

 path:"~/maxima/batch/julia/spine/basilica/"$ /*  if empty then file is in a home dir */

 /* if graphic  file is empty (= 0 bytes) then run draw2d command again */
 
 draw2d(
  terminal  = 'svg,
  file_name = sconcat(path,"BasilicaSpine"),
  user_preamble="set size square;set key top right",
  title= concat("Dynamical plane for fc(z)=z*z",string(c)),
  dimensions = [iX_max, iY_max],
  yrange = [ZyMin,ZyMax],
  xrange = [ZxMin,ZyMax],
  xlabel     = "Z.re ",
  ylabel     = "Z.im",
  point_type = filled_circle,
  points_joined =true,
  point_size    =  0.2,
  color         = red,
    
  
  
  points_joined =false,
  color         = black,
  key = "backward orbit of z=beta",
  ToPoints(xy),
  
  
  
 
  points_joined =true,
  point_size    =  0.2,
  color         = red,
  key = "external ray 0",
  ToPoints(eRayZero),
  
  
  key = "external ray 1/2",
  color = magenta,
  ToPoints(eRay1o2),
  
  
  
  
  points_joined =true,
  point_size    =  0.8,
  color         = gray,
  key = "spine",
  spine, 
  
  
  

  points_joined =false,
  
  color         = black,
  point_size    =  1.4,
  key = "critical point z = 0.0",
  ToPoint(0.0),
  
  
  color         = red,
  point_size    =  1.4,
  key = "repelling fixed point z= beta",
  ToPoint(Beta),
  
  color = magenta,
  key = "minus beta",
  ToPoint(-Beta),
  
  
  color         = yellow,
  key = "parabolic fixed point z= alfa",
  ToPoint(alfa)
 
 )$

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 4.0 International 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.

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current07:56, 25 October 2020Thumbnail for version as of 07:56, 25 October 20201,000 × 1,000 (775 KB)Soul windsurferchange attracting to parabolic
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