This plot was created with Gnuplot by n.

/*  
this is batch file for Maxima  5.13.0
http://maxima.sourceforge.net/
tested in wxMaxima 0.7.1
using draw package ( interface to gnuplot ) to draw on the screen


draws  critical orbit = orbit of critical point
 

Adam Majewski fraktal.republika.pl
with help of 
Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates
*/


/*  */

c:%i*0.21687214+0.37496784;


/* define function ( map) for dynamical system z(n+1)=f(zn,c)  */
f(z,c):=expand (z*z + c); /* expand speed up  computations and fix the stack overflow problem. Robert Dodier */

/* maximal number of iterations */
iMax:100000; /* to big couses bind stack overflow */
EscapeRadius:10;

/* define z-plane ( dynamical ) */
zxMin:-0.8;
zxMax:0.2;
zyMin:-0.2;
zyMax:0.8;

/* resolution is proportional to number of details and time of drawing */
iXmax:1000;
iYmax:1000;


/* compute critical point */
zcr:rhs(solve(diff(f(z,c),z,1)));
/* save critical point to 2 lists */
xcr:makelist (realpart(zcr), i, 1, 1); /* list of re(z) */
ycr:makelist (imagpart(zcr), i, 1, 1); /* list of im(z) */	

/* compute attracting fixed point */
fixed:solve([z=f(z,c)], [z]);
zfa:if cabs(2*rhs(fixed[1]))<1 then rhs(fixed[1]) else rhs(fixed[2]);
/* check multipliers of fixed points */
float(cabs(2*rhs(fixed[1])));
float(cabs(2*rhs(fixed[2])));
/* save attracting fixed point to 2 lists */
xfa:makelist (realpart(zfa), i, 1, 1); /* list of re(z) */
yfa:makelist (imagpart(zfa), i, 1, 1); /* list of im(z) */	


/* ------------------- compute forward orbit of critical point in 6 subsets ----------*/
z:zcr; /* first point  */
for i:1 thru 6 step 1 do
block
(
  z:f(z,c),
  if i=1 then orbit1:[z]
	elseif i=2 then orbit2:[z]
	elseif i=3 then orbit3:[z]
	elseif i=4 then orbit4:[z]
	elseif i=5 then orbit5:[z]
	elseif i=6 then orbit6:[z]
  );

for i:6 thru iMax step 1 do
block
(
  z:f(z,c),
  
  if abs(z)>EscapeRadius 
	then exit(i) 
	else (
		j:mod(i,6),
		if j=0 then orbit1:endcons(z,orbit1)
		elseif j=1 then orbit2:endcons(z,orbit2)
		elseif j=2 then orbit3:endcons(z,orbit3)
		elseif j=3 then orbit4:endcons(z,orbit4)
		elseif j=4 then orbit5:endcons(z,orbit5)
		elseif j=5 then orbit6:endcons(z,orbit6)
		
		)
  );

z1:makelist(i*6,i,1,length(orbit1));
z2:makelist((i+1)*6,i,1,length(orbit2));
z3:makelist((i+2)*6,i,1,length(orbit3))$
z4:makelist((i+3)*6,i,1,length(orbit4))$
z5:makelist((i+4)*6,i,1,length(orbit5))$
z6:makelist((i+5)*6,i,1,length(orbit6))$


/* draw reversed orbit of beta  using draw package */
load(draw);
draw3d(
    title= concat(" dynamical plane for f(z,c):=z*z+",string(c)),
	user_preamble = "set size square;set key out;set key top right",
	file_name = "cr_6spiral_3d",
    terminal  = 'png,
	pic_width  = iXmax,
    pic_height = iYmax,
	/* yrange = [zyMin,zyMax],
    xrange = [zxMin,zxMax],
	*/
	xlabel     = "Z.re ",
    ylabel     = "Z.im",
    point_type    = filled_circle,
	points_joined = false,
	
	point_size    = 0.8,
	key = "critical point",
	color		  =magenta,
	points(xcr,ycr,[0]),
	
	point_size    = 0.2,
	
	color		  =red,
	key = "1-spiral of critical orbit",
	points(map (realpart,orbit1),map (imagpart,orbit1),z1),
	
	color		  =orange,
	key = "2-spiral of critical orbit",
	points(map (realpart,orbit2),map (imagpart,orbit2),z2),
	
	color		  =yellow,
	key = "3-spiral of critical orbit",
	points(map (realpart,orbit3),map (imagpart,orbit3),z3),
	
	color		  =green,
	key = "4-spiral of critical orbit",
	points(map (realpart,orbit4),map (imagpart,orbit4),z4),
	
	color		  =cyan,
	key = "5-spiral of critical orbit",
	points(map (realpart,orbit5),map (imagpart,orbit5),z5),
	
	color		  =blue,
	key = "6-spiral of critical orbit",
	points(map (realpart,orbit6),map (imagpart,orbit6),z6),
	
	point_size    = 0.8,
	key = "attracting fixed point",
	color		  =black,
	points(xfa,yfa,[iMax+1])
 );

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