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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "The Mandelbrot set " }}{PARA 0 "
" 0 "" {TEXT -1 42 "by Peter Stone, Nanaimo, B.C., Canada     " }}
{PARA 0 "" 0 "" {TEXT -1 17 "Version: 6.3.2007" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 5 "load " }{TEXT 0 11 "fractalplot" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "fractal.
m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 11 "f
ractalplot" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "
" {TEXT -1 122 "It can be read into a Maple session by a command simil
ar to the one that follows, where the file path gives its location. " 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "read \"K:\\\\MapleMath/pro
cdrs/fractal.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 28 "What is the Mandelbrot set? " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 537 "If a real number
 greater than one is squared, the resulting number is larger than the \+
original number. If you repeat the process of squaring over and over a
gain, a sequence of progressively larger numbers is produced. Even sta
rting with a number \"close\" to 1, like 1.0001, eventually very large
 numbers are obtained. On the other hand, if you start with a number b
etween zero and 1, and carry out the same process, eventually numbers \+
very close to 0 are obtained. This happens even if you start very clos
e to 1. \nNow take a fixed number " }{TEXT 263 1 "k" }{TEXT -1 90 ", a
nd consider the repeated (or iterated) process of squaring and adding \+
the fixed number " }{TEXT 264 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 291 136 
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46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "For example, if " }{TEXT 265 1 "k" }{TEXT -1 63 " is 0, t
his is the same process as before. Now vary the number " }{TEXT 266 1 
"k" }{TEXT -1 44 " but always start with the number 0. Taking " }
{XPPEDIT 18 0 "k = -2" "6#/%\"kG,$\"\"#!\"\"" }{TEXT -1 78 " we get th
e sequence -2, 2, 2, 2, . . .  which stays constant, whereas taking " 
}{XPPEDIT 18 0 "k = -2.1" "6#/%\"kG,$-%&FloatG6$\"#@!\"\"F*" }{TEXT 
-1 141 " gives the sequence -2.1, 2.31, 3.2361, 8.37234, 67.9961, 4621
.37 and then the values become huge; that is they \"escape to infinity
\".\nTaking " }{XPPEDIT 18 0 "k = -1" "6#/%\"kG,$\"\"\"!\"\"" }{TEXT 
-1 100 " gives the sequence -1, 0, -1, 0, -1, 0, -1, . .  which oscill
ates forever between -1 and 0. Taking " }{XPPEDIT 18 0 "k = -1.3" "6#/
%\"kG,$-%&FloatG6$\"#8!\"\"F*" }{TEXT -1 461 " gives the sequence -1.3
, 0.39,  -1.1479, 0.0176744,  -1.29969, 0.389188, -1.14853, 0.0191275,
 -1.29963, 0.389049,  -1.14864, 0.0193761, -1.29962, 0.389024, -1.1486
6,  0.0194205, -1.29962, 0.38902, -1.14866, . . . which now cycles for
ever around the last 4 numbers.  Thus some sequences escape to infinit
y and some are trapped forever.\nExtending this process out from the r
eal axis into the complex plane, the Mandelbrot set consists of all th
e complex numbers " }{TEXT 267 1 "k" }{TEXT -1 145 " which do not prod
uce an \"escape to infinity\" under this procedure. The number -1.3 si
ts near the centre of the left hand black circular region. " }}{PARA 
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0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 322 "For the points whic
h do escape, the usual practice is to provide colours to indicate the \+
number of iterations needed for the sequence to escape further than a \+
specified given fixed distance away from 0. If a number has not escape
d after a specified maximum number of iterations, the corresponding po
int is coloured black. " }}{PARA 0 "" 0 "" {TEXT -1 448 "It is not alw
ays easy to determine a suitable choice for the maximum number of iter
ations ahead of time. If it is set too low, points get coloured black \+
when further iterations would show that they really don't belong to th
e Mandelbrot set. On the other hand, if it is set to a high value, the
 constuction of the fractal may take a while. For initial quick explor
ations it's a good idea not to set the maximum number of iterations to
 a high value. " }}{PARA 0 "" 0 "" {TEXT -1 126 "The structure of the \+
Mandelbrot set and the region of escape points close to the boundary c
an be investigated by \"zooming in\"." }}{PARA 0 "" 0 "" {TEXT -1 59 "
The following picture has centre at the point -0.72 + 0.33 " }{TEXT 
268 1 "i" }{TEXT -1 33 ", and has a width of 0.175 units." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{BITMAP 240 
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t[YRB?CCAgFkyDSS^WfO1:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 282 "If you do some exploring, you will s
oon discover that these coloured regions have an is incredibly complic
ated fractal structure, that is, similar structures appear at all scal
es when zooming inwards. The structure also seems to become more intri
cate the further one zooms inwards.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "Algorithms
 for drawing regions of the Mandelbrot fractal " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Prel
iminary algorithm " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 178 "The following flow diagram indicates the
 computation used to determine the colour to be associated with the po
int in the complex plane corresponding to a point (complex number)  " 
}{TEXT 269 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 210 "The m
aximum number of iterations used to decide whether a point does not es
cape and therefore belongs to the Mandelbrot set is arbitrarily set at
 100, but this should be increased when finer detail is required." }}
{PARA 0 "" 0 "" {TEXT -1 97 "Similarly, the escape distance is set at \+
4, but it should not be necessary to change this value. " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{GLPLOT2D 293 470 470 {PLOTDATA 2 "6;-%'CURVES
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#[FirQ.or~~n~>~100~?F[sF[vFbs-Fdr6&7$$FduFir$\"#JFirQ$yesF[s-F56&F7F*$
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ermine~theF[sF_xFbs-Fdr6&7$F+F\\wQ-colour~for~kF[sF_xFbs-%+AXESLABELSG
6$Q!F[sF_y-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFgy" 1 2 0 1 10 0 
2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" 
"Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" }}
{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 123 "The following prelima
ry Maple procedure can be used to determine the colour to be associate
d with a point (complex number) " }{TEXT 270 1 "k" }{TEXT -1 2 ". " }}
{PARA 257 "" 0 "" {TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 231 "Mandelbrot := proc(x,y)\n   local z,n,k;\n   k := x+y*I;\n   \+
z := k;\n   n := 0;\n   do\n      z := z^2+k;\n      n := n+1;\n      \+
if abs(z)>4 or n>50 then break end if;   \n   end do;\n   if n<=50 the
n evalf(n/10) else 3.3 end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The colour function which can b
e used with the Maple procedure " }{TEXT 0 6 "plot3d" }{TEXT -1 51 " p
rovides a method of drawing Mandelbrot fractals. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot3d(0,-2
.1..1.1,-1.6..1.6,orientation=[-90,0], grid=[50,50],style=patchnogrid,
scaling=constrained,color=Mandelbrot);" }}{PARA 13 "" 1 "" {GLPLOT3D 
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^2;\n   ys := yy^2;\n   for k from 0 to _maxiter while xs+ys<16 do\n  \+
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i := floor(h);\n\011   \011\011f := h-i;\n\011   \011 p := v*(1-s);\n
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 elif i=2 then\n         A[1] := p; A[2] := v; A[3] := t;\n      elif \+
i=3 then\n         A[1] := p; A[2] := q; A[3] := v;\n      elif i=4 th
en\n         A[1] := t; A[2] := p; A[3] := v;\n\011\011\011   else # i
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The following sequence o
f \"zoomed\" pictures, are located near the \"neck\" of the \"upper pr
otuberance\"." }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 89 ": In
 order to choose a centre point for a new zoom you can click on the pl
ot and use the " }{TEXT 276 4 "Axes" }{TEXT -1 112 " menu to obtain wh
at would be boxed, framed or normal  axes if you viewed the image as a
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
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he \"upper protuberance\"." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "1st \+
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{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x = -0" "6#/%\"xG,$\"\"!!\"\"" }
{TEXT -1 10 ".1616 and " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG\"\"!" }
{TEXT -1 45 ".6519.  Width and height of picture:  0.012. " }}{PARA 0 
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qq^hxd`nyD>[cWw=GhEn]Qvgfv_pYa^qnPXhyGqc`nCfhSqvQ^\\LB>[r=Yr=YJB@kB`r`
\\jq`PoqLeMUTUX<qQwhqw\\QuLQtPQwhttpKi]K\\uNNeqGLohiWYtqOitsmtNUtOaWQY
TNiQcmV^EVAir<XUihXaDxhIMjqwh=XF\\vxLkEakjuwi=Xoajg\\TM]tgAXOqTjENjtKl
YuQYulqQw`arCr[id;WTorh_Iq=y@;fXkEX\\U`XqXlOs@XEDKB<kAtZGB=SbWWEFWsAiR
JuxpdWgmWvPPSYKhaWf]gIN[En__yobpn_hNE?\\tBuNeIWcMWCDMAHtguw]DQBQKiAORi
tNexjmtSIRjaTYiqwXQuTapku\\qf]v]3;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 92 "Determining the cycle length asociated with small disco
nnected regions of the Mandelbrot set" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "The point (complex \+
number) " }{XPPEDIT 18 0 "k=-0" "6#/%\"kG,$\"\"!!\"\"" }{TEXT -1 1 ".
" }{XPPEDIT 18 0 "16161869 + 0" "6#,&\")p=;;\"\"\"\"\"!F%" }{TEXT -1 
1 "." }{XPPEDIT 18 0 "65452044*i" "6#*&\")W?Xl\"\"\"%\"iGF%" }{TEXT 
-1 181 " lies inside the miniature disconnected region of the Mandelbr
ot set which appears in the 3rd zoomed picture in the previous section
. The cycle length associated with this value of " }{TEXT 273 1 "k" }
{TEXT -1 9 " is 103. " }}{PARA 0 "" 0 "" {TEXT -1 35 "This can be dete
rmined as follows. " }}{PARA 0 "" 0 "" {TEXT -1 52 "Firstly a large nu
mber of iterations of the formula " }{XPPEDIT 18 0 "z := z^2+k" "6#>%
\"zG,&*$F$\"\"#\"\"\"%\"kGF(" }{TEXT -1 70 " are performed in order to
 ensure convergence to the associated cycle." }}{PARA 0 "" 0 "" {TEXT 
-1 19 "The final value of " }{TEXT 272 1 "z" }{TEXT -1 50 " after thes
e iterations is recorded in a variable " }{TEXT 271 1 "w" }{TEXT -1 6 
", say." }}{PARA 0 "" 0 "" {TEXT -1 59 "Subsequent iterates z are then
 compared until a match with " }{TEXT 274 1 "w" }{TEXT -1 102 " is fou
nd. The number of iterations needed to obtain this match is the length
 of the associated cycle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 421 "kk := -0.16161869 + 0.65452044*I;
\nDigits := 14:\neps := Float(1,-Digits+2):\nzz := kk:\nn := 0:\n# ini
tial iterations to obtain convergence to the cycle\ndo\n   zz := zz^2+
kk;\n   n := n+1;\n   if n>500000 then break end if;\nend do:\nww := z
z;\nsmall := eps*abs(ww):\nn := 0:\n# iterations to determine the cycl
e length\ndo\n   zz := zz^2+kk;\n   n := n+1;\n   if abs(ww-zz)<small \+
or n>500 then break end if;\nend do:\nn;\nzz;\nDigits := 10:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#kkG^$$!)p=;;!\")$\")W?XlF(" }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#>%#wwG^$$!/V=g8A3I!#9$\"/^(oABWk(
F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$.\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#^$$!/V=g8A3I!#9$\"/^(oABWk(F&" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 48 "For another example, the point (complex number) " }
{XPPEDIT 18 0 "k= -0" "6#/%\"kG,$\"\"!!\"\"" }{TEXT -1 1 "." }
{XPPEDIT 18 0 "161617140232233 + 0" "6#,&\"0LABSrhh\"\"\"\"\"\"!F%" }
{TEXT -1 1 "." }{XPPEDIT 18 0 "65451813597249*i" "6#*&\"/\\sf8=Xl\"\"
\"%\"iGF%" }{TEXT -1 145 "  is the centre of the last zoomed fractal i
n the previous section and lies inside another miniature disconnected \+
region of the Mandelbrot set.  " }}{PARA 0 "" 0 "" {TEXT -1 36 "The as
sociated cycle length is 406. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "kk := -0.161617140232233 + \+
0.65451813597249*I;\nDigits := 14:\neps := Float(1,-Digits+3):\nzz := \+
kk:\nn := 0:\n# initial iterations to obtain convergence to the cycle
\ndo\n   zz := zz^2+kk;\n   n := n+1;\n   if n>500000 then break end i
f;\nend do:\nww := zz;\nsmall := eps*abs(ww):\nn := 0:\n# iterations t
o determine the cycle length\ndo\n   zz := zz^2+kk;\n   n := n+1;\n   \+
if abs(ww-zz)<small or n>500 then break end if;\nend do:\nn;\nzz;\nDig
its := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#kkG^$$!0LABSrhh\"!#:$
\"/\\sf8=Xl!#9" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#>%#ww
G^$$!/EcJq:]@!#9$\"/5N[@GUjF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$1%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!/$*fJq:]@!#9$\"/?P[@GUjF&" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The calcu
lation can be speeded up by using hardware floating point arithmetic. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 642 "kk := -0.161617140232233 + 0.65451813597249*I;\nDigits := 14:
\neps := Float(1,-Digits+3):\nxx := Re(kk): yy := Im(kk):\nuu := xx: v
v := yy:\n# initial iterations to obtain convergence to the cycle\nfor
 n to 500001 do\n   prev_uu := uu;\n   uu := evalhf(uu^2-vv^2+xx);\n  \+
 vv := evalhf(2*prev_uu*vv+yy);\nend do:\nsaved_uu := uu:\nsaved_vv :=
 vv:\nuu + vv*I;\nsmall := evalhf(eps*sqrt(uu^2+vv^2)):\nn := 0:\n# it
erations to determine the cycle length\ndo\n   prev_uu := uu;\n   uu :
= evalhf(uu^2-vv^2+xx);\n   vv := evalhf(2*prev_uu*vv+yy);\n   n := n+
1;\n   if abs(saved_uu-uu)+abs(saved_vv-vv)<small or n>800 then break \+
end if;\nend do:\nn;\nuu + vv*I;\nDigits := 10:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#kkG^$$!0LABSrhh\"!#:$\"/\\sf8=Xl!#9" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#^$$!3[3%o:.d,:#!#=$\"3@SyP[@GUjF&" }}{PARA 11 "" 
1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#\"$1%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#^$$!3[3%o:.d,:#!#=$\"3@SyP[@GUjF&" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "
" {TEXT -1 33 "Code for drawing 1st flow diagram" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "p1 := plot(
[[[-4.5,2],[-8,2],[-8,0],[8,0],[8,2],[4.5,2]],\n   [[-4.5,1.2],[-4.5,2
.8],[4.5,2.8],[4.5,1.2],[-4.5,1.2]],\n   [[.5,-.3],[-.5,0],[.5,.3]]],
\n   color=COLOR(RGB,0,.6,0),thickness=[2,2,3]):\nt1 := plots[textplot
]([0,2.05,`square and add k`],\n   font=[HELVETICA,BOLD,12],color=COLO
R(RGB,.4,0,.9)):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "Code for drawing 2nd flow diagram
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1173 "p1 := plot([[[-2,0],[-2,2],[2,2],[2,0],[-2,0]],\n   [[0,3.8]
,[-3,5],[0,6.2],[3,5],[0,3.8]],\n   [[-2,7.7],[-2,9.7],[2,9.7],[2,7.7]
,[-2,7.7]],\n   [[-1.5,12],[-1.5,13.5],[1.5,13.5],[1.5,12],[-1.5,12]],
\n   [[0,12],[0,9.7]],[[0,7.7],[0,6.2]],[[0,3.8],[0,2]],\n   [[-3,5],[
-5,5],[-5,10.75],[0,10.75]],\n   [[-5.2,7.9],[-5,8.2],[-4.8,7.9]],\n  \+
 [[-.2,7.1],[0,6.8],[.2,7.1]]],\n   color=COLOR(RGB,0,0,0),thickness=[
2,2,2]):\nt1 := plots[textplot]([[0,13.1,`z := k`],[0,12.6,`n := 0`],
\n    [0,9.1,`z := z  + k`],[0,8.4,`n := n + 1`]],\n   font=[HELVETICA
,BOLD,10],color=COLOR(RGB,.4,0,.9)):\nt2 := plots[textplot]([.22,9.28,
`2`],\n   font=[HELVETICA,BOLD,8],color=COLOR(RGB,.4,0,.9)):\nt3 := pl
ots[textplot]([[0,5.4,`Is |z| > 4`],[0,4.8,`or  n > 100 ?`]],\n   font
=[HELVETICA,BOLD,10],color=COLOR(RGB,0,0,.4)):\nt4 := plots[textplot](
[.8,3.1,`yes`],\n   font=[HELVETICA,BOLD,10],color=COLOR(RGB,0,.6,0)):
\nt5 := plots[textplot]([-3.7,5.5,`no`],\n   font=[HELVETICA,BOLD,10],
color=COLOR(RGB,1,0,0)):\nt6 := plots[textplot]([[0,1.6,`Use n to`],\n
   [0,1.1,`determine the`],[0,.6,`colour for k`]],\n   font=[HELVETICA
,BOLD,10],color=COLOR(RGB,.6,.1,0)):\nplots[display]([p1,t1,t2,t3,t4,t
5,t6],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 37 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }