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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 10 "L-syst\217mes" }}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Felice Rong
a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 24 "Ve
rsion du 12 avril 2002" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 12 "Intro
duction" }}{PARA 0 "" 0 "" {TEXT -1 348 "Le formalisme des L-systemes \+
est du a Aristid Lindenmayer (1968), dont le but premier etait de mode
liser la croissance des plantes. Il s'est revele par la suite que les \+
L-systemes permettait d'engendrer une large classe d'objets fractals, \+
c'est-a-dire de figures (dans le plan ou dans l'espace) dont toute par
tie a une structure semblable au tout. " }}{PARA 0 "" 0 "" {TEXT -1 
246 "Nous proposons ici un algorithme issu des L-systemes, qui convien
t pour engendrer des figures fractales dans le plan, puis une variante
 qui convient pour modeliser la croissance des plantes (qui seront aus
si representees par des figures planes). " }}{PARA 0 "" 0 "" {TEXT -1 
239 "Le mechanisme de base consiste a se donner des regles pour engend
rer une chaine de caracteres (ou mot) comportant un certain nombre de \+
symboles de base (alphabet), puis une interpretation des mots de facon
s a engendrer une image graphique." }}{PARA 0 "" 0 "" {TEXT -1 30 "Plu
s precisement on se donne :" }}{PARA 0 "" 0 "" {TEXT -1 35 "   1) un e
nsemble fini de symboles " }{TEXT 258 1 "V" }{TEXT -1 25 " (alphabet);
 par exemple " }{TEXT 259 9 "V=\{f,p,m\}" }}{PARA 0 "" 0 "" {TEXT -1 
90 "   2) un mot de depart, appele axiome ou \"seed\"(=graine, semence
 en anglais); par exemple " }{TEXT 260 13 "f p f p f p f" }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 44 "   3) des regles de reecriture; par
 exemple " }{TEXT 261 32 "f -> f p f m f m f f p f p f m f" }}{PARA 0 
"" 0 "" {TEXT -1 142 "   4) le nombre d'etapes, chaque etape consistan
t a appliquer la ou les regles de reecriture, en commencant par le mot
 de depart donne sous 2)" }}{PARA 0 "" 0 "" {TEXT -1 170 "   5) l'inte
rpretation graphique d'un mot. Generalement, cela se fait sous forme d
'ordres a donner a une tortue qui se deplace sur une feuille de dessin
 ou sur un ecran. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Par exemple : " }}
{PARA 0 "" 0 "" {TEXT 262 44 "f => avancer d'une distance fixee en tra
cant" }{TEXT -1 136 " (cette distance dependra du nombre d'etapes afin
 de ne pas sortir de la feuille de dessin - cela se fait automatiqueme
nt avec Maple).  " }}{PARA 0 "" 0 "" {TEXT 263 45 "p => rotation d'un \+
angle alpha donne d'avance" }{TEXT -1 73 " (p pour plus, le symbole + \+
pouvant etre problematique dans un programme)" }}{PARA 0 "" 0 "" 
{TEXT 264 31 "m => rotation d'un angle -alpha" }{TEXT -1 16 " (m pour \+
moins)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
172 "L'image est engendree en transformant un certain nombre de fois l
e mot de depart selon les regles de reecriture, puis a faire interpret
er par la tortue le mot ainsi obtenu." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 132 "References: \"Lindenmayer Systems\",
 par P. Prusinkiewicz et J. Hanan, springer Verlag, Lecture Notes in B
iomathematics no. 79 (1989)." }}{PARA 0 "" 0 "" {TEXT -1 163 "Le code \+
Maple a ete tire en partie de : \"Maple, a comprehensive introduction
\", par Roy Nicolaides et Noel Walkington, Cambridge University Press,
 Cambridge (1996)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 266 10 "Exercice 1" }{TEXT -1 62 ". Reprendre l'exemple donne ci
-dessus en cours d'explication :" }}{PARA 0 "" 0 "" {TEXT -1 8 "axiome
: " }{TEXT 265 13 "f p f p f p f" }{TEXT -1 13 "     regle : " }{TEXT 
267 58 "f -> f p f m f m f f p f p f m f,   alpha=Pi/2 (90 degres)" }}
{PARA 0 "" 0 "" {TEXT -1 188 "Ecrire le mot obtenu en appliquant la re
gle 1, voire 2 fois. Tracer sur une feuille de papier le dessin corres
pondant (on conseille de faire d'abord un dessin pour la regle de reec
riture)." }}{PARA 0 "" 0 "" {TEXT -1 46 "Quelle est la longueur du mot
 apres k etapes ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 
"" 0 "" {TEXT 268 20 "I. Figures fractales" }}{PARA 0 "" 0 "" {TEXT 
-1 45 "On prend l'alphabet de l'exemple ci-dessus : " }{TEXT 280 9 "V=
\{f,p,m\}" }{TEXT -1 179 ". Un peu plus loin dans ce paragraphe on uti
lisera encore 2 lettres, x et y, qui ne seront pas interpretees par la
 tortue, mais serviront uniquement pour les regles de reecriture." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 241 "Voici un
e procedure qui prend comme parametres le mot de depart, la regle de s
ubstitution, le nombre d'etapes et l'angle alpha et qui retourne le de
ssin correspondant. Notez que si le mot comporte des lettres qui ne so
nt pas dans l'alphabet " }{TEXT 281 1 "V" }{TEXT -1 38 ", elles sont t
out simplement ignorees." }}{PARA 0 "" 0 "" {TEXT -1 12 "La variable \+
" }{TEXT 269 3 "mot" }{TEXT -1 126 " est une liste; elle represente le
 mot, qui est l'axiome,  dans lequel on opere les substitutions prescr
ites par la procedure " }{TEXT 270 5 "regle" }{TEXT -1 27 " un nombre \+
d'etapes egal a " }{TEXT 271 6 "etapes" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 44 "Efforcez-vous de comprendre cette procedure." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "dessine:=proc(axiome::list,
regle::procedure,etapes::integer,alpha::realcons)\nlocal i,mot,pts,tt,
xx,yy;\nmot:=axiome;\nfor i to etapes do\nmot:=map(regle,mot)\nod;\ntt
:=0;\nxx:=0;\nyy:=0;\npts:=[xx,yy];\nfor i in mot do\nif i=p then tt:=
tt+alpha;\n elif i=m then tt:=tt-alpha;\n  elif i=f then \n   xx:=xx+c
os(tt);\n   yy:=yy+sin(tt);\n   pts:=pts,[xx,yy]\nfi;\nod;\nplot([pts]
,axes=none,scaling=CONSTRAINED)\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%(dessineGf*6&'%'axiomeG%%listG'%&regleG%*procedureG'%'etapesG%(in
tegerG'%&alphaG%)realconsG6(%\"iG%$motG%$ptsG%#ttG%#xxG%#yyG6\"F:C*>8%
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-%$cosG6#FJFA>FO,&FOFA-%$sinGF`oFA>FQ6$FQFR-%%plotG6%7#FQ/%%axesG%%non
eG/%(scalingG%,CONSTRAINEDGF:F:F:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "regle1:=a->subs(f=(f,p,f,m,f,f,p,f,p,f,m,f),a);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'regle1Gf*6#%\"aG6\"6$%)operatorG%&a
rrowGF(-%%subsG6$/%\"fG6.F0%\"pGF0%\"mGF0F0F2F0F2F0F3F09$F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Par exemple :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 39 "dessine([f,p,f,p,f,p,f],regle1,1,Pi/2);" }}
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>F'-%'COLOURG6&%$RGBG$\"#5F<F(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,C
ONSTRAINEDG-%+AXESLABELSG6$Q!6\"FP-%%VIEWG6$%(DEFAULTGFU" 1 2 0 1 10 
0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dessine([f,p,f,p,f,p,f],regle1,2,Pi
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\"&F)F]o7$F\\p$!\"*F)7$F\\p$!#5F)7$$\"\"'F)Fbp7$FepF_p7$$\"\"(F)F_p7$$
\"\")F)F_p7$F\\qF]o7$$\"\"*F)F]o7$$\"#5F)F]o7$FcqFjn7$F`qFjn7$F`qFR7$F
\\qFR7$F\\qFjn7$FipFjn7$FepFjn7$FepF]o7$FipF]oFhpF[qF^qF_qFbqFeqFfqFgq
7$FcqFR7$FcqFO7$$\"#6F)FO7$$\"#7F)FO7$FdrFK7$FarFK7$FarFE7$FcqFE7$FcqF
K7$F`qFK7$F\\qFK7$F\\qFO7$F`qFOFgqF^rF_rF`rFcrFfrFgrFhr7$FdrFE7$FdrFB7
$$\"#8F)FB7$$\"#9F)FB7$FesF;7$FbsF;7$FbsF(7$FbsF+7$FdrF+7$FdrF/7$FdrF2
7$FarF27$FarF/7$FcqF/7$FcqF+7$FarF+7$FarF(7$FarF;7$FdrF;7$FdrF(FisFjsF
[tF\\tF]tF^tF_tF`t7$FcqF27$F`qF27$F`qFbo7$F`qF\\p7$F\\qF\\p7$F\\qFbo7$
FipFbo7$FipF27$F\\qF27$F\\qF/7$F\\qF+7$F`qF+7$F`qF/F`tFgtFhtFitFjtF[uF
\\uF]u7$FipF\\p7$FepF\\p7$FepFep7$FepFip7$F\\pFip7$F\\pFep7$FboFep7$F2
Fep7$F2F\\p7$F/F\\p7$F+F\\p7$F+Fbo7$F/FboF67$F2F27$F2Fbo7$FboFbo7$F\\p
Fbo7$F\\pF\\p7$FboF\\pFjuF[vF\\vF]vF^vF_vF`vF6F7F8F9F:F=F>F'-%'COLOURG
6&%$RGBG$FdqF<F(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%+
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 49 "Encore un exemple relativement simple a mediter :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "regle0:=a->subs(f=(f,f,p,f,p
,f,p,f,p,f,f),a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'regle0Gf*6#%\"
aG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"fG6-F0F0%\"pGF0F2F0F2F0F2F0F
09$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dessine([f,p,f
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0 "" {TEXT 273 10 "Exercice 2" }{TEXT -1 62 ". Dessinez l'exemple de l
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{TEXT -1 11 " ci dessus." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 124 "La courbe de Von Koch est obtenue par la
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, comme suit :" }}}{EXCHG {PARA 0 "" 0 "" {GLPLOT2D 220 153 153 
{PLOTDATA 2 "6%-%'CURVESG6$7$7$\"\"!F(7$$\"\"\"F(F(-%'COLOURG6&%$RGBG$
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{GLPLOT2D 207 185 185 {PLOTDATA 2 "6%-%'CURVESG6$7'7$\"\"!F(7$$\"\"\"F
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}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "puis on reco
mmence avec chacun des 4 nouveaux segments. Cela donne apres 3 iterati
ons :" }}}{EXCHG {PARA 0 "" 0 "" {GLPLOT2D 246 246 246 {PLOTDATA 2 "6%
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{TEXT 274 10 "Exercice 3" }{TEXT -1 70 ". Trouver un axiome et une reg
le qui decrivent la courbe de Von Koch. " }}{PARA 0 "" 0 "" {TEXT -1 
55 "Meme question pour la courbe de type Peano ci-dessous :" }}}
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "L'exemple suivant fait interven
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g[lFjo7$Fc[lFjo7$Fc[lFfo7$F_[lFfo7$F_[lFbo7$F[[lFbo7$F[[lFfo7$F_zFfo7$
F_zFjo7$F[zFjo7$F[zFfo7$F(Ffo7$F(Fbo7$F[zFbo7$F[zF^o7$F_zF^o7$F_zFjn7$
F[zFjn7$F[zFT7$F(FT7$F(FP7$F[zFP7$F[zFK7$F_zFK7$F_zFP7$F[[lFP7$F[[lFT7
$F_[lFT7$F_[lFP7$Fc[lFP7$Fc[lFK7$Fg[lFK7$Fg[lF@7$Fc[lF@7$Fc[lF<7$F_[lF
<7$F_[lF87$F[[lF87$F[[lF<7$F_zF<7$F_zF@7$F[zF@7$F[zF<7$F(F<7$F(F87$F[z
F87$F[zF37$F_zF37$F_zF/7$F[zF/7$F[zF+7$F(F+F'-%'COLOURG6&%$RGBG$F[oF\\
zF(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$
Q!6\"F]gl-%%VIEWG6$%(DEFAULTGFbgl" 1 2 0 1 10 0 2 9 1 1 1 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
130 "L'exemple suivant est la variante de Hilbert de la courbe de Pean
o. Il utilise l'alphabet \{f,p,m,x,y\} et 2 regles de reecritures :" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "regle3:=a->subs(\{x=(m,y,f
,p,x,f,x,p,f,y,m),y=(p,x,f,m,y,f,y,m,f,x,p)\},a);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'regle3Gf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%%subsG6$
<$/%\"xG6-%\"mG%\"yG%\"fG%\"pGF1F5F1F6F5F4F3/F46-F6F1F5F3F4F5F4F3F5F1F
69$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dessine([y],re
gle3,4,Pi/2);" }}{PARA 13 "" 1 "" {GLPLOT2D 256 256 256 {PLOTDATA 2 "6
'-%'CURVESG6$7\\[l7$$\"\"!F)F(7$$\"\"\"F)F(7$F+F+7$F(F+7$F($\"\"#F)7$F
($\"\"$F)7$F+F37$F+F07$F0F07$F0F37$F3F37$F3F07$F3F+7$F0F+7$F0F(7$F3F(7
$$\"\"%F)F(7$F@F+7$$\"\"&F)F+7$FDF(7$$\"\"'F)F(7$$\"\"(F)F(7$FKF+7$FHF
+7$FHF07$FKF07$FKF37$FHF37$FDF37$FDF07$F@F07$F@F37$F@F@7$F@FD7$FDFD7$F
DF@7$FHF@7$FKF@7$FKFD7$FHFD7$FHFH7$FKFH7$FKFK7$FHFK7$FDFK7$FDFH7$F@FH7
$F@FK7$F3FK7$F0FK7$F0FH7$F3FH7$F3FD7$F3F@7$F0F@7$F0FD7$F+FD7$F+F@7$F(F
@7$F(FD7$F(FH7$F+FH7$F+FK7$F(FK7$F($\"\")F)7$F($\"\"*F)7$F+Fep7$F+Fbp7
$F0Fbp7$F3Fbp7$F3Fep7$F0Fep7$F0$\"#5F)7$F3F^q7$F3$\"#6F)7$F0Fbq7$F+Fbq
7$F+F^q7$F(F^q7$F(Fbq7$F($\"#7F)7$F+Fjq7$F+$\"#8F)7$F(F^r7$F($\"#9F)7$
F($\"#:F)7$F+Fer7$F+Fbr7$F0Fbr7$F0Fer7$F3Fer7$F3Fbr7$F3F^r7$F0F^r7$F0F
jq7$F3Fjq7$F@Fjq7$FDFjq7$FDF^r7$F@F^r7$F@Fbr7$F@Fer7$FDFer7$FDFbr7$FHF
br7$FHFer7$FKFer7$FKFbr7$FKF^r7$FHF^r7$FHFjq7$FKFjq7$FKFbq7$FKF^q7$FHF
^q7$FHFbq7$FDFbq7$F@Fbq7$F@F^q7$FDF^q7$FDFep7$F@Fep7$F@Fbp7$FDFbp7$FHF
bp7$FHFep7$FKFep7$FKFbp7$FbpFbp7$FbpFep7$FepFep7$FepFbp7$F^qFbp7$FbqFb
p7$FbqFep7$F^qFep7$F^qF^q7$FbqF^q7$FbqFbq7$F^qFbq7$FepFbq7$FepF^q7$Fbp
F^q7$FbpFbq7$FbpFjq7$FepFjq7$FepF^r7$FbpF^r7$FbpFbr7$FbpFer7$FepFer7$F
epFbr7$F^qFbr7$F^qFer7$FbqFer7$FbqFbr7$FbqF^r7$F^qF^r7$F^qFjq7$FbqFjq7
$FjqFjq7$F^rFjq7$F^rF^r7$FjqF^r7$FjqFbr7$FjqFer7$F^rFer7$F^rFbr7$FbrFb
r7$FbrFer7$FerFer7$FerFbr7$FerF^r7$FbrF^r7$FbrFjq7$FerFjq7$FerFbq7$Fer
F^q7$FbrF^q7$FbrFbq7$F^rFbq7$FjqFbq7$FjqF^q7$F^rF^q7$F^rFep7$FjqFep7$F
jqFbp7$F^rFbp7$FbrFbp7$FbrFep7$FerFep7$FerFbp7$FerFK7$FbrFK7$FbrFH7$Fe
rFH7$FerFD7$FerF@7$FbrF@7$FbrFD7$F^rFD7$F^rF@7$FjqF@7$FjqFD7$FjqFH7$F^
rFH7$F^rFK7$FjqFK7$FbqFK7$FbqFH7$F^qFH7$F^qFK7$FepFK7$FbpFK7$FbpFH7$Fe
pFH7$FepFD7$FbpFD7$FbpF@7$FepF@7$F^qF@7$F^qFD7$FbqFD7$FbqF@7$FbqF37$Fb
qF07$F^qF07$F^qF37$FepF37$FbpF37$FbpF07$FepF07$FepF+7$FbpF+7$FbpF(7$Fe
pF(7$F^qF(7$F^qF+7$FbqF+7$FbqF(7$FjqF(7$F^rF(7$F^rF+7$FjqF+7$FjqF07$Fj
qF37$F^rF37$F^rF07$FbrF07$FbrF37$FerF37$FerF07$FerF+7$FbrF+7$FbrF(7$Fe
rF(-%'COLOURG6&%$RGBG$F_q!\"\"F(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%
,CONSTRAINEDG-%+AXESLABELSG6$Q!6\"Fb^l-%%VIEWG6$%(DEFAULTGFg^l" 1 2 0 
1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 223 "Efforcez-vous de comprendre cet e
xemple. Cela aide de partir des mots [x] et [y] et de faire une ou deu
x etapes seulement, en appliquant pour commencer seulement la regle de
 substitution pour x, respectivement celle pour y." }}}{SECT 1 {PARA 
257 "" 0 "" {TEXT 275 11 "Exercice 4." }}{PARA 0 "" 0 "" {TEXT -1 89 "
Trouver le codage de la courbe suivante, qui est une variante du trian
gle de Sierpinski :" }}{EXCHG {PARA 0 "" 0 "" {GLPLOT2D 246 246 246 
{PLOTDATA 2 "6%-%'CURVESG6$7fhm7$\"\"!F(7$$\"\"\"F(F(7$$\"1+++++++:!#:
$\"1'QWy.a-m)!#;7$F*$\"1x)ov!30K<F/7$F-$\"1;LN6i2)f#F/7$$\"1+++++++DF/
F77$$\"\"$F(F47$F:F07$F=F(7$$\"\"%F(F(7$$\"1+++++++XF/F07$$\"1+++++++b
F/F07$$\"\"'F(F(7$$\"\"(F(F(7$$\"1+++++++vF/F07$FNF47$FKF47$FHF77$FK$
\"1ax8:;5kMF/7$FH$\"1$>A*=q7IVF/7$FEFZ7$FBFW7$F=FW7$F:FZ7$F=$\"1KmqAC:
'>&F/7$FBF[o7$FE$\"1q5\\Ey<igF/7$FB$\"14bFIK?GpF/7$FE$\"1Z*fSjGUz(F/7$
FHFeo7$FKFbo7$FNFbo7$FQFeo7$FN$F1F/7$FKF\\p7$FH$\"1D)G;Wzi_*F/7$FK$\"1
E8a%[I#R5!#97$FNFbp7$FQ$\"1q(>\\-Le7\"Fdp7$FN$\"19#)HlbV77Fdp7$FQ$\"1e
mn0\"Q!*H\"Fdp7$$\"1+++++++&)F/F]q7$$\"\"*F(Fjp7$F`qFgp7$FcqFbp7$$\"#5
F(Fbp7$$\"1++++++]5FdpF_p7$FhqF\\p7$FcqF\\p7$F`qFeo7$FcqFbo7$FhqFbo7$F
[rFeo7$$\"1++++++]6FdpFeo7$$\"#7F(Fbo7$FdrF_o7$FgrF[o7$$\"#8F(F[o7$$\"
1++++++]8FdpFZ7$F\\sFW7$FgrFW7$FdrFZ7$F[rFZ7$FhqFW7$F[rF77$FhqF47$FcqF
47$F`qF07$FcqF(7$FhqF(7$F[rF07$FdrF07$FgrF(7$F\\sF(7$F_sF07$F\\sF47$F_
sF77$$\"1++++++]9FdpF77$$\"#:F(F47$FdtF07$FgtF(7$$\"#;F(F(7$$\"1++++++
];FdpF07$$\"1++++++]<FdpF07$$\"#=F(F(7$$\"#>F(F(7$$\"1++++++]>FdpF07$F
huF47$FeuF47$FbuF77$FeuFW7$FhuFW7$F[vFZ7$FhuF[o7$F[vF_o7$$\"1++++++]?F
dpF_o7$$\"#@F(F[o7$FfvFZ7$FivFW7$$\"#AF(FW7$$\"1++++++]AFdpF77$F^wF47$
FivF47$FfvF07$FivF(7$F^wF(7$FawF07$$\"1++++++]BFdpF07$$\"#CF(F(7$$\"#D
F(F(7$$\"1++++++]DFdpF07$F`xF47$FcxF77$$\"1++++++]EFdpF77$$\"#FF(F47$F
hxF07$F[yF(7$$\"#GF(F(7$$\"1++++++]GFdpF07$$\"1++++++]HFdpF07$$\"#IF(F
(7$$\"#JF(F(7$$\"1++++++]JFdpF07$F\\zF47$FiyF47$FfyF77$FiyFW7$FfyFZ7$F
cyFZ7$F`yFW7$F[yFW7$FhxFZ7$F[yF[o7$F`yF[o7$FcyF_o7$F`yFbo7$F[yFbo7$Fhx
Feo7$F[yF\\p7$FhxF_p7$FcxF_p7$F`xF\\p7$FcxFeo7$F`xFbo7$F]xFbo7$FjwFeo7
$FawFeo7$F^wFbo7$FivFbo7$FfvFeo7$FivF\\p7$F^wF\\p7$FawF_p7$F^wFbp7$Faw
Fgp7$FjwFgp7$F]xFbp7$F`xFbp7$FcxFgp7$F`xFjp7$F]xFjp7$FjwF]q7$F]x$\"1-^
0Y1k&Q\"Fdp7$Fjw$\"1YNV'=VAZ\"Fdp7$FawF\\]l7$F^wFi\\l7$FivFi\\l7$FfvF
\\]l7$Fiv$\"1*)>\"osX)e:Fdp7$F^wFc]l7$Faw$\"1L/>n#[ak\"Fdp7$F^w$F5Fdp7
$FivFj]l7$Ffv$\"1@t%zM`'==Fdp7$Fiv$\"1ldK))eD0>Fdp7$Ffv$\"14UqG%e=*>Fd
p7$F[vFc^l7$FhuF`^l7$F[vF]^l7$FhuFj]l7$FeuFj]l7$FbuFg]l7$FeuFc]l7$FhuF
c]l7$F[vF\\]l7$FhuFi\\l7$FeuFi\\l7$FbuF\\]l7$F_uF\\]l7$F\\uFi\\l7$FgtF
i\\l7$FdtF\\]l7$FgtFc]l7$FdtFg]l7$F_sFg]l7$F\\sFc]l7$F_sF\\]l7$F\\sFi
\\l7$FgrFi\\l7$FdrF\\]l7$F[rF\\]l7$FhqFi\\l7$FcqFi\\l7$F`qF\\]l7$FcqFc
]l7$FhqFc]l7$F[rFg]l7$FhqFj]l7$F[rF]^l7$FdrF]^l7$FgrFj]l7$F\\sFj]l7$F_
sF]^l7$F\\sF`^l7$FgrF`^l7$FdrFc^l7$Fgr$\"1`E3p4Yy?Fdp7$F\\sF^al7$F_s$
\"1(4h%4N1l@Fdp7$F\\s$\"1S&R)\\gm^AFdp7$F_s$\"1%)z@!fo#QBFdp7$FdtFhal7
$FgtFeal7$FdtFbal7$FgtF^al7$F\\uF^al7$F_uFbal7$FbuFbal7$FeuF^al7$FhuF^
al7$F[vFbal7$FhuFeal7$FeuFeal7$FbuFhal7$Feu$\"1GkfI6([U#Fdp7$Fbu$\"1s[
(4nt9^#Fdp7$F_uF[cl7$F\\uFhbl7$FgtFhbl7$FdtF[cl7$Fgt$F8Fdp7$F\\uFbcl7$
F_u$\"1g<t^(yYo#Fdp7$F\\u$\"1/-6#H\"GrFFdp7$F_u$\"1Z')[KQ)y&GFdp7$FbuF
[dl7$FeuFhcl7$FhuFhcl7$F[vF[dl7$Fhu$\"1\"4nGP'[WHFdp7$FeuFbdl7$Fbu$\"1
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1n3QMl*3H$Fdp7$F[v$\"16$fZ2*\\xLFdp7$FfvFcel7$FivF`el7$FfvF]el7$FivFid
l7$F^wFidl7$FawFfdl7$F^wFbdl7$FivFbdl7$FfvF[dl7$FivFhcl7$F^wFhcl7$FawF
[dl7$FjwF[dl7$F]xFhcl7$F`xFhcl7$FcxF[dl7$F`xFbdl7$FcxFfdl7$FhxFfdl7$F[
yFbdl7$FhxF[dl7$F[yFhcl7$F`yFhcl7$FcyF[dl7$FfyF[dl7$FiyFhcl7$F\\zFhcl7
$F_zF[dl7$F\\zFbdl7$FiyFbdl7$FfyFfdl7$FiyFidl7$FfyF]el7$FcyF]el7$F`yFi
dl7$F[yFidl7$FhxF]el7$F[yF`el7$F`yF`el7$FcyFcel7$F`y$FXFdp7$F[yF^hl7$F
hx$\"1)>;b:/2b$Fdp7$F[y$\"1UY*ep1tj$Fdp7$Fhx$\"1'3tiB4Rs$Fdp7$FcxFghl7
$F`xFdhl7$FcxFahl7$F`xF^hl7$F]xF^hl7$FjwFahl7$FawFahl7$F^wF^hl7$FivF^h
l7$FfvFahl7$FivFdhl7$F^wFdhl7$FawFghl7$F^w$\"1I:lw<^5QFdp7$Faw$\"1u*Hq
J9r*QFdp7$FjwFjil7$F]xFgil7$F`xFgil7$FcxFjil7$F`x$\"1<%3u&or$)RFdp7$F]
xFajl7$Fjw$\"1ioy(R>.2%Fdp7$F]x$\"11`;Q>#p:%Fdp7$F`xFhjl7$Fcx$\"1\\Pay
W_VUFdp7$F`x$FenFdp7$Fcx$\"1P1If&HnT%Fdp7$FhxFa[m7$F[yF_[m7$FhxF\\[m7$
F[yFhjl7$F`yFhjl7$FcyF\\[m7$FfyF\\[m7$FiyFhjl7$F\\zFhjl7$F_zF\\[m7$F\\
zF_[m7$FiyF_[m7$FfyFa[m7$Fiy$\"1\"3z'*4KL]%Fdp7$Ffy$\"1Dv0SY$**e%Fdp7$
FcyFd\\m7$F`yFa\\m7$F[yFa\\m7$FhxFd\\m7$F[y$\"1ofV!=Pln%Fdp7$F`yF[]m7$
Fcy$\"17W\"3sRJw%Fdp7$F`y$\"1cG>hAu\\[Fdp7$Fcy$\"1+8d,[MO\\Fdp7$FfyFe]
m7$FiyFb]m7$F\\zFb]m7$F_zFe]m7$F\\z$\"1W(\\>MZH-&Fdp7$FiyF\\^m7$Ffy$\"
1)=GB))\\&4^Fdp7$Fiy$\"1JmqAC:'>&Fdp7$F\\zFc^m7$F_z$\"1v]3j\\v#G&Fdp7$
F\\z$\"1?NY.vNp`Fdp7$F_z$\"1j>%Q/gfX&Fdp7$$\"1++++++]KFdpF]_m7$$\"#LF(
Fj^m7$F`_mFg^m7$Fc_mFc^m7$$\"#MF(Fc^m7$$\"1++++++]MFdpF`^m7$Fh_mF\\^m7
$Fc_mF\\^m7$F`_mFe]m7$Fc_mFb]m7$Fh_mFb]m7$F[`mFe]m7$$\"1++++++]NFdpFe]
m7$$\"#OF(Fb]m7$Fd`mF_]m7$Fg`mF[]m7$$\"#PF(F[]m7$$\"1++++++]PFdpFd\\m7
$F\\amFa\\m7$Fg`mFa\\m7$Fd`mFd\\m7$F[`mFd\\m7$Fh_mFa\\m7$F[`mFa[m7$Fh_
mF_[m7$Fc_mF_[m7$F`_mF\\[m7$Fc_mFhjl7$Fh_mFhjl7$F[`mF\\[m7$Fd`mF\\[m7$
Fg`mFhjl7$F\\amFhjl7$F_amF\\[m7$F\\amF_[m7$F_amFa[m7$$\"1++++++]QFdpFa
[m7$$\"#RF(F_[m7$FdbmF\\[m7$FgbmFhjl7$$\"#SF(Fhjl7$$\"1++++++]SFdpFejl
7$F\\cmFajl7$FgbmFajl7$FdbmFjil7$FgbmFgil7$F\\cmFgil7$F_cmFjil7$$\"1++
++++]TFdpFjil7$$\"#UF(Fgil7$FhcmFghl7$F[dmFdhl7$$\"#VF(Fdhl7$$\"1+++++
+]VFdpFahl7$F`dmF^hl7$F[dmF^hl7$FhcmFahl7$F_cmFahl7$F\\cmF^hl7$FgbmF^h
l7$FdbmFahl7$FgbmFdhl7$FdbmFghl7$F_amFghl7$F\\amFdhl7$F_amFahl7$F\\amF
^hl7$Fg`mF^hl7$Fd`mFcel7$Fg`mF`el7$F\\amF`el7$F_amF]el7$F\\amFidl7$Fg`
mFidl7$Fd`mF]el7$F[`mF]el7$Fh_mFidl7$F[`mFfdl7$Fh_mFbdl7$Fc_mFbdl7$F`_
mF[dl7$Fc_mFhcl7$Fh_mFhcl7$F[`mF[dl7$Fd`mF[dl7$Fg`mFhcl7$F\\amFhcl7$F_
amF[dl7$F\\amFbdl7$F_amFfdl7$FdbmFfdl7$FgbmFbdl7$FdbmF[dl7$FgbmFhcl7$F
\\cmFhcl7$F_cmF[dl7$FhcmF[dl7$F[dmFhcl7$F`dmFhcl7$FcdmF[dl7$F`dmFbdl7$
F[dmFbdl7$FhcmFfdl7$F[dmFidl7$F`dmFidl7$FcdmF]el7$F`dmF`el7$FcdmFcel7$
$\"1++++++]WFdpFcel7$$\"#XF(F`el7$F\\hmF]el7$F_hmFidl7$$\"#YF(Fidl7$$
\"1++++++]YFdpFfdl7$FdhmFbdl7$F_hmFbdl7$F\\hmF[dl7$F_hmFhcl7$FdhmFhcl7
$FghmF[dl7$$\"1++++++]ZFdpF[dl7$$\"#[F(Fhcl7$F`imFecl7$FcimFbcl7$$\"#
\\F(Fbcl7$$\"1++++++]\\FdpF[cl7$FhimFhbl7$FcimFhbl7$F`imF[cl7$FghmF[cl
7$FdhmFhbl7$FghmFhal7$FdhmFeal7$F_hmFeal7$F\\hmFbal7$F_hmF^al7$FdhmF^a
l7$FghmFbal7$F`imFbal7$FcimF^al7$FhimF^al7$F[jmFbal7$FhimFeal7$F[jmFha
l7$$\"1++++++]]FdpFhal7$$\"#^F(Feal7$F`[nFbal7$Fc[nF^al7$$\"#_F(F^al7$
$\"1++++++]_FdpFc^l7$Fh[nF`^l7$Fc[nF`^l7$F`[nF]^l7$Fc[nFj]l7$Fh[nFj]l7
$F[\\nF]^l7$$\"1++++++]`FdpF]^l7$$\"#aF(Fj]l7$Fd\\nFg]l7$Fg\\nFc]l7$$
\"#bF(Fc]l7$$\"1++++++]bFdpF\\]l7$F\\]nFi\\l7$Fg\\nFi\\l7$Fd\\nF\\]l7$
F[\\nF\\]l7$Fh[nFi\\l7$Fc[nFi\\l7$F`[nF\\]l7$Fc[nFc]l7$F`[nFg]l7$F[jmF
g]l7$FhimFc]l7$F[jmF\\]l7$FhimFi\\l7$FcimFi\\l7$F`imF\\]l7$FghmF\\]l7$
FdhmFi\\l7$F_hmFi\\l7$F\\hmF\\]l7$F_hmFc]l7$FdhmFc]l7$FghmFg]l7$FdhmFj
]l7$F_hmFj]l7$F\\hmF]^l7$F_hmF`^l7$F\\hmFc^l7$FcdmFc^l7$F`dmF`^l7$Fcdm
F]^l7$F`dmFj]l7$F[dmFj]l7$FhcmFg]l7$F[dmFc]l7$F`dmFc]l7$FcdmF\\]l7$F`d
mFi\\l7$F[dmFi\\l7$FhcmF\\]l7$F_cmF\\]l7$F\\cmFi\\l7$F_cmF]q7$F\\cmFjp
7$FgbmFjp7$FdbmFgp7$FgbmFbp7$F\\cmFbp7$F_cmFgp7$FhcmFgp7$F[dmFbp7$Fhcm
F_p7$F[dmF\\p7$F`dmF\\p7$FcdmFeo7$F`dmFbo7$F[dmFbo7$FhcmFeo7$F_cmFeo7$
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246 246 {PLOTDATA 2 "6%-%'CURVESG6$7&7$\"\"!F(7$$\"\"\"F(F(7$$\"1+++++
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246 {PLOTDATA 2 "6%-%'CURVESG6$7,7$\"\"!F(7$$\"\"\"F(F(7$$\"1+++++++:!
#:$\"1'QWy.a-m)!#;7$F*$\"1x)ov!30K<F/7$F-$\"1;LN6i2)f#F/7$$\"1+++++++D
F/F77$$\"\"$F(F47$F:F07$F=F(7$$\"\"%F(F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(
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1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "C'est en fait un exercice cerebr
al assez ardu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 44 "Voici encore d'autres exemples interessants." }}
{PARA 0 "" 0 "" {TEXT -1 30 "La courbe de Peano originale :" }}}
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FcelF.7$F\\elF.7$F^alF.7$F^alF37$F\\elF3Fa^nFb^nFc^n7$Fi`lF37$Fi`lF67$
Fi`lF97$F^alF97$F^alF6Ff^nFg^nFh^n7$F^alF>7$F\\elF>7$FcelF>7$FcelF97$F
\\elF9F[_nF\\_nF]_nF[^nF\\^nF]^nF^^nF_^nF\\^nF]^nF^^nF`^nFa^nFb^nFc^nF
d^nFa^nFb^nFc^nFe^n7$Fi`lF.7$Fi`lF+7$F(F+7$F(F.F__nF`_nFa_nF'-%'COLOUR
G6&%$RGBG$FfnFj`lF(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG
-%+AXESLABELSG6$Q!6\"Fc`n-%%VIEWG6$%(DEFAULTGFh`n" 1 2 0 1 10 0 2 9 1 
1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT 276 25 "II. Croissance de plantes" }}{PARA 0 "" 0 "" 
{TEXT -1 696 "Dans le paragraphe precedent, tous les dessins obtenus s
ont des courbes d'un seul trait, puisque chaque deplacement de la tort
ue est tracant. Cela ne convient par pour simuler la croissance de pla
ntes, qui presentent des branchements. Pour cela il est necessaire d'i
ntroduire deux nouveaux symboles, le crochet gauche et droit, notes re
spectivement l et r, et une nouvelle regle pour leur interpretation : \+
lorsqu'on trouve un crochet gauche, on met sur une pile la position et
 l'orientation de la tortue; lorsqu'on rencontre un crochet droit, la \+
position et orientation du crochet gauche correspondant sont enleves d
e la pile, et la tortue se met, sans tracer, dans cette position et or
ientation." }}{PARA 0 "" 0 "" {TEXT -1 35 "Voici une variante de la pr
ocedure " }{TEXT 279 7 "dessine" }{TEXT -1 70 " qui permet de traiter \+
les crochets; efforcez-vous de la comprendre : " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 694 "machin:=proc(axiome::list,rule::procedure,eta
pes::integer,theta::realcons)\nlocal i,lst, plante,pts,tt,xx,yy,xyt,pi
le;\n\nplante:=axiome;\nfor i to etapes do\nplante:=map(rule,plante)\n
od;\ntt:=Pi/2;\nxx:=0;\nyy:=0;\npts:=[xx,yy];\nlst:=NULL;\npile:=[];\n
for i in plante do\nif i=p then tt:=tt+theta;\n elif i=m then tt:=tt-t
heta;\nelif i=l then pile:=[[xx,yy,tt],op(pile)];\nelif i=r then if no
ps(pile)<1 then ERROR('Pelato');\n fi;\n               \n   xyt:=pile[
1];\n   pile:=subsop(1=NULL,pile);\n   xx:=xyt[1];\n   yy:=xyt[2];\n  \+
 tt:=xyt[3];\n   lst:=lst,[pts];\n   pts:=[xx,yy];\n elif i=f then \n \+
  xx:=xx+cos(tt);\n   yy:=yy+sin(tt);\n   pts:=pts,[xx,yy]\nfi;\nod;\n
plot(\{lst,[pts]\},axes=NONE,scaling=CONSTRAINED)\nend;" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%'machinGf*6&'%'axiomeG%%listG'%%ruleG%*procedu
reG'%'etapesG%(integerG'%&thetaG%)realconsG6+%\"iG%$lstG%'planteG%$pts
G%#ttG%#xxG%#yyG%$xytG%%pileG6\"F=C,>8&9$?(8$\"\"\"FD9&%%trueG>F@-%$ma
pG6$9%F@>8(,$%#PiG#FD\"\"#>8)\"\"!>8*FT>8'7$FSFV>8%%%NULLG>8,7\"?&FCF@
FF@,/FC%\"pG>FM,&FMFD9'FD/FC%\"mG>FM,&FMFDF`o!\"\"/FC%\"lG>Fhn7$7%FSFV
FM-%#opG6#Fhn/FC%\"rGC*@$2-%%nopsGF]pFD-%&ERRORG6#.%'PelatoG>8+&Fhn6#F
D>Fhn-%'subsopG6$/FDFfnFhn>FS&F[qF]q>FV&F[q6#FQ>FM&F[q6#\"\"$>Fen6$Fen
7#FX>FXFY/FC%\"fGC%>FS,&FSFD-%$cosG6#FMFD>FV,&FVFD-%$sinGFgrFD>FX6$FXF
Y-%%plotG6%<$F^rFen/%%axesG%%NONEG/%(scalingG%,CONSTRAINEDGF=F=F=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rule1:=a->subs(f=(f,l,p,f,r,
f,l,m,f,r,f),a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule1Gf*6#%\"aG
6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"fG6-F0%\"lG%\"pGF0%\"rGF0F2%\"
mGF0F4F09$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "machin(
[f],rule1,2,Pi/7);" }}{PARA 13 "" 1 "" {GLPLOT2D 256 256 256 
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1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur
ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}}{EXCHG {PARA 0 "
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3" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "
Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curv
e 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42
" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "C
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 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61
" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "C
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" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "C
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05" "Curve 106" "Curve 107" "Curve 108" "Curve 109" "Curve 110" "Curve
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ve 117" "Curve 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "C
urve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" 
"Curve 129" "Curve 130" "Curve 131" "Curve 132" "Curve 133" "Curve 134
" "Curve 135" "Curve 136" "Curve 137" "Curve 138" "Curve 139" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 10 "Exercice 5" }
{TEXT -1 62 ". Trouver le codage du dessin suivant, obtenu apres 3 eta
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urve 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" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Cur
ve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 3
4" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "
Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curv
e 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53
" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "C
urve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve
 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72
" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "C
urve 79" "Curve 80" "Curve 81" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 
10 "Exercice 6" }{TEXT -1 34 ". Trouver le code de cette plante." }}}}
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"" {TEXT -1 0 "" }{TEXT 284 3 "FIN" }}}}{MARK "1 22 0" 54 }{VIEWOPTS 
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