1 | (* Content-type: application/mathematica *) |
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2 | |
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3 | (*** Wolfram Notebook File ***) |
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4 | (* http://www.wolfram.com/nb *) |
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5 | |
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6 | (* CreatedBy='Mathematica 6.0' *) |
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7 | |
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8 | (*CacheID: 234*) |
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9 | (* Internal cache information: |
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10 | NotebookFileLineBreakTest |
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11 | NotebookFileLineBreakTest |
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17 | WindowFrame->Normal |
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18 | ContainsDynamic->False*) |
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19 | |
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20 | (* Beginning of Notebook Content *) |
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21 | Notebook[{ |
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22 | |
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23 | Cell[CellGroupData[{ |
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24 | Cell["Pavages du plan", "Title", |
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25 | Evaluatable->False, |
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26 | AspectRatioFixed->True], |
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27 | |
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28 | Cell[TextData[{ |
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29 | "Il existe partout des dessins obtenus par r\[EAcute]p\[EAcute]titions d'un \ |
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30 | motif unique: carreaux d'une fen\[EHat]tre, tuiles d'une toiture, lattes de \ |
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31 | bois d'une cl\[OHat]ture, briques d'un mur, papier peint, tissu,... On ne les \ |
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32 | trouve pas seulement dans des objets produits par l'homme: le rayon de cire \ |
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33 | d'abeille, la corolle des fleurs, le cristal de sel,...\nLa couverture d'un \ |
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34 | plan par r\[EAcute]p\[EAcute]titions d'un m\[EHat]me motif, sans \ |
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35 | chevauchements ni trous, s'appelle un ", |
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36 | StyleBox["pavage", |
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37 | FontWeight->"Bold"], |
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38 | ". Comme c'est le cas pour toutes les t\[AHat]ches \ |
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39 | r\[EAcute]p\[EAcute]titives, l'informatique s'av\[EGrave]re tr\[EGrave]s \ |
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40 | utile pour produire des pavages." |
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41 | }], "Text", |
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42 | Evaluatable->False, |
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43 | PageBreakAbove->False, |
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44 | TextAlignment->Left, |
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45 | TextJustification->0, |
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46 | AspectRatioFixed->True], |
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47 | |
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48 | Cell[CellGroupData[{ |
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49 | |
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50 | Cell["D\[EAcute]finition du motif", "Subsubtitle"], |
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51 | |
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52 | Cell[TextData[{ |
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53 | "Pour d\[EAcute]finir une forme, on proc\[EGrave]de comme ci-dessous. On \ |
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54 | donne un nom sans espaces ou symboles ayant une signification \ |
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55 | sp\[EAcute]ciale dans ", |
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56 | StyleBox["Mathematica", |
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57 | FontSlant->"Italic"], |
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58 | " (comme - par exemple). Vient ensuite le signe =. Enfin, entre accolades, \ |
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59 | on indique la couleur, puis, apr\[EGrave]s une virgule, viennent la ou les \ |
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60 | primitives graphiques." |
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61 | }], "Text", |
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62 | Evaluatable->False, |
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63 | AspectRatioFixed->True], |
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64 | |
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65 | Cell[BoxData[{ |
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66 | RowBox[{ |
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67 | RowBox[{"carr\[EAcute]", "=", |
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68 | RowBox[{"{", |
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69 | RowBox[{ |
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70 | RowBox[{"RGBColor", "[", |
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71 | RowBox[{"1", ",", "0", ",", "0"}], "]"}], ",", |
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72 | RowBox[{"Polygon", "[", |
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73 | RowBox[{"{", |
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74 | RowBox[{ |
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75 | RowBox[{"{", |
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76 | RowBox[{ |
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78 | RowBox[{"-", "1"}]}], "}"}], ",", |
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79 | RowBox[{"{", |
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80 | RowBox[{ |
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81 | RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", |
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82 | RowBox[{"{", |
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83 | RowBox[{"1", ",", "1"}], "}"}], ",", |
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84 | RowBox[{"{", |
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85 | RowBox[{"1", ",", |
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86 | RowBox[{"-", "1"}]}], "}"}]}], "}"}], "]"}]}], "}"}]}], |
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87 | ";"}], "\n", |
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88 | RowBox[{ |
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89 | RowBox[{"croix", "=", |
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90 | RowBox[{"{", |
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91 | RowBox[{ |
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92 | RowBox[{"RGBColor", "[", |
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94 | RowBox[{"Thickness", "[", "0.01", "]"}], ",", |
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95 | RowBox[{"Line", "[", |
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96 | RowBox[{"{", |
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97 | RowBox[{ |
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98 | RowBox[{"{", |
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110 | RowBox[{"{", |
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111 | RowBox[{ |
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113 | "}"}]}], ";"}]}], "Input", |
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114 | AspectRatioFixed->True], |
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115 | |
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116 | Cell["\<\ |
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117 | Une fois toutes les parties du dessin d\[EAcute]finies, on indique l'ordre \ |
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118 | dans lequel elles doivent \[EHat]tre dessin\[EAcute]es:\ |
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119 | \>", "Text", |
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120 | Evaluatable->False, |
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121 | AspectRatioFixed->True], |
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122 | |
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123 | Cell[BoxData[ |
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124 | RowBox[{ |
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125 | RowBox[{"pav\[EAcute]", "=", |
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126 | RowBox[{"{", |
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127 | RowBox[{"carr\[EAcute]", ",", "croix"}], "}"}]}], ";"}]], "Input", |
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128 | AspectRatioFixed->True], |
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129 | |
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130 | Cell[TextData[{ |
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131 | StyleBox["Mathematica", |
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132 | FontSlant->"Italic"], |
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133 | " dessinera d'abord le carr\[EAcute] puis la croix. Changez l'ordre pour \ |
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134 | voir la diff\[EAcute]rence ! \nRien ne sera dessin\[EAcute] \[AGrave] l'\ |
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135 | \[EAcute]cran tant que vous ne direz pas \[AGrave] ", |
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136 | StyleBox["Mathematica", |
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137 | FontSlant->"Italic"], |
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138 | " de le faire ! Voici l'instruction \[AGrave] utiliser:" |
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139 | }], "Text", |
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140 | Evaluatable->False, |
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141 | AspectRatioFixed->True], |
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142 | |
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143 | Cell[BoxData[ |
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144 | RowBox[{"Show", "[", |
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145 | RowBox[{ |
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146 | RowBox[{"Graphics", "[", "pav\[EAcute]", "]"}], ",", |
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147 | RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input", |
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148 | AspectRatioFixed->True], |
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149 | |
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150 | Cell["\<\ |
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151 | (on peut r\[EAcute]duire ou agrandir de dessin en cliquant dessus et en \ |
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152 | manipulant les poign\[EAcute]es).\ |
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153 | \>", "Text"] |
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154 | }, Open ]], |
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155 | |
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156 | Cell[CellGroupData[{ |
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157 | |
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158 | Cell["R\[EAcute]p\[EAcute]tition du motif", "Subsubtitle"], |
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159 | |
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160 | Cell[TextData[{ |
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161 | "Le motif g\[EAcute]n\[EAcute]rique de pavage sera d\[EAcute]fini comme \ |
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162 | suit. On doit d'abord donner les deux directions de translation (pour couvrir \ |
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163 | un plan il faut deux directions). On peut faire une analogie avec les \ |
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164 | machines \[AGrave] \[EAcute]crire. En tapant sur une touche, la touche \ |
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165 | \"avance\" automatiquement d'un cran vers la droite; cela correspond pour \ |
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166 | nous \[AGrave] notre direction ", |
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167 | StyleBox["u", |
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168 | FontFamily->"Courier", |
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169 | FontWeight->"Bold"], |
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170 | ". \[CapitalAGrave] la fin d'une ligne, on descend d'un cran avant de \ |
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171 | recommencer une nouvelle ligne; c'est notre direction ", |
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172 | StyleBox["v", |
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173 | FontFamily->"Courier", |
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174 | FontWeight->"Bold"], |
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175 | ". Remarquez que les vecteurs ", |
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176 | StyleBox["u", |
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177 | FontFamily->"Courier", |
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178 | FontWeight->"Bold"], |
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179 | " et ", |
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180 | StyleBox["v", |
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181 | FontFamily->"Courier", |
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182 | FontWeight->"Bold"], |
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183 | " n'ont pas besoin d'\[EHat]tre perpendiculaires; ils doivent seulement \ |
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184 | \[EHat]tre non colin\[EAcute]aires." |
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185 | }], "Text", |
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186 | Evaluatable->False, |
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187 | AspectRatioFixed->True], |
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188 | |
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189 | Cell[TextData[{ |
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190 | "Comme notre dessin a une largeur de 2, on se d\[EAcute]calera de deux unit\ |
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191 | \[EAcute]s vers la droite (", |
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192 | StyleBox["u", |
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193 | FontFamily->"Courier", |
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194 | FontWeight->"Bold"], |
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195 | "). Comme on a envie que nos dessins soient r\[EAcute]p\[EAcute]t\[EAcute]s \ |
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196 | en quinconce, la direction ", |
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197 | StyleBox["v", |
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198 | FontFamily->"Courier", |
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199 | FontWeight->"Bold"], |
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200 | " indique que pour recommencer une ligne, on se d\[EAcute]calera d'un cran \ |
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201 | vers la droite et de deux vers le haut par rapport \[AGrave] la ligne pr\ |
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202 | \[EAcute]c\[EAcute]dente." |
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203 | }], "Text", |
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204 | Evaluatable->False, |
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205 | AspectRatioFixed->True], |
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206 | |
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207 | Cell[BoxData[ |
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208 | RowBox[{ |
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209 | RowBox[{"u", "=", |
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210 | RowBox[{"{", |
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211 | RowBox[{"2", ",", "0"}], "}"}]}], ";", |
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212 | RowBox[{"v", "=", |
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213 | RowBox[{"{", |
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214 | RowBox[{"1", ",", "2"}], "}"}]}], ";"}]], "Input", |
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215 | AspectRatioFixed->True], |
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216 | |
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217 | Cell["\<\ |
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218 | On d\[EAcute]finit maintenant le motif g\[EAcute]n\[EAcute]rique qui \ |
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219 | permettra de paver notre plan:\ |
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220 | \>", "Text", |
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221 | Evaluatable->False, |
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222 | AspectRatioFixed->True], |
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223 | |
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224 | Cell[BoxData[ |
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225 | RowBox[{ |
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226 | RowBox[{"motif", "=", |
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227 | RowBox[{"ReplaceAll", "[", |
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228 | RowBox[{"pav\[EAcute]", ",", |
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229 | RowBox[{ |
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230 | RowBox[{"{", |
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231 | RowBox[{ |
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232 | RowBox[{"x_", "?", "NumberQ"}], ",", |
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233 | RowBox[{"y_", "?", "NumberQ"}]}], "}"}], "\[Rule]", |
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234 | RowBox[{"{", |
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235 | RowBox[{ |
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236 | RowBox[{"x", "+", |
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237 | RowBox[{ |
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238 | RowBox[{"u", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], |
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239 | " ", "dx"}], "+", |
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240 | RowBox[{ |
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241 | RowBox[{"v", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], |
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242 | " ", "dy"}]}], ",", |
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243 | RowBox[{"y", "+", |
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244 | RowBox[{ |
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245 | RowBox[{"u", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], |
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246 | " ", "dx"}], "+", |
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247 | RowBox[{ |
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248 | RowBox[{"v", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], |
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249 | " ", "dy"}]}]}], "}"}]}]}], "]"}]}], ";"}]], "Input", |
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250 | AspectRatioFixed->True], |
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251 | |
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252 | Cell[TextData[{ |
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253 | "La fonction ", |
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254 | StyleBox["ReplaceAll", |
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255 | FontFamily->"Courier", |
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256 | FontWeight->"Bold"], |
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257 | " remplace les coordonn\[EAcute]es ", |
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258 | StyleBox["x", |
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259 | FontFamily->"Courier", |
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260 | FontWeight->"Bold"], |
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261 | " par ", |
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262 | StyleBox["x+u[[1]]dx+v[[1]]dy", |
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263 | FontFamily->"Courier", |
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264 | FontWeight->"Bold"], |
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265 | " et les coordonn\[EAcute]es ", |
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266 | StyleBox["y", |
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267 | FontFamily->"Courier", |
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268 | FontWeight->"Bold"], |
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269 | " par ", |
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270 | StyleBox["y+u[[2]]dx+v[[2]]dy", |
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271 | FontFamily->"Courier", |
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272 | FontWeight->"Bold"], |
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273 | ". On peut maintenant cr\[EAcute]er une liste en faisant varier ", |
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274 | StyleBox["dx", |
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275 | FontFamily->"Courier", |
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276 | FontWeight->"Bold"], |
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277 | " et ", |
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278 | StyleBox["dy", |
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279 | FontFamily->"Courier", |
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280 | FontWeight->"Bold"], |
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281 | " entre 0 et le nombre de r\[EAcute]p\[EAcute]titions moins une que l'on \ |
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282 | veut en horizontal et en vertical. Dans notre exemple, on a d\[EAcute]cid\ |
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283 | \[EAcute] de r\[EAcute]p\[EAcute]ter 5 fois selon la direction ", |
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284 | StyleBox["u", |
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285 | FontFamily->"Courier", |
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286 | FontWeight->"Bold"], |
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287 | " et 4 fois selon la direction ", |
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288 | StyleBox["v", |
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289 | FontFamily->"Courier", |
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290 | FontWeight->"Bold"], |
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291 | ":" |
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292 | }], "Text", |
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293 | PageWidth->WindowWidth, |
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294 | Evaluatable->False, |
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295 | AspectRatioFixed->True], |
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296 | |
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297 | Cell[BoxData[ |
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298 | RowBox[{ |
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299 | RowBox[{"pavage", "=", |
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300 | RowBox[{"Table", "[", |
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301 | RowBox[{"motif", ",", |
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302 | RowBox[{"{", |
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303 | RowBox[{"dx", ",", "0", ",", "4"}], "}"}], ",", |
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304 | RowBox[{"{", |
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305 | RowBox[{"dy", ",", "0", ",", "3"}], "}"}]}], "]"}]}], ";"}]], "Input", |
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306 | AspectRatioFixed->True], |
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307 | |
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308 | Cell["Dessinons enfin ce pavage :", "Text", |
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309 | Evaluatable->False, |
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310 | AspectRatioFixed->True], |
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311 | |
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312 | Cell[BoxData[ |
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313 | RowBox[{"Show", "[", |
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314 | RowBox[{ |
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315 | RowBox[{"Graphics", "[", "pavage", "]"}], ",", |
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316 | RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input", |
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317 | AspectRatioFixed->True], |
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318 | |
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319 | Cell["\<\ |
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320 | Vous trouverez ci-dessous trois pavages de difficult\[EAcute]s croissantes.\ |
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321 | \>", "Text"], |
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322 | |
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323 | Cell[CellGroupData[{ |
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324 | |
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325 | Cell["Exemple 1 : le carrelage", "Subsection", |
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326 | Evaluatable->False, |
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327 | PageBreakAbove->True, |
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328 | AspectRatioFixed->True], |
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329 | |
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330 | Cell["\<\ |
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331 | D'apr\[EGrave]s le travail de Pauline Bart, classe 1BD', mars 1997.\ |
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332 | \>", "Text", |
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333 | Evaluatable->False, |
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334 | AspectRatioFixed->True], |
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335 | |
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336 | Cell[BoxData[{ |
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337 | RowBox[{ |
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338 | RowBox[{"fond", "=", |
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339 | RowBox[{"{", |
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340 | RowBox[{ |
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341 | RowBox[{"RGBColor", "[", |
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342 | RowBox[{"0", ",", "0.8`", ",", "0.8`"}], "]"}], ",", |
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343 | RowBox[{"Rectangle", "[", |
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344 | RowBox[{ |
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345 | RowBox[{"{", |
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346 | RowBox[{ |
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347 | RowBox[{"-", "1"}], ",", |
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348 | RowBox[{"-", "1"}]}], "}"}], ",", |
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349 | RowBox[{"{", |
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350 | RowBox[{"2", ",", "2"}], "}"}]}], "]"}]}], "}"}]}], ";"}], "\n", |
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351 | RowBox[{ |
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352 | RowBox[{"carr\[EAcute]", "=", |
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353 | RowBox[{"{", |
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354 | RowBox[{ |
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355 | RowBox[{"RGBColor", "[", |
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356 | RowBox[{"1", ",", "0.5`", ",", "0"}], "]"}], ",", |
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357 | RowBox[{"Rectangle", "[", |
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358 | RowBox[{ |
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360 | RowBox[{"0", ",", "0"}], "}"}], ",", |
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361 | RowBox[{"{", |
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362 | RowBox[{"1", ",", "1"}], "}"}]}], "]"}]}], "}"}]}], ";"}], "\n", |
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363 | RowBox[{ |
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364 | RowBox[{"triangleGauche", "=", |
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365 | RowBox[{"{", |
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366 | RowBox[{ |
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367 | RowBox[{"RGBColor", "[", |
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380 | ";"}], "\n", |
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382 | RowBox[{"triangleHaut", "=", |
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384 | RowBox[{ |
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387 | RowBox[{"Polygon", "[", |
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388 | RowBox[{"{", |
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389 | RowBox[{ |
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394 | RowBox[{"{", |
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396 | ";"}], "\n", |
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398 | RowBox[{"triangleDroit", "=", |
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403 | RowBox[{"Polygon", "[", |
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404 | RowBox[{"{", |
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410 | RowBox[{"{", |
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412 | ";"}], "\n", |
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430 | ";"}], "\n", |
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436 | ",", "triangleDroit", ",", "triangleBas"}], "}"}]}], ";"}], "\n", |
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441 | AspectRatioFixed->True], |
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442 | |
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443 | Cell[BoxData[{ |
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