[419] | 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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| 12 | NotebookDataPosition[ 145, 7] |
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| 13 | NotebookDataLength[ 36391, 1211] |
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| 14 | NotebookOptionsPosition[ 33151, 1099] |
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| 15 | NotebookOutlinePosition[ 34539, 1147] |
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| 16 | CellTagsIndexPosition[ 34496, 1144] |
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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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| 77 | RowBox[{"-", "1"}], ",", |
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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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| 93 | RowBox[{"1", ",", "1", ",", "0"}], "]"}], ",", |
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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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| 99 | RowBox[{ |
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| 100 | RowBox[{"-", "0.5"}], ",", |
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| 101 | RowBox[{"-", "0.5"}]}], "}"}], ",", |
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| 102 | RowBox[{"{", |
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| 103 | RowBox[{"0.5", ",", "0.5"}], "}"}]}], "}"}], "]"}], ",", |
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| 104 | RowBox[{"Line", "[", |
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| 105 | RowBox[{"{", |
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| 106 | RowBox[{ |
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| 107 | RowBox[{"{", |
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| 108 | RowBox[{"0.5", ",", |
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| 109 | RowBox[{"-", "0.5"}]}], "}"}], ",", |
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| 110 | RowBox[{"{", |
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| 111 | RowBox[{ |
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| 112 | RowBox[{"-", "0.5"}], ",", "0.5"}], "}"}]}], "}"}], "]"}]}], |
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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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| 359 | 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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| 368 | RowBox[{"0.1`", ",", "0.1`", ",", "0.7`"}], "]"}], ",", |
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| 369 | RowBox[{"Polygon", "[", |
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| 370 | RowBox[{"{", |
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| 371 | RowBox[{ |
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| 372 | RowBox[{"{", |
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| 373 | RowBox[{ |
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| 374 | RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", |
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| 375 | RowBox[{"{", |
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| 376 | RowBox[{ |
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| 377 | RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", |
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| 378 | RowBox[{"{", |
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| 379 | RowBox[{"0", ",", "1"}], "}"}]}], "}"}], "]"}]}], "}"}]}], |
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| 380 | ";"}], "\n", |
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| 381 | RowBox[{ |
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| 382 | RowBox[{"triangleHaut", "=", |
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| 383 | RowBox[{"{", |
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| 384 | RowBox[{ |
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| 385 | RowBox[{"RGBColor", "[", |
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| 386 | RowBox[{"0.1`", ",", "0.1`", ",", "0.7`"}], "]"}], ",", |
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| 387 | RowBox[{"Polygon", "[", |
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| 388 | RowBox[{"{", |
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| 389 | RowBox[{ |
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| 390 | RowBox[{"{", |
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| 391 | RowBox[{"0", ",", "2"}], "}"}], ",", |
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| 392 | RowBox[{"{", |
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| 393 | RowBox[{"1", ",", "2"}], "}"}], ",", |
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| 394 | RowBox[{"{", |
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| 395 | RowBox[{"1", ",", "1"}], "}"}]}], "}"}], "]"}]}], "}"}]}], |
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| 396 | ";"}], "\n", |
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| 397 | RowBox[{ |
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| 398 | RowBox[{"triangleDroit", "=", |
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| 399 | RowBox[{"{", |
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| 400 | RowBox[{ |
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| 401 | RowBox[{"RGBColor", "[", |
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| 402 | RowBox[{"0.1`", ",", "0.1`", ",", "0.7`"}], "]"}], ",", |
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| 403 | RowBox[{"Polygon", "[", |
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| 404 | RowBox[{"{", |
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| 405 | RowBox[{ |
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| 406 | RowBox[{"{", |
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| 407 | RowBox[{"1", ",", "0"}], "}"}], ",", |
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| 408 | RowBox[{"{", |
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| 409 | RowBox[{"2", ",", "0"}], "}"}], ",", |
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| 410 | RowBox[{"{", |
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| 411 | RowBox[{"2", ",", "1"}], "}"}]}], "}"}], "]"}]}], "}"}]}], |
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| 412 | ";"}], "\n", |
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| 413 | RowBox[{ |
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| 414 | RowBox[{"triangleBas", "=", |
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| 415 | RowBox[{"{", |
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| 416 | RowBox[{ |
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| 417 | RowBox[{"RGBColor", "[", |
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| 418 | RowBox[{"0.1`", ",", "0.1`", ",", "0.7`"}], "]"}], ",", |
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| 419 | RowBox[{"Polygon", "[", |
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| 420 | RowBox[{"{", |
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| 421 | RowBox[{ |
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| 422 | RowBox[{"{", |
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| 423 | RowBox[{"0", ",", "0"}], "}"}], ",", |
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| 424 | RowBox[{"{", |
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| 425 | RowBox[{"0", ",", |
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| 426 | RowBox[{"-", "1"}]}], "}"}], ",", |
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| 427 | RowBox[{"{", |
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| 428 | RowBox[{"1", ",", |
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| 429 | RowBox[{"-", "1"}]}], "}"}]}], "}"}], "]"}]}], "}"}]}], |
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| 430 | ";"}], "\n", |
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| 431 | RowBox[{ |
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| 432 | RowBox[{"planelle", "=", |
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