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FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " of all the line segments that the turtle traverses on her journey." }], "Text", CellChangeTimes->{{3.398075468306027*^9, 3.398075481368414*^9}}], Cell[CellGroupData[{ Cell["Implementation", "Exercise"], Cell[TextData[{ "We begin with a program", StyleBox[" ", Background->GrayLevel[1]], StyleBox["init", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " that initializes the turtle ", StyleBox["position", FontSlant->"Italic"], " to the origin, the turtle ", StyleBox["direction", FontSlant->"Italic"], " to the unit vector pointing in the ", StyleBox["x", FontSlant->"Italic"], "-direction, and the ", StyleBox["turtlepath", FontSlant->"Italic"], " to the empty list." }], "Text", CellChangeTimes->{{3.39807550973925*^9, 3.398075580548501*^9}, { 3.398075858162675*^9, 3.398075859303315*^9}}], Cell["\<\ To implement this procedure, click you mouse anywhere in the orange box and \ hit SHIFT-RETURN (or hit ENTER if you have a 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changes the direction of the turtle, but leaves its position unchanged. \n\ The", StyleBox[" forward", FontSlant->"Italic"], " command is a function of a single variable, ", StyleBox["distance", FontSlant->"Italic"], ". 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", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "current"], TraditionalForm]]], StyleBox["sin", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(\[Alpha])\n\t", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "new"], TraditionalForm]]], StyleBox[" = v", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["", "current"], " "}], TraditionalForm]]], StyleBox["cos", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(\[Alpha]) + ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["u", "current"], TraditionalForm]]], StyleBox["sin", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(\[Alpha])\nTypically we shall represent these equations in matrix \ form:\n\t(", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["u", "new"], TraditionalForm]]], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "new"], TraditionalForm]]], StyleBox[") = (", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["u", "current"], " "}], TraditionalForm]]], " ", StyleBox["v", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["", "current"], TraditionalForm]]], StyleBox[") ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}], StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}, { RowBox[{"-", StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}], StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], ".\nVectors have both direction and length. ", StyleBox["The", FontVariations->{"CompatibilityType"->0}], StyleBox[" turn", FontSlant->"Italic"], " command changes the turtle's heading by rotating the turtle's direction \ vector, but we can also change the turtle's step size by changing the length \ of the turtle's direction vector. The ", StyleBox["resize", FontSlant->"Italic"], " command is a function of a single variable, the ", StyleBox["scalefactor", FontSlant->"Italic"], ". The ", StyleBox["resize", FontSlant->"Italic"], " command rescales the turtle step size by setting:\n\t", StyleBox["new direction = scalefactor \[Cross] current direction", FontSlant->"Italic"], StyleBox["\nAgain we can write this transformation in matrix notation:\n\t(", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["u", "new"], TraditionalForm]]], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "new"], TraditionalForm]]], StyleBox[") = (", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["u", "current"], " "}], TraditionalForm]]], StyleBox["v", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["", "current"], TraditionalForm]]], StyleBox[") ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["0", FontVariations->{"CompatibilityType"->0}]}, {"0", StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], ".\nSometimes, we may want to simultaneously change the turtle direction and \ the turtle step size -- that is, we may want to combine the ", StyleBox["turn", FontSlant->"Italic"], " and ", StyleBox["resize ", FontSlant->"Italic"], StyleBox["commands.", FontVariations->{"CompatibilityType"->0}], " Since ", StyleBox["turn", FontSlant->"Italic"], " and ", StyleBox["resize", FontSlant->"Italic"], " are both represented by 2\[Times]2 matrices, we can combine these \ operations by matrix multiplication into a single matrix.\n\t", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}], StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}, { RowBox[{"-", StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}], StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["0", FontVariations->{"CompatibilityType"->0}]}, {"0", StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{" ", StyleBox[ RowBox[{ StyleBox["scalefactor", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}]}], FontVariations->{"CompatibilityType"->0}]}], RowBox[{" ", StyleBox[ RowBox[{ StyleBox["scalefactor", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}]}], FontVariations->{"CompatibilityType"->0}]}]}, { RowBox[{ RowBox[{"-", StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]}], StyleBox[" ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}], RowBox[{" ", StyleBox[ RowBox[{ StyleBox["scalefactor", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}]}], FontVariations->{"CompatibilityType"->0}]}]} }], ")"}], TraditionalForm]]], ".\nThe advantage of this approach is that it replaces two matrix \ multiplications by a single matrix multiplication. Notice if we reverse the \ order in which we multiply the two matrices, we get the same result. This \ observation corresponds to the fact that if we ", StyleBox["turn ", FontSlant->"Italic"], "first and then ", StyleBox["resize", FontSlant->"Italic"], ", we get the same result as when we ", StyleBox["resize", FontSlant->"Italic"], " first and then ", StyleBox["turn", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell["Implementation", "Exercise"], Cell[TextData[{ StyleBox["Below are four turtle functions ", Background->GrayLevel[1]], StyleBox["forward", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[", ", Background->GrayLevel[1]], StyleBox["move", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[", ", Background->GrayLevel[1]], StyleBox["turn", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[", and ", Background->GrayLevel[1]], StyleBox["resize", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[". THESE ARE THE ONLY PERMISSIBLE TURTLE COMMANDS. A command to \ move to a specific (x,y) location is not allowed, except in the INIT \ command.\nThe ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command takes as input a single parameter, ", Background->GrayLevel[1]], StyleBox["distance", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". The result of the ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is to move the turtle to its new location and to update \ the list in ", Background->GrayLevel[1]], StyleBox["turtlepath", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[" by inserting the new line segment into ", Background->GrayLevel[1]], StyleBox["turtlepath", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[".\nThe ", Background->GrayLevel[1]], StyleBox["move", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is the same as the ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command, except that the ", Background->GrayLevel[1]], StyleBox["move", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command does not update the list in ", Background->GrayLevel[1]], StyleBox["turtlepath", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[".\nThe ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command takes as input a single parameter, ", Background->GrayLevel[1]], StyleBox["angle, ", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox["measured in degrees (see ", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox["Degree", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox[" in the Help browser). ", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox["The result of the ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is to change the turtle's direction by rotating the \ turtle direction vector by the given ", Background->GrayLevel[1]], StyleBox["angle.\n", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox["The ", Background->GrayLevel[1]], StyleBox["resize", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command takes as input a single parameter, ", Background->GrayLevel[1]], StyleBox["scalefactor", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". ", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox["The result of the ", Background->GrayLevel[1]], StyleBox["resize", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is to change the turtle's step size by scaling the turtle \ direction vector by the given ", Background->GrayLevel[1]], StyleBox["scalefactor.", FontSlant->"Italic", Background->GrayLevel[1]] }], "Text", CellChangeTimes->{{3.398075643780702*^9, 3.398075682066102*^9}}], Cell["\<\ To implement these procedurse, click you mouse anywhere in the orange box and \ hit SHIFT - RETURN (or hit ENTER if you have a Mac).\ \>", "Text", CellChangeTimes->{{3.3980757241109657`*^9, 3.398075726834382*^9}, 3.3980757656883087`*^9}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"forward", "[", "dist_", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"oldposition", "=", "position"}], "}"}], ",", RowBox[{ RowBox[{"position", "=", RowBox[{"oldposition", " ", "+", " ", RowBox[{"dist", RowBox[{"(", "direction", ")"}]}]}]}], ";", RowBox[{"AppendTo", "[", RowBox[{"turtlepath", ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{"oldposition", ",", "position"}], "}"}], "]"}]}], "]"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"move", "[", "dist_", "]"}], ":=", RowBox[{"position", "=", RowBox[{"position", " ", "+", " ", RowBox[{"dist", RowBox[{"(", "direction", ")"}]}]}]}]}], "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"turn", "[", "angle_", "]"}], ":=", RowBox[{"direction", "=", RowBox[{"N", "[", " ", RowBox[{"direction", " ", ".", RowBox[{"(", GridBox[{ { RowBox[{"Cos", "[", RowBox[{"angle", " ", "Degree"}], "]"}], RowBox[{"Sin", "[", RowBox[{"angle", " ", "Degree"}], "]"}]}, { RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"angle", " ", "Degree"}], "]"}]}], RowBox[{"Cos", "[", RowBox[{"angle", " ", "Degree"}], "]"}]} }], ")"}]}], "]"}]}]}], "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"resize", "[", "scalefactor_", "]"}], ":=", RowBox[{"direction", "=", RowBox[{"N", "[", " ", RowBox[{"direction", " ", ".", RowBox[{"(", GridBox[{ {"scalefactor", "0"}, {"0", "scalefactor"} }], ")"}]}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.39807574108724*^9, 3.3980757421838703`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Implementation", "Exercise"], Cell[TextData[{ StyleBox["Later we shall want to display the path of the turtle, so here we \ write two programs ", Background->GrayLevel[1]], StyleBox["displayturtlepath", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " and ", StyleBox["displayturtlecolor", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " ", StyleBox["to display the path of the turtle by displaying the list of line \ segment in the list ", Background->GrayLevel[1]], StyleBox["turtlepath", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". The axes are turned off, since we are not interested in seeing \ the coordinate system.", Background->GrayLevel[1]] }], "Text"], Cell["\<\ Again to implement these procedurse, click you mouse anywhere in the orange \ box and hit SHIFT - RETURN (or hit ENTER if you have a Mac).\ \>", "Text", CellChangeTimes->{{3.3980757241109657`*^9, 3.398075726834382*^9}, { 3.3980757656883087`*^9, 3.398075781587788*^9}, {3.398075813892446*^9, 3.3980758159956512`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"displayturtlepath", "[", "shape_", "]"}], ":=", RowBox[{"Show", "[", RowBox[{ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", ".0025", "]"}], ",", "turtlepath"}], "}"}], "]"}], ",", RowBox[{"Axes", "\[Rule]", "False"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"ImageSize", "\[Rule]", "200"}]}], "]"}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"displayturtlecolor", "[", "color_", "]"}], ":=", RowBox[{"Show", "[", RowBox[{ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Hue", "[", "color", "]"}], ",", RowBox[{"Thickness", "[", ".0035", "]"}], ",", "turtlepath"}], "}"}], "]"}], ",", RowBox[{"Axes", "\[Rule]", "False"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"ImageSize", "\[Rule]", "230"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.394542824417403*^9, 3.394542827637383*^9}, { 3.394542859291737*^9, 3.394542884966013*^9}, {3.394543887656597*^9, 3.394543888151785*^9}, {3.394543921865273*^9, 3.394543924126443*^9}, 3.394547519003417*^9, 3.394547698137738*^9, 3.3945478097337646`*^9, { 3.3980107387756777`*^9, 3.398010767215773*^9}, {3.398010801406927*^9, 3.398010835598792*^9}, 3.39801273521838*^9, 3.3980127684381332`*^9, 3.398013268500551*^9, {3.398013324159527*^9, 3.3980133269508333`*^9}, 3.398013393633882*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Polygons", "Section"], Cell[TextData[{ StyleBox["Using the ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" and ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" commands, you can now make simple shapes by ordering the turtle \ around. For example, the following program will move the turtle along a \ square path:", Background->GrayLevel[1]] }], "Text"], Cell["\<\ {To implement any of the subsequent procedure, click you mouse anywhere in \ the orange box and hit SHIFT-RETURN (or hit ENTER if you have a Mac)}.\ \>", "Text", CellChangeTimes->{{3.39807550973925*^9, 3.398075580548501*^9}, { 3.398075865232924*^9, 3.398075892963479*^9}, {3.398075927752124*^9, 3.398075943731093*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"makesquare", "[", "size_", "]"}], ":=", RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"forward", "[", "size", "]"}], ";", RowBox[{"turn", "[", "90", "]"}]}], ",", RowBox[{"{", "4", "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.394542992867387*^9, 3.394543054073539*^9}, { 3.39454325239855*^9, 3.394543262079855*^9}, 3.39454330236826*^9}], Cell[TextData[{ StyleBox["You can test the ", Background->GrayLevel[1]], StyleBox["makesquare", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" program with the ", Background->GrayLevel[1]], StyleBox["displayturtlepath", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" program to make sure that ", Background->GrayLevel[1]], StyleBox["makesquare", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" really does generate a square. Notice that we start with the ", Background->GrayLevel[1]], StyleBox["init", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[" routine to initialize the position and location of the turtle and \ to clear out any previous turtle paths. 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One informal definition is that fractals are \ simply ", Background->GrayLevel[1]], StyleBox["recursion made visible", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". Here we will write simple recursive turtle programs to \ generate several interesting kinds of fractals, including gaskets, \ snowflakes, and general bump fractals.", Background->GrayLevel[1]] }], "Text"], Cell[CellGroupData[{ Cell["Gaskets", "Exercise"], Cell[TextData[{ StyleBox["The first type of fractal we shall investigate is called a ", Background->GrayLevel[1]], StyleBox["gasket", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". 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Don't panic. You don't have to give \ the turtle thousands of ", StyleBox["forward", FontSlant->"Italic"], " and ", StyleBox["turn", FontSlant->"Italic"], " commands to generate this figure. A simple recursive program will do the \ job.\nThe key observation is that the big gasket is made up of three \ identical smaller gaskets. Conceptually this gasket is almost the same as a \ spiral. A spiral is a line joined to a smaller spiral. A Sierpinski gasket \ is just three smaller Sierpinski gaskets joined together. ", StyleBox["Again this definition of a gasket would be circular, if we did not \ have a base case at which to stop. So we say that a very small gasket is \ just a small triangle.\nThe structure of the ", Background->GrayLevel[1]], StyleBox["makesierp", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[" program must be something like the following:\n\tTo make a large \ gasket, \n\t\tMake a smaller gasket at one of the corners.\n\t\tThen move to \ another corner and make another small gasket.\n\t\tThen move to another \ corner and make another small gasket.\n\t\tThen return to the turtle to its \ starting position.\nThe moves can be made with ", Background->GrayLevel[1]], StyleBox["move", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" and ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" commands; the rescaling is achieved with ", Background->GrayLevel[1]], StyleBox["resize", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" commands. To make the smaller gaskets require three recursive \ calls to the ", Background->GrayLevel[1]], StyleBox["makesierp", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[" program. ", Background->GrayLevel[1]], StyleBox["Form follows function!", FontWeight->"Bold", Background->GrayLevel[1]], StyleBox[" By the way, why do you think you have to return to the turtle \ to the state in which it started?\nBelow is the ", Background->GrayLevel[1]], StyleBox["Mathematica", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" program for ", Background->GrayLevel[1]], StyleBox["makesierp", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[". 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C7vq5JPi/J00NGDGPalECbs3o6MbO9Tldl3a1YC0tnypYQ7KkVIddpBXhJPV cndaXO4EyvVhf7k5CFH0TIqzMGvFWZiNtwN8zGxxSnGqmjIrt/nA7y49Zh2d Z49IoJ30SHSEh+i1XD8gsqYCsrUC8HJ0wYB5vFfhIZ8LFHJ8ecS8ElX/uwzQ 0fUdCyAtlTvu3Rsp3Q9iuMBk08n67EIUwwUrGdzsopQ2x8vwHC+xcbxKh6A7 3KaKdC0baDA27bxcKOotScWtZ9wi2lFWcetZiNmKW898HbINy9bF9ZSi8GRz E6pts5n3dS2mzqYlCbR8icREeIie3PvxXAZzvBU5zWuJ07z1edNciNuyuMFE i3qzt3ozYP0PzZMjmLoUWv+jt5mndam54Rr1e07Wv08DW84uI19br8iIZhHv UnTvq+xFWWy6rD65RthOpOQok7cKIQ1tk9IZT6FtSy5n0VAxN3wEKIU0GIzs E8D7qRRlUXM+9kdXqkjG+JnZ/MimLOVzXX7v0LcdkHO7NLQLqLJU5GU9m3Cp 53/lZPpG\ \>"], ImageRangeCache->{{{0, 359}, {414.625, 0}} -> {0, -8, 0, 0}}], "\nThere is a simple trick for generating the Koch snowflake and other \ snowflakes as well. Consider the following simple bump: \n", Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .28868 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 4.7619 0.00687322 4.7619 [ [ 0 0 0 0 ] [ 1 .28868 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .28868 L 0 .28868 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash .02381 .00687 m .34127 .00687 L .5 .2818 L .65873 .00687 L .97619 .00687 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 83}, ImageMargins->{{184.75, 0}, {0, 421.5}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["CompressedBitmap", "\<\ eJydmPtLU3EYh98xREokQiRCokVEhIhIhISIiJRESEREhIgchrgRYix/yltq ampqat7zfst76R+4dvaOCnlA+BzGOd89z3M++30NQUeipS3oSMaDWH0qeJ9I xj/EnranMigaMYskzKw1ZuE5nTnmbunMJ50MbyH5d74aPlJh0v73VOCqM/uI mr+ffcnagH30vAvUO2CdUt4NKgnMZ60HVCuwbinvBdUCrEfKP4GKA+uV8j5Q ATCftX5QzcD6pHwAVBOwfin/DKoR2ICUD4J6C8xnbQjUG2CDUj4M6jWwISn/ AuoVsGEpHwH1EpjP2iioF8BGpHwMVAOwUSn/Cuo5sDEpHwf1DJjP2gSoemDj Uj4J6gmwCSn/BqoO2KSUT4GqBeazNg2qBtiUlM+AqgY2LeXfQVUBm5HyWVCP gfmszYGqBDYr5fOgHgGbk/IFUA+BzUv5IqgKYD5rS6DKgS1K+TKoMmBLUv4D VCmwZSlfAfUAmM/aKqj7wFakfA3UPWCrUr4O6i6wNSnfAHUHmM/aJqjbwDak fAvULWCbUr4NqgTYlpTvgLoJzGdtF9QNYDtSvgeqGNiulP8EVQRsT8r3QV0H 5rN2AOoasH0pPwRVCOxAyo9AFQA7lPJjUFeA+aydgMoHdizlp6DygJ1I+S9Q UWCnUv479/P/q6zIXGd+OrvwdnidX6rO4Tdzg9kv8EePRf4AZQebew==\ \>"], ImageRangeCache->{{{0, 359}, {102.75, 0}} -> {-0.00590658, -0.00144368, 0.000737512, 0.000737512}}], "\nTo generate one side of the Koch snowflake, replace every line in this \ figure by this bump, scaled to 1/3 its size. Now you have:\n", Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .28868 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 3.1746 0.00687322 3.1746 [ [ 0 0 0 0 ] [ 1 .28868 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .28868 L 0 .28868 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash .02381 .00687 m .12963 .00687 L .18254 .09852 L .23545 .00687 L .34127 .00687 L .39418 .09852 L .34127 .19016 L .44709 .19016 L .5 .2818 L .55291 .19016 L .65873 .19016 L .60582 .09852 L .65873 .00687 L .76455 .00687 L .81746 .09852 L .87037 .00687 L .97619 .00687 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 83}, ImageMargins->{{189.5, 0}, {0, 536.438}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["CompressedBitmap", "\<\ eJzNmftLFFEYhs+yRJREREREiBuxWUSZmqalphZWhEREREjIsIi7iBjmb3m/ 3/obMjVNTS39A7eZOYPa8szsN18EOyy7x/d95n1n9jK759jpDGV7+p2hXMZJ PRt0PmZzmU+pjoFBV0omjElkjTG9KeON8+4wuMu7t3zOu/OU4/FZ7+GGh6S9 UaU3GjgapY/cMotX+Q9JYzP9IHNdqPX/w753bf09sK6BlgatT8hRXlVJ1FeD VQFaJWg5IUd59rRNDVjloN0U1hNHedXx6m+B1ivkIuprwboK2m3QeoQc5dWU RP19sK6Adge0jJCjvNqSqK8D6zJodHV0hBzl2dM29WBdAo0uT1RPHOXVxaun y1O3kIuofwDWxdB3avF64iivPl49fUA/CLmI+gawLoS+VYrXE0d59rRNI1jn Q4/2b61LyFFeQ7x6epHeC7mI+odgnQvFi9cTR3mN8erpWXon5CLqH4FVBhod JtUTF54nrifurZCLqG8C6wxoxFG9NM8epmkG6zRoLaC9EXKU11QS9YSfAq0V tNdCjvKaS6L+MVhJ0NpBeyXkKK8lXv0T0DqFXER9a/D0nLR8w906hJVPIzLa 7KitYDdvew5RL0ErPAxva/+vsfQSUuwLIRc8CV+EEcsWXxaeyJIKXxK+uIsW XxS++xdU+AJY9NGeV+HzYNFlc06Fz4FFX0mzKnwWLPq6n7H4DFj022hahU+D Rb87p1T4FFj0o3pShU+C1Q3ahMUnwKLZ3bgKHweLZs5jKnwMLFqVGFXho2DR is+ICh8Bi5azhi0+DBat/X1W4f7D8XKtPwqKV2AvWkr9avFvYNHa44oKXwWr AjQba9bAorW4VRW+Dhatda2p8O9g0VrSugrfAIuWimys2QSL1lY2VPgPsGjt YlOFb4FFSxM21myDRXP5LRW+AxbNlbdV+E+waC66o8J3waKppo01e2DR3GxX he+DRXOfPRX+CyyaW+yr8N9B/UnLN9ztwI4OCvb2tsOi1iF0BoH+H/CPOpP4 A8QS52c=\ \>"], ImageRangeCache->{{{0, 359}, {102.75, 0}} -> {-0.00885988, -0.00216552, 0.00110627, 0.00110627}}], "\nRepeat this process again and again until the bumps are very small -- \ that is, replace each line by a bump 1/3 the size of the line. 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", Background->GrayLevel[1]], StyleBox["makekoch", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " for this part of the Koch snowflake. Easy. There are four recursive \ calls.\n", StyleBox["\tTo make a large snowflake, \n\t\tMake a smaller snowflake along \ one of the edges of the original figure.\n\t\tThen move to the next edge and \ make another small snowflake.\n\t\tThen move to the next edge and make \ another small snowflake.\n\t\tThen move to the last edge and make another \ small snowflake.\nBelow is the ", Background->GrayLevel[1]], StyleBox["Mathematica", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" program for ", Background->GrayLevel[1]], StyleBox["makekoch", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[". Notice that the base case is simply a straight line. 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