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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1278313, 58302]*) (*NotebookOutlinePosition[ 1278999, 58327]*) (* CellTagsIndexPosition[ 1278955, 58323]*) (*WindowFrame->Normal*) Notebook[{ Cell["MATHEMATICA IN ACTION", "Subtitle"], Cell[CellGroupData[{ Cell["The Traveling Salesman and the Turtle", "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Combining a two-dimensional turtle with Mathematica\[CloseCurlyQuote]s \ three-dimensional capabilities gives us a new way to look at space-filling \ curves."], "Subtitle", Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ Turtle.m package and code for figures to be added to the Electronic \ Supplement\ \>", "Subsection", FontColor->RGBColor[1, 0, 0]], Cell[CellGroupData[{ Cell["The Turtle.m Package ", "Subsubsection"], Cell["\<\ Off[General::spell1]; Remove[FractalTurtle, StartDirection, TerminalSubstitution, TurtleStep, StartPosition, TurtlePath] BeginPackage[\"Turtle`\", {\"Utilities`FilterOptions`\"}];\ \>", "Input"], Cell["\<\ CurveDepth::usage = \"CurveDepth is an option to TravelingSalesmanTour that \ specifies the depth of the space-filling curve approximation to use when \ estimating the position of points on the curve.\"; FractalTurtle::usage = \"FractalTurtle[r, s, d, a] creates (in \\\"TurtlePath\ \\\") and shows the list of points that a turtle would make if it starts from \ s and then applies the rewrite rule d times. The size of the turns is set by \ a (assumed to be in degrees). Strings can involve F, B, +, -, and i, which \ instruct the turtle to move forward, move back, turn left, turn right, and \ roll over, respectively. The characters [ (push) and ] (pop) can also be used \ to store a location in a stack and pop back to it.\"; ShowPointLabels::usage = \"ShowPointLabels is an option to \ TravelingSalesmanTour that causes the points to be large, with labels inside \ them.\"; LineStyle::usage = \"LineStyle is an option to TravelingSalesmanTour that \ specifies the style of the line forming the tour.\"; PointSwitching::usage = \"PointSwitching is an option to \ TravelingSalesmanTour that follows up the space-filling heuristic with a \ three passes of a check on switches of consecutive pairs and triples to find \ improvements.\"; PointStyle::usage = \"PointStyle is an option to TravelingSalesmanTour that \ specifies the style of the points.\"; RandomPoints::usage = \"RandomPoints[n] gives n random points in the unit \ square.\"; ShowDistance::usage = \"ShowDistance is an option to TravelingSalesmanTour \ that asks the distance of the tour to be used as a plot label.\"; ShowLabeledPoints::usage = \"ShowLabeledPoints[p] shows the points in p in a \ large size with random soft colors and the labels inside the points.\"; ShowIndices::usage = \"ShowIndices is an option to TravelingSalesmanTour that \ causes the index list to be returned.\"; StartPosition::usage = \"StartPosition is an option to FractalTurtle that \ specifies the point at which the turtle starts its journey.\"; StartDirection::usage = \"StartDirection is an option to FractalTurtle that \ specifies the initial direction the turtle faces. It must be a unit \ vector.\"; TerminalSubstitution::usage = \"TerminalSubstitution is an option to \ FractalTurtle that causes one additional substitution (or many if it is set \ to be a list) to be made after the specified depth is reached.\"; TravelingSalesmanTour::usage = \"TravelingSalesmanTour[p] uses a Hilbert \ space-filling curve to come up with a traveling salesman tour through the \ given points that is, on average, not too far from optimal.\"; TSPDistance::usage = \"TSPDistance[data, tour] gives the sum of the \ straight-line lengths corresponding to a trip through the points in data in \ the order determined by tour. The last entry in tour must be the same as the \ first.\"; TurtleStep::usage = \"TurtleStep is an option to FractalTurtle that specifies \ the length of each turtle step.\"; TurtlePath::usage = \"TurtlePath is where the points of the turtle's path are \ stored by FractalTurtle.\"; TSPDistance::bddata = \"A traveling salesman tour must have its last point \ being the same as its first point.\"; SecondaryStepSize::usage = \"SecondaryStepSize is an option to FractalTurtle \ that specifies the step size for a secondary set of moves, corrsponding to \ the instructions \\\"f\\\" and \\\"b\\\".\";\ \>", "Input"], Cell["\<\ Begin[\"`Private`\"]; \ \>", "Input"], Cell["\<\ Options[FractalTurtle] = {StartPosition -> {0, 0}, StartDirection -> {1, 0}, TurtleStep -> 1, TerminalSubstitution -> None, PlotStyle -> AbsoluteThickness[0.4]}; FractalTurtle[r_, s_String, depths_List, angle_, opts___] := Show[GraphicsArray[ Map[FractalTurtle[r,s,#,angle,opts, DisplayFunction->Identity]&, depths, {Depth[depths]-1}]]] FractalTurtle[rewrite_, start_String, depth_Integer, angle_, opts___] := Module[{ turtle, sdir, scale, i = 1, rotateleft, rotateright, stack,gg}, inds = {1}; {startingPt, dir, ts, term, sty, secsize} = {StartPosition, StartDirection, TurtleStep, TerminalSubstitution, PlotStyle, SecondaryStepSize} /. {opts} /. Options[FractalTurtle]; TurtlePath = {last = startingPt}; stack = {{startingPt, dir}}; rotateleft = N[ {{Cos[angle Degree], -Sin[angle Degree]}, {Sin[angle Degree], Cos[angle Degree]}}]; rotateright = Transpose[rotateleft]; If[!ListQ[sty], sty = {sty} /. {Automatic} ->{}]; s = Nest[StringReplace[#, rewrite]&, start, depth]; If[term =!= None, s = StringReplace[s, term]]; Characters[s] /. { \"+\" :> (dir = rotateleft . dir;), \"-\" :> (dir = rotateright . dir;), \"F\" :> (i++;TurtlePath = {TurtlePath, last += ts*dir};), \"B\" :> (i++;TurtlePath = {TurtlePath,last -= ts*dir};), \"f\" :> (i++;TurtlePath = {TurtlePath, last += secsize*dir};), \"b\" :> (i++;TurtlePath = {TurtlePath,last -= secsize*dir};), \"i\" :> ({rotateleft, rotateright} = {rotateright, rotateleft};), \"[\" :> (stack = Flatten[{{{last, dir}},stack},1];), \"]\" :> (i++; dir = stack[[1,2]]; TurtlePath = {TurtlePath, last=stack[[1,1]]}; stack = Rest[stack]; inds = Flatten[{inds,i}];)}; TurtlePath = Partition[Flatten[TurtlePath], 2]; gg = Graphics[If[inds == {1}, {Append[sty, Line[TurtlePath]]}, TurtlePath = (TurtlePath[[Range[#[[1]], #[[2]]-1]]] & /@ Partition[inds, 2, 1]); Join[sty, Line /@ TurtlePath]], FilterOptions[Graphics, opts], AspectRatio -> Automatic, PlotRange -> All]; {{x0,x1}, {y0,y1}} = PlotRange[gg]; Show[gg, Evaluate@FilterOptions[Graphics, opts], PlotRange -> {{x0,x1}+{-.1, .1} (x1-x0), {y0 ,y1} + {-.1, .1} (y1-y0)}, Frame->True, FrameTicks->None]] dC = Compile[{x0,y0,x1,y1},Sqrt[(x0-x1)^2+(y0-y1)^2]]; distance[points_] := Plus @@ Apply[dC, Flatten /@ Partition[points,2,1],{1}] TSPDistance[data_, tour_] := (If[Last[tour] != First[tour], Message[TSPDistance::bddata]; Return[HoldForm[TSPDistance[tour]]]]; distance[data[[tour]]]) position[{x_, y_}, k_Integer] := Which[ x < 0.5 && y < 0.5, \tMod[0.875 + position[{1-2 x, 1- 2y}, k-1]/4, 1], x < 0.5 && y >= 0.5, 0.125 + position[{2y-1,1-2x},k-1]/4, x >= 0.5 && y >= 0.5, 0.375 + position[ {2x-1,2y-1},k-1]/4, True, 0.625 + position[{1-2y,2x-1},k-1]/4 ] position[{x_, y_}, 0] := Which[ x < 0.5 && y < 0.5, \t 0, x < 0.5 && y >= 0.5, 0.25, x >= 0.5 && y > 0.5, 0.5, True, \t\t\t\t 0.75] tsp[pointList_, k_Integer] := First@Thread@Sort[ Map[{#, position[#, k]}&, pointList], #1[[2]] < #2[[2]] &] \t\t\t improve[path_] := Module[{ path1 = path, len = Length[path]}, adj[s_] := Mod[s,len] /. {0->len}; Do[If[distance[path1[[adj@{i-1,i,i+1,i+2}]]]> distance[path1[[adj@{i-1,i+1,i,i+2}]]], path1 = ReplacePart[ReplacePart[path1, path1[[adj[i+1]]],i], path1[[i]],adj[i+1]]], {i, len}]; Do[If[distance[path1[[adj@Range[i-1,i+3]]]]> distance[path1[[adj@{i-1,i+2,i+1,i,i+3}]]], path1 = ReplacePart[ReplacePart[path1, path1[[adj[i+2]]],i], path1[[i]],adj[i+2]]], {i, len}]; path1] \t\t\t Options[TravelingSalesmanTour] = { CurveDepth -> 10, PointSwitching->True, ShowDistance->True, PointStyle->PointSize[0.015], LineStyle -> {}, ShowIndices -> False, ShowPointLabels->False}; TravelingSalesmanTour[pts_, opts___] := Module[ {dep, impQ, disQ, tour, tourQ}, {dep, impQ, disQ, lsty, psty, tourQ, labQ} = {CurveDepth, PointSwitching, ShowDistance, LineStyle, PointStyle, ShowIndices,ShowPointLabels} /. {opts} /. Options[TravelingSalesmanTour]; If[!ListQ[lsty], lsty = {lsty}]; If[!ListQ[psty], psty = {psty}]; tour = tsp[pts,dep]; If[impQ, tour = Nest[improve,tour,3]]; ss=Show[Graphics[{Thickness[0.0005], Append[lsty, Line[realtour=Append[tour, First[tour]]]], \tIf[labQ, {{AbsolutePointSize[16], {RGBColor @@ Array[Random[Real, {0.4,1}]&,3], Point[#]}& /@ pts}, MapIndexed[Text[#2[[1]], #1+{.005,0}]&, pts]}, \tAppend[psty, Point /@ pts]]}], Evaluate[FilterOptions[Graphics, opts]], PlotLabel->If[disQ, StringForm[\"Length is ``\", NumberForm[distance[realtour],6]], None], PlotRange->{{-0.1,1.1},{-0.1,1.1}}, AspectRatio->Automatic, DefaultFont->{\"Times-Bold\",12}]; If[!tourQ, ss, {ss, Flatten[ Position[pts, #]& /@ Append[tour, First[tour]]]}]] RandomPoints[n_] := Array[{Random[], Random[]}&, n] ShowLabeledPoints[pts_] := Show[Graphics[{ {AbsolutePointSize[16], {RGBColor @@ Array[Random[Real, {0.4,1}]&,3], Point[#]}& /@ pts}, MapIndexed[Text[#2[[1]], #1+{.005,0}]&, pts]}], AspectRatio->Automatic, 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CellMargins->{{Inherited, Inherited}, {18, Inherited}}, Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True, FontFamily->"Helvetica"], Cell[TextData[{ "Readers familiar with space-filling curves may well be intrigued by the \ problem of using ", StyleBox["Mathematica", FontSlant->"Italic"], " to generate them. These objects are complicated, and it is not \ immediately clear what the best approach is. In this column I will show how \ to simulate a turtle and use it to generate approximations to a space-filling \ curve. As often happens, ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s superb 2- and 3-dimensional graphics allow us to \ generate some new ways of visualizing these monstrous objects. Of course, \ there is ample opportunity to learn some tricks to save time and memory." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["1. A Literate Turtle", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The turtle, an idealized traveler on a computer screen, will be familiar \ to readers who have experience with the LOGO language: it is an abstract \ object that wanders about in the plane. We think of it as occupying a point \ and able to accomplish certain simple tasks: a step forward or backward, or a \ right or left turn through a fixed angle. Because ", StyleBox["Mathematica", FontSlant->"Italic"], " generates entire images at once, we cannot easily show the real-time \ motion of the turtle. Instead, we will generate a complete sequence of moves, \ and then show the turtle's path as a sequence of line segments.\n\nOur \ approach is based on the idea of string rewriting (also known as \ \[OpenCurlyDoubleQuote]L-systems\[CloseCurlyDoubleQuote], after A. \ Lindenmayer, who used them to generate images that simulate plant growth; see \ [Prusinkiewicz and Lindenmeyer 1990] and chapter 7 of [Wagon 1998]). The idea \ is to attach motions to certain characters: ", StyleBox["\"F\"", FontFamily->"Courier", FontWeight->"Bold"], " means forward, ", StyleBox["\"B\"", FontFamily->"Courier", FontWeight->"Bold"], " means backward, and ", StyleBox["\"+\"", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["\"-\"", FontFamily->"Courier", FontWeight->"Bold"], " denote a right and left turn, respectively. We can start with a simple \ string, such as ", StyleBox["\"F\"", FontFamily->"Courier", FontWeight->"Bold"], ", and then repeatedly apply a rule such as ", StyleBox["\"F\"\[MediumSpace]\[Rule]\[MediumSpace]\"F+F--F+F\"", FontFamily->"Courier", FontWeight->"Bold"], ". Such a substitution is simple to implement using ", StyleBox["StringReplace", FontFamily->"Courier", FontWeight->"Bold"], "; we very quickly generate long strings." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Nest[StringReplace[#, "\" \[Rule] "\"]\ &, "\", 3] \)], "Input", GroupPageBreakWithin->Automatic, AspectRatioFixed->True], Cell[BoxData[ \("F+F--F+F+F+F--F+F--F+F--F+F+F+F--F+F+F+F--F+F+F+F--F+F--F+F--F+F+F+F--\ F+F--F+F--F+F+F+F--F+F--F+F--F+F+F+F--F+F+F+F--F+F+F+F--F+F--F+F--F+F+F+F--F+\ F"\)], "Output"] }, Open ]], Cell[TextData[{ "The next step would be to have our turtle scan the characters of the \ string and perform the indicated moves. On the face of it, this seems simple. \ Break down the string into its characers and have the turtle, as it scans \ each character, either add a point to a list of points it has visited (if it \ sees an ", StyleBox["\"F\"", FontFamily->"Courier", FontWeight->"Bold"], " or ", StyleBox["\"B\"", FontFamily->"Courier", FontWeight->"Bold"], ") or change its heading if it sees ", StyleBox["\"+\"", FontFamily->"Courier", FontWeight->"Bold"], " or ", StyleBox["\"-\"", FontFamily->"Courier", FontWeight->"Bold"], ". But there are a couple of complications.\n\nRecall that ", StyleBox["AppendTo", FontFamily->"Courier", FontWeight->"Bold"], " is a natural way to add a point to a list. This command has the advantage \ that we need not know in advance how long the list will ultimately be. This \ is indeed convenient, but ", StyleBox["AppendTo", FontFamily->"Courier", FontWeight->"Bold"], " turns out to be quite slow, so much so that it should never be used to \ build long lists. It is much better to build the list as a massively nested \ object and then flatten it appropriately. Here is an example. I leave it the \ reader to compare this technique to others on examples using thousands of \ points. I believe this is the fastest way to do it." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(l = {}; \)\), \(\(Do[l = {l, {Random[], Random[]}}, {10}]; \)\), \(Partition[Flatten[l], 2]\)}], "Input"], Cell[BoxData[ \({{0.739265858791156205`, 0.443989009690431668`}, { 0.185754443835868237`, 0.400523458839680479`}, { 0.948026168304630267`, 0.322357521024391457`}, { 0.261487374363617419`, 0.218652038971364026`}, { 0.127952998810742357`, 0.894639402061352484`}, { 0.0500878507086698121`, 0.190500037952679655`}, { 0.627320799897264791`, 0.654159123527496699`}, { 0.175667333608784197`, 0.925753768141190391`}, {0.95548840953062335`, 0.307522119698182017`}, {0.272591728821405743`, 0.302793547750544966`}}\)], "Output"] }, Open ]], Cell[TextData[{ "With this preamble, the details are straightforward. We maintain the \ current direction in ", StyleBox["dir", FontFamily->"Courier", FontWeight->"Bold"], " and the current path-end in ", StyleBox["last", "Input"], "; ", StyleBox["path", FontFamily->"Courier", FontWeight->"Bold"], " starts out as the empty set and grows appropriately. The key pieces of \ the code are the definition of ", StyleBox["turtle", FontFamily->"Courier", FontWeight->"Bold"], ", which takes a string, looks at the characters, and carries out the \ proper instructions. In this example we assume that the turtle starts at the \ origin, turns through 60\:02da, and has steps having length 1." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(turtle[s_String]\ := \ \((\[Theta] = 60 \[Degree]; \n\ \ \ dir = {1, 0}; \n\ \ \ rotateleft = {{Cos[\[Theta]], \(-Sin[\[Theta]]\)}, {Sin[\[Theta]], Cos[\[Theta]]}}; \n\ \ \ rotateright = Transpose[rotateleft]; \n \ \ \ last = {0., 0.}; \n\ \ \ path = {last}; \n\t Characters[s] /. {\n\t\t\ "\<+\>" \[RuleDelayed] \((dir = rotateleft.dir; )\), \n\t\t\ "\<-\>" \[RuleDelayed] \((dir = rotateright.dir; )\), \n\t\t\ "\" \[RuleDelayed] \((path = {path, last += dir}; )\), \n \t\t\ "\" \[RuleDelayed] \((path = {path, last -= dir}; )\)}; \t\t\t\t\t\t\n\tpath = Partition[Flatten[path], 2])\)\)], "Input", GroupPageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Now we start with the string ", StyleBox["\"F++F++F\"", FontFamily->"Courier", FontWeight->"Bold"], ", which instructs the turtle to trace out a counterclockwise equilateral \ triangle. When we then use the rewrite rule ", StyleBox["\"F\"\[MediumSpace]\[Rule]\[MediumSpace]\"F-F++F-F\"", FontFamily->"Courier", FontWeight->"Bold"], ", we are asking the turtle to replace each single step by a four-step \ walk, as illustrated in Figure 1. Of course, repeated application of this \ recursive idea leads to the Koch snowflake, perhaps the most famous of the \ family of curves known as fractals." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ FIGURE 1: A graphic view of the rewriting step that produces a \ Koch snowflake.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .20968 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 9 scalefont setfont % Scaling calculations 0.483871 0.16129 0.16129 0.16129 [ [ 0 0 0 0 ] [ 1 .20968 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .20968 L 0 .20968 L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 0 0 ] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript 0 g .5 Mabswid [ ] 0 setdash .25806 .16129 m .37097 .16129 L s % Start of user PostScript -1.4 0 -0.7 0 MAarrowhead1 % End of user PostScript 2 Mabswid 0 .16129 m .16129 .16129 L s .48387 .16129 m .64516 .16129 L .72581 .02161 L .80645 .16129 L .96774 .16129 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 60.375}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgol00;go00<<02?o00<<03[o002mo`03300So`03300jo`00_?l0 0``09Ol00``0>Ol00;_o00<<02Oo00<<03So001Lo`03001Lo`03300Wo`03300ho`00G?l705Oo00<< 02Wo00<<03Oo001Lo`/0Dol00``0:Ol00``0=ol002l0;Ol?01oo<@0/oc402?l002l06olQ01oo<@0/ oc402?l005co302ho`00G?l80;co001Lo`040030o`00oolQo`00oolQo`00oolQo`00oolQo`00oolQ o`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00\ \>"], ImageRangeCache->{{{0, 287}, {59.375, 0}} -> {-3.000666, -0.978468, 0.021532, 0.021532}}], Cell["\<\ To continue, we define the starting strong and the rewrite rule, \ and then iterate to get longer strings to feed to the turtle.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(rewrite = "\" \[Rule] "\"; \)\), \(start = "\"; \n Show[Graphics[ Line[turtle[Nest[StringReplace[#, \ rewrite]\ &, start, 1]]], \n\t\t AspectRatio \[Rule] Automatic]]; \n\)}], "Input", GroupPageBreakWithin->Automatic, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.1547 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 9 scalefont setfont % Scaling calculations 0.02381 0.31746 0.302422 0.31746 [ [ 0 0 0 0 ] [ 1 1.1547 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1.1547 L 0 1.1547 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash .02381 .30242 m .34127 .30242 L .5 .02749 L .65873 .30242 L .97619 .30242 L .81746 .57735 L .97619 .85228 L .65873 .85228 L .5 1.12721 L .34127 .85228 L .02381 .85228 L .18254 .57735 L .02381 .30242 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{249.375, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgol0 0c`0G?l005go00So00Go007o00o`03?03Io`03?00= o`003ol00c`0f?l00c`03Ol000oo00o`004?l00c`0eOl0 0c`03ol0017o00 o`003ol00c`0f?l00c`03Ol000ko00o`03?03Jo`03?003o00?o00Oo00oe001_l005Go00ol00c`0G?l005ko00"], ImageRangeCache->{{{0, 248.375}, {287, 0}} -> {-0.075002, -0.940293, 0.012632, 0.012632}}] }, Open ]], Cell["\<\ FIGURE 2: A single application of the rewriting rule leads to \ instructions that cause the turtle to trace out a six-pointed star.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The electronic supplement contains a fleshier version of this code, called \ ", StyleBox["FractalTurtle", FontFamily->"Courier", FontWeight->"Bold"], ". 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These \ images can be generated by judicious use of two string rewriting rules.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Evidently, we need a recursive scheme that turns each U-shape, in \ any of four orientations, into a complicated shape made up of four linked \ U-shapes, as shown in Figure 5.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ FIGURE 5: A graphic view of the main recursive step to construct a \ space-filling curve.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .54113 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 9 scalefont setfont % Scaling calculations 0.460833 0.164914 0.012884 0.164914 [ [ 0 0 0 0 ] [ 1 .54113 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .54113 L 0 .54113 L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 0 0 ] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript 0 g .5 Mabswid [ ] 0 setdash .02381 .05411 m .02381 .21903 L .18872 .21903 L .18872 .05411 L s .48145 .0335 m .64636 .0335 L .64636 .19841 L .48145 .19841 L .48145 .36333 L .48145 .52824 L .64636 .52824 L .64636 .36333 L .81128 .36333 L .81128 .52824 L .97619 .52824 L .97619 .36333 L .97619 .19841 L .81128 .19841 L .81128 .0335 L .97619 .0335 L s .24644 .14482 m .39487 .14482 L s % Start of user PostScript -1.3 0.8 -0.4 0.8 MAarrowhead1 % End of user PostScript % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 155.8125}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgSo00Ol00c`0@Ol00>So00Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol0 0c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l0 0c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol0 0c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l0 0c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0=_l506Oo00Ol00c`0@Ol000So00?l00c`0=_lA05_o00Ol00c`0@Ol0 00So00?l00c`0=_l<063o00Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol0 0c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l0 0c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0X_l00c`0>Ol00c`0@Ol000So00?l00c`0Iol00c`0[ol00c`01_l000So00?l00c`0Iol00c`0[ol00c`01_l000So00?l00c`0>Ol00c`0>?l0 0c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l0 0c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l0 0c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l0 0c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l0 0c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l0 0c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l0 0c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go00?l0 0c`0>Ol00c`0>?l00c`01_l00:go00?l00c`0>Ol00c`0>?l00c`01_l00:go?00koc`02?l00?ooJOl0 0?ooJOl00?ooJOl00?ooJOl00001\ \>"], ImageRangeCache->{{{0, 359}, {193.75, 0}} -> {-2.795268, -0.061277, 0.021061, 0.021061}}], Cell[TextData[{ "The following rewrite rules do the job:\n\n", StyleBox[ " \"+F-F-F+\" \[Rule] \"+-F+F+F-F-+F-F-F+F+F-F-F+-F-F+F+F-+\"\n \"-F+F+F-\ \" \[Rule] \"-+F-F-F+F+-F+F+F-F-F+F+F-+F+F-F-F+-\"\n\n", FontFamily->"Courier", FontWeight->"Bold"], "where the starting string is ", StyleBox["\"+F-F-F+\"", FontFamily->"Courier", FontWeight->"Bold"], " . The code that follows generates the depth-3 approximation, with the ", StyleBox["TurtleStep", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["StartPosition", FontFamily->"Courier", FontWeight->"Bold"], " options used to scale and center the path. 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But Have We Really Filled Space?", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "So far we have not been too precise about what a space-filling curve is \ and what the shapes in Figure 4 have to do with a space-filling curve. The \ definition is simple: a space-filling curve is a continuous function that \ maps the unit interval onto the unit square. G. Peano, in 1890, was the first \ to construct such a beast. We present here an example due to David Hilbert. \ Here is how one can rigorously construct Hilbert\[CloseCurlyQuote]s \ space-filling curve ", StyleBox["H", FontWeight->"Bold"], ". Given a real number t in the unit interval, we can define ", StyleBox["H", FontWeight->"Bold"], "(t) by writing t in base 4. First divide the square into four equal \ squares and label them, with 0, 0.1, 0.2, 0.3 as in Figure 7(a). Then \ subdivide each of these squares and label them with 0, 0.01, 0.02, . . . , \ 0.33 as in Figure 7(b). Note how a path through the centers of the 16 labeled \ squares in order of the labels corresponds to our 4-U object. Continue \ subdividing and labeling in this way forever. Now, every real number t has a \ base-4 representation and considering initial segments leads to a nested \ sequence of squares. Since nested sequences of closed sets whose sizes \ approach zero converge to a unique point, we have defined the point that will \ be ", StyleBox["H", FontWeight->"Bold"], "(t). Thus, for example, 0\[CloseCurlyQuote]s representation consists of \ all 0s, and so its nested sequence converges to the point (0, 0).We will omit \ the details of the proof that ", StyleBox["H", FontWeight->"Bold"], " is continuous. 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Given a number t, use initial segments \ of its base-4 representation to locate points on the approximations. These \ points converge to ", StyleBox["H", FontWeight->"Bold"], "(t).\n\nThere are a few misleading aspects of this point of view, however. \ First, all the approximations are one-to-one. But the ideal function ", StyleBox["H", FontWeight->"Bold"], " is definitely not one-to-one: there is no continuous one-to-one function \ from an ineterval onto a square. And second, note that there is little point \ in buying more memory and generating deeper approximations to ", StyleBox["H", FontWeight->"Bold"], "; one would soon end up with an uninteresting square that is completely \ black. We will address these two issues in the next section." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["4. The True Graph of a Space-Filler", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "One shortcoming of viewing the space-filling curve in the target square is \ that deeper images of the approximation are not visually enlightening. But \ remember that a function from an interval to the square has a graph that \ lives in 3-space. It is very simple for us to view an approximation of this \ graph: we simply generate the planar paths as before, and then attach the \ proper t-coordinate to each point. 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Another Viewpoint", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "There is another way of looking at Hilbert\[CloseCurlyQuote]s curve. We \ could look at the exact values of, say, ", StyleBox["H", FontWeight->"Bold"], "(0), ", StyleBox["H", FontWeight->"Bold"], "(", StyleBox["1", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], "/", StyleBox["2", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], "), and ", StyleBox["H", FontWeight->"Bold"], "(1) and connect them linearly. Then we could do the same with ", StyleBox["H", FontWeight->"Bold"], "(0), ", StyleBox["H", FontWeight->"Bold"], "(", StyleBox["1", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], "/", StyleBox["4", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], "), ", StyleBox["H", FontWeight->"Bold"], "(", StyleBox["1", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], "/", StyleBox["2", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], "), ", StyleBox["H", FontWeight->"Bold"], "(", StyleBox["3", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], "/", StyleBox["4", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], "), ", StyleBox["H", FontWeight->"Bold"], "(1); and so on. \n\nFirst we need to improve the turtle. There are many \ ways to enhance a turtle, some of which are discussed in chapter 5 of [Wagon, \ 1991]. One simple enhancement is to add a flip command (we use ", StyleBox["\"i\"", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], " for inversion). Upon seeing an ", StyleBox["\"i\"", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], " the turtle flips onto its back, thus switching left and right. ", StyleBox["FractalTurtle", FontFamily->"Courier", FontWeight->"Bold"], " has this ability. One can then generate the Hilbert paths using only one \ substitution rule as follows." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["FractalTurtle[\"U\" -> \"+iUiF-UFU-FiUi+\", \"U\", 3, 90];", "Input", GroupPageBreakWithin->Automatic, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 9 scalefont setfont % Scaling calculations 0.103175 0.113379 0.103175 0.113379 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 1 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 0 m 1 0 L s 0 0 m 0 1 L s 0 1 m 1 1 L s 1 0 m 1 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath .4 Mabswid .10317 .10317 m .10317 .21655 L .21655 .21655 L .21655 .10317 L .32993 .10317 L .44331 .10317 L .44331 .21655 L .32993 .21655 L .32993 .32993 L .44331 .32993 L .44331 .44331 L .32993 .44331 L .21655 .44331 L .21655 .32993 L .10317 .32993 L .10317 .44331 L .10317 .55669 L .21655 .55669 L .21655 .67007 L .10317 .67007 L .10317 .78345 L .10317 .89683 L .21655 .89683 L .21655 .78345 L .32993 .78345 L .32993 .89683 L .44331 .89683 L .44331 .78345 L .44331 .67007 L .32993 .67007 L .32993 .55669 L .44331 .55669 L .55669 .55669 L .67007 .55669 L .67007 .67007 L .55669 .67007 L .55669 .78345 L .55669 .89683 L .67007 .89683 L .67007 .78345 L .78345 .78345 L .78345 .89683 L .89683 .89683 L .89683 .78345 L .89683 .67007 L .78345 .67007 L .78345 .55669 L .89683 .55669 L .89683 .44331 L .89683 .32993 L Mistroke .78345 .32993 L .78345 .44331 L .67007 .44331 L .55669 .44331 L .55669 .32993 L .67007 .32993 L .67007 .21655 L .55669 .21655 L .55669 .10317 L .67007 .10317 L .78345 .10317 L .78345 .21655 L .89683 .21655 L .89683 .10317 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg;o00;o00;o00;o00;o00;o00;o00;o00;o00;o00ol00c`07_l00c`0G_l00c`07_l00c`0>ol1000000ol00c`07_l00c`0G_l0 0c`07_l00c`0>ol1000000ol00c`07_l00c`0G_l00c`07_l00c`0>ol1000000ol00c`07_l00c`0G_l00c`07_l00c`0>ol1000000ol00c`07_l0 0c`0G_l00c`07_l00c`0>ol1000000ol00c`07_l00c`0G_l00c`07_l00c`0>ol1 000000ol00c`07_l00c`0G_l00c`07_l00c`0>ol1000000ol0 0c`07_l00c`0G_l00c`07_l00c`0>ol1000000ol00c`07_l00c`0G_l00c`07_l0 0c`0>ol1000000ol00c`07_l00c`0G_l00c`07_l00c`0>ol1000000"], ImageRangeCache->{{{0, 287}, {287, 0}} -> {-0.913876, -0.883225, 0.030652, 0.030652}}] }, Open ]], Cell[TextData[{ "Another enhancement is the ability to generate long words using \ meaningless letters, and then, at the end, replace some of the letters with a \ string of letters corresponding to motions (i.e., a string such as ", StyleBox["\"F++F++\"", FontFamily->"Courier", FontWeight->"Bold"], "). We call this a terminal substitution, and it is implemented as an \ option to ", StyleBox["FractalTurtle", FontFamily->"Courier", FontWeight->"Bold"], ". We leave it to the reader to work out why the rules in the following \ code do the job; they do generate the approximations to Hilbert\ \[CloseCurlyQuote]s curve that are perfectly correct on the appropriate \ binary fractions as discussed above. Although they look quite different than \ the traditional approximations of Figure 4, a comparison of the 3-dimensional \ view derived from this approach with Figure 8 will confirm that thay approach \ the same limit." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ image[d_?OddQ] := FractalTurtle[\"U\" -> \"iU+iU+Ui+++Ui++\", \"+U\", (d-1)/2, 45, TerminalSubstitution -> (\"U\" -> \"F--F+\"), TurtleStep -> 2^(-(d-1)/2-0.5), PathStyle -> AbsoluteThickness[2], Axes -> True, Ticks -> {Range[0, 1, 1/4], Range[0, 1, 1/4]}, PlotRange -> {{0, 1}, {0, 1}}] image[d_?EvenQ] := FractalTurtle[\"U\" -> \"+iUi--U+UiUi-\", \"U\", d/2 - 1, 90, TerminalSubstitution -> (\"U\" -> \"+F-FF-F\"), TurtleStep -> 2^(-(d/2)), PathStyle -> AbsoluteThickness[2], Axes->True, Ticks -> {Range[0, 1, 1/4], Range[0, 1, 1/4]}, PlotRange -> {{0, 1}, {0, 1}}] Show[GraphicsArray[Block[{$DisplayFunction = Identity}, Map[image, {{1, 2, 3}, {4, 5, 6}}, {2}]], GraphicsSpacing -> {.15, 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Such loops \ can guide a traveling salesman through a region so that he visits points in \ an order that is close to optimal as far as total length is concerned.\ \>", "Text"], Cell[TextData[{ "For more details see [Bartholdi and Platzman 1988] or [Wagon 1998]; the \ latter contains a ", StyleBox["Mathematica", FontSlant->"Italic"], " implementation. In fact, their method has been used to deliver \ meals-on-wheels in Atlanta! A great advantage is that a computer is not \ needed; it is sufficient to have a space-filling curve superimposed on a city \ map! And of course it is very easy to add or remove addresses from the list. \ I am not a great believer in supporting all kinds of abstract mathematical \ research on the grounds that one never knows what will be useful in the \ future. That said, it is indeed remarkable that one of the most abstract \ constructions of the 19th century has found its way to a real-world problem \ 100 years later." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["REFERENCES", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Bartholdi, J. J., and L. K. Platzman, Heuristics based on space-filling \ curves for combinatorial problems in Euclidean space, ", StyleBox["Management Science", FontSlant->"Italic"], " 34 (1988) 291\[Dash]305.\nPrusinkiewicz, P., and A. Lindenmayer, ", StyleBox["The Algorithmic Beauty of Plants", FontSlant->"Italic"], ", Springer, New York, 1990.\nWagon, S., ", StyleBox["Mathematica in Action", FontSlant->"Italic"], ", Springer/TELOS, New York, 1988." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["ABOUT THE AUTHOR", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Stan Wagon uses ", StyleBox["Mathematica", FontSlant->"Italic"], " extensively in his research, teaching, and exposition, and is especially \ appreciative of the new ways ", StyleBox["Mathematica", FontSlant->"Italic"], " allows us to look at mathematical objects. He is author or coauthor of \ several books, including ", StyleBox["Mathematica in Action", FontSlant->"Italic"], ", ", StyleBox["Animating Calculus", FontSlant->"Italic"], ", ", StyleBox["VisualDSolve", FontSlant->"Italic"], ", and ", StyleBox["Which Way Did the Bicycle Go?", FontSlant->"Italic"], ". He also teaches a high-altitude course, ", StyleBox["Rocky Mountain Mathematica", FontSlant->"Italic"], ", every summer in the mountains of Colorado.\n\nStan Wagon\nDepartment of \ Mathematics and Computer Science\nMacalester College\nSt. Paul, MN 55105\n\ wagon@macalester.edu" }], "Text"], Cell[BoxData[ \(\ \)], "Input"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, AutoGeneratedPackage->None, WindowSize->{776, 721}, WindowMargins->{{Automatic, 211}, {Automatic, 86}}, Magnification->1.25 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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