(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 3597306, 102405]*) (*NotebookOutlinePosition[ 3598301, 102439]*) (* CellTagsIndexPosition[ 3598257, 102435]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["A Wood Of Trees.", "Subtitle", TextAlignment->Center, TextJustification->0], Cell["\<\ inspired by : Scientific American, june 1976 pg 120, \"Mathematical Games\" \ (Martin Gardner).\ \>", "Subtitle", FontSize->10], Cell["\<\ An amateurs view on a set of pretty combinatorial objects. Symmetries galore! Wouter Meeussen, 02.10.1997\ \>", "SmallText", FontColor->GrayLevel[0.666667]], Cell[TextData[{ StyleBox["Forestry, a long term hobby", "Subsubtitle", FontSize->9, Background->RGBColor[1, 1, 0]], StyleBox["..", "Subsubtitle", FontSize->8, Background->RGBColor[1, 1, 0]], StyleBox[" ", "Subsubtitle", FontSize->8, Background->GrayLevel[0.900008]] }], "Text"], Cell[CellGroupData[{ Cell["Operations R(everse), F(lip) and T(urn) on a tree.", "Section"], Cell[CellGroupData[{ Cell["Initialisations", "Subsection", InitializationCell->True], Cell[BoxData[ \(\(SetOptions[TableForm, TableSpacing \[Rule] {0, 1}];\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(cat[n_] := \(\((2\ n)\)!\)\/\(\(n!\)\ \(\((n + 1)\)!\)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(b2d[l_List] := Fold[2\ #1 + #2 &, 0, l]\)], "Input", InitializationCell->True], Cell[BoxData[ \(d2b[n_Integer] := IntegerDigits[n, 2]\)], "Input", InitializationCell->True], Cell[BoxData[ \(p100[t_] := Max[Table[ If[t\[LeftDoubleBracket]{i, i + 1, i + 2}\[RightDoubleBracket] == {1, 0, 0}, i, 0], {i, 1, Length[t] - 2}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \( (*\ nexttree[t_] := b2d[Module[{tt = d2b[t], p}, Flatten[{{{Take[tt, {1, \((p = p100[tt])\) - 1}], {0, 1}}, Table[1, {Count[Take[tt, \(-\((Length[tt] - 2 - p)\)\)], 1]}]}, Table[0, {1 + Count[Take[tt, \(-\((Length[tt] - 2 - p)\)\)], 0]}]}]]]\ *) \)], "Input"], Cell["\<\ nexttree[t_]:=Flatten[Reverse[t]/. {a___,0,0,1,b___}:> Reverse[{Sort[{a,0}]//Reverse,1,0,b}]]\ \>", "Input", InitializationCell->True], Cell[BoxData[ \(wood[n_ /; n < 13] := b2d /@ Reverse[NestList[nexttree, tree[n], cat[n] - 1]\ ]\)], "Input", InitializationCell->True], Cell[BoxData[ \(r[t_] := b2d[Abs[1 - Reverse[d2b[t]]]]\)], "Input", InitializationCell->True], Cell["\<\ bracket[tree_]:=(Flatten[{tree,0}]/. \ 0->{0})//.{1,z___,1,a_List,b_List,y___}:>{1,z,{1,a,b},y}\ \>", "Input", InitializationCell->True], Cell["flip[{1,a_List,b_List}]:={1,flip[b],flip[a]};flip[{0}]={0};", "Input", InitializationCell->True], Cell["f[dec_]:=b2d[Drop[Flatten[flip[bracket[d2b[dec]]]],-1]]", "Input", InitializationCell->True], Cell[BoxData[ \( (*\ f[t_] := Module[{tt = d2b[t], lit, long, it, C, S, N, E, W}, long = {C}; lit = Flatten[{1, tt, 0, 0}]; For[i = 1, i \[LessEqual] Length[lit], \(i++\), If[lit\[LeftDoubleBracket]i\[RightDoubleBracket] == 1, long = Flatten[ ReplacePart[ long, {S, C, N, E, C, W}, \(Position[long, C]\)\[LeftDoubleBracket]1\[RightDoubleBracket]]], long = ReplacePart[long, Null, \(Position[long, C]\)\[LeftDoubleBracket]1\[RightDoubleBracket]]]]; long = DeleteCases[Flatten[long], Null]; long = Reverse[long]; it = Drop[ DeleteCases[ Partition[long, 2, 1] /. {{W, W} \[Rule] 1, {W, E} \[Rule] 0, {E, N} \[Rule] Null, {N, W} \[Rule] 1, {N, S} \[Rule] 0, {S, E} \[Rule] Null, {S, S} \[Rule] Null}, Null], \(-1\)]; it = Drop[it, 1]; b2d[it]]\ *) \)], "Input"], Cell[BoxData[ \(<< "\"\)], "Input", InitializationCell->True], Cell["\<\ SetOptions[ListPlot,PlotJoined->True,PlotStyle->AbsolutePointSize[5],\ DisplayFunction->Identity,Axes->False,FrameTicks->None,Frame->True]; disp=DisplayFunction:>$DisplayFunction;\ \>", "Input", InitializationCell->True], Cell["\<\ mountainplot[t_Integer]:=ListPlot[FoldList[Plus,0,2 # \ -1],PlotRange->{0,Length[#]/2}]&@d2b@t\ \>", "Input", InitializationCell->True], Cell["\<\ A helper function, analogous to StringReplace, but limited to the first \ occurence :\ \>", "Text"], Cell[BoxData[ \(myrep[str_, from_, to_] := Module[{p = \(StringPosition[str, from, 1]\)\[LeftDoubleBracket]1, 2\[RightDoubleBracket]}, StringReplace[StringTake[str, p], from \[Rule] to] <> StringDrop[str, p]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(graf[ mytree_] := \((\((Fold[ If[#2 === 1, myrep[#1, "\", "\"], myrep[#1, "\", "\<\>"]] &, "\", #1] &)\)[ Join[#1, {0}]] &)\)[mytree]\)], "Input", InitializationCell->True], Cell["\<\ mytreeplot[tree_Integer]:=Module[{dna,rna,vect,down,deep,branchdepth,slim,\ trail}, dna=d2b@tree; rna=Characters[graf[dna]]; vect=rna/.Thread[Rule[{\"S\",\"N\",\"E\",\"W\"},{-1-I,1+I,1-I,-1+I}]]; down=rna/.Thread[Rule[{\"S\",\"N\",\"E\",\"W\"},{1,-1,1,-1}]]; deep=FoldList[Plus,0,down]; branchdepth=Max/@Partition[deep,2,1]-1; slim=I Im[vect] +Re[vect] 2^(-branchdepth); trail=FoldList[Plus,0,slim ]; ListPlot[{{0,1}}~Join~({Re[#],Im[#]}&/@ trail ) \ ,PlotJoined->True,Axes->False]]\ \>", "Input", InitializationCell->True], Cell[BoxData[ \(my[x_, a_, b_] := If[x \[LessEqual] a, b, x]\)], "Input", InitializationCell->True], Cell[BoxData[ \(mytocycles[perm_List] := DeleteCases[ Table[Module[{predi = \(Position[perm, i]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket], fpl}, fpl = If[i > 1 && predi < i, Null, Drop[FixedPointList[ my[perm\[LeftDoubleBracket]#1\[RightDoubleBracket], i, predi] &, i], \(-1\)]]; If[fpl === Null || perm\[LeftDoubleBracket]fpl\[LeftDoubleBracket]\(-Min[ Length[fpl], 2]\)\[RightDoubleBracket]\[RightDoubleBracket] =!= fpl\[LeftDoubleBracket]\(-1\)\[RightDoubleBracket] || fpl\[LeftDoubleBracket]1\[RightDoubleBracket] =!= Min[fpl], Null, fpl]], {i, 1, Length[perm]}], Null]\)], "Input", InitializationCell->True], Cell[BoxData[ \(mytocyclesshort[perm_List] := Module[{c = {}, len, predi}, \n\t\tlen = Length[perm]; \n\t\t\t\tDo[ If[FreeQ[c, i], \n\t\t\t\t\tpredi = \(Position[perm, i]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]; c = Join[ c, {Drop[ FixedPointList[ If[perm[\([#1]\)]\ <= \ i, \ \n\ \ \ \ \ \ \ \ \ \ \ \ predi, \ perm[\([#1]\)]] &, i], \(-1\)]}]; \n\t\t\t\tContinue[]], {i, 1, len}]; c]\)], "Input", InitializationCell->True], Cell[BoxData[ \(mytocyclesmixed[perm_List] := Module[{c = {}, cyc, len, predi, theRest, i}, \n\t\tlen = Length[perm]; \n\t\ttheRest = Range[len]; \n\t\t\t\tWhile[ theRest =!= {}\ && FreeQ[c, i = theRest[\([1]\)]]\ , \n\t\t\t\t\tpredi = \(Position[perm, i]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]; \n\t\t\t\t\tc = Join[c, cyc = {Drop[ FixedPointList[ If[perm[\([#1]\)]\ <= \ i, predi, \ perm[\([#1]\)]] &, i], \(-1\)]}]; \n\t\t\t\t\ttheRest = Complement[theRest, Flatten[cyc]]\ \ \n\t\t\t\t\t\t\t\t\t]; \n\t\t\tc]\)], "Input", InitializationCell->True], Cell["\<\ This one produces a \"L1\" resp. \"L3\" for connections to south- resp. \ east-pointing leaves, and \"x\" for all other connections. It starts with \"x\ \" from the root, and ends with \"L\" because the root should be counted as a \ potential leaf (after rotation).\ \>", "Text"], Cell[BoxData[ \( (*framed[tree_] := "\" <> StringReplace[graf[tree], Thread[{"\", "\", "\", "\", "\", "\"} \ \[Rule] {"\", "\", "\", "\", "\", "\"}]] <> \ \*"\""\(L \*"\"\< \>"\)*) \)], "Input"], Cell["The next one rotates the string upto the first leaf :", "Text"], Cell[BoxData[ \( (*\ dash[tree_] := StringDrop[ argu = framed[tree], \(StringPosition[argu, "\", 1]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]] <> StringTake[ argu, \(StringPosition[argu, "\", 1]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]]\ *) \)], "Input"], Cell["\<\ Last, a somewhat cryptic function, that digs two levels deeper each time it \ encounters anything non-leafy (and adds a \"1\" to the output), and then \ climbs one level, except that it climbs up one level each time it encounters \ a leaf (and adds a \"0\").\ \>", "Text"], Cell[BoxData[ \( (*\ dig[argu_, pos_] := Module[{}, p = pos; If[StringTake[argu, {p = p + 1}] === "\", myout = myout <> "\<0\>"; Return[argu]]; myout = myout <> "\<1\>"; dig[argu, p]; dig[argu, p = p + 1]; p = p + 1; Return[myout]]\ *) \)], "Input"], Cell[BoxData[ \( (*\ rotate[tree_] := Module[{}, myout = "\<1\>"; Return[b2d[ Drop[Drop[ ToExpression[ Characters[\n\t\t\t\t\t\t\t\tdig[dash[d2b[tree]], 0]\t\t\t\t\t\t\t]], 1], \(-1\)]]]]\ *) \)], "Input"], Cell["A faster way to rotate is:", "Text"], Cell["\<\ t[dec_Integer]:=Module[{bin,n,carry,brat}, bin=d2b@dec; n=First[Position[bin,0]][[1]] -2; brat=bracket[bin]; carry=Reverse[ brat[[Sequence@@#]]&/@Table[Append[Table[2,{i}],3],{i,n}]]; b2d@Flatten[{Table[{1,carry[[i]]},{i,n}],1,brat[[3]]}] ]\ \>", "Input", InitializationCell->True], Cell["\<\ The following a a representation of trees as a double coloumn, sorted both \ horizontally and vertically: chessboard path coordinates.\ \>", "Text"], Cell["\<\ bilist[dec_]:=Module[{tree,w},tree=d2b@dec;w=(Range[Length[tree]] #)& \ /@{tree,1-tree};Transpose[DeleteCases[w,0,{-1}]]]\ \>", "Input", InitializationCell->True, Background->RGBColor[0, 1, 1]], Cell["And now, clumsily but workable, the Euler triangulation :", "Text"], Cell["\<\ diags[test_]:=Module[{ngon=Length[d2b@test]/2+2,z,res},ac={};p=.; sides=Apply[li,Partition[p/@Range[ngon],2,1],{1}]; (euler[d2b@test]/.x_Integer:>sides[[x]])//.{u_li,v_li}:>chord[u,v]; res=(Flatten[ac]/.p:>(circum[ngon][[#]]&))/. li[a_,b_]->Line[{{Re[a],Im[a]},{Re[b],Im[b]}}]; ac=.; Graphics[{res,Line[{Re[#],Im[#]}&/@circum[ngon]]},AspectRatio->Automatic] \ ]\ \>", "Input", InitializationCell->True], Cell["and an other helper function for closing the triangles :", "Text"], Cell["\<\ chord[i_li,j_li]:=Module[{z}, Flatten[li[i,j]]/.li[a_,b_,b_,c_]:>(z=li[a,c];ac={ac,z};z)] \ \>", "Input", InitializationCell->True], Cell["a polygon with horizontal top:", "Text"], Cell["\<\ circum[n_]:=circum[n]=NestList[# E^(2 I Pi/n)&,I E^(I Pi /n),n]\ \>", "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["eulerrule={a___,1,x_?ListQ,y_?ListQ,b___}:>{a,{x,y},b}", "Input", InitializationCell->True], Cell[BoxData[ \({a___, 1, x_?ListQ, y_?ListQ, b___} \[RuleDelayed] {a, {x, y}, b}\)], "Output"] }, Open ]], Cell["\<\ a Euler list : representing the couples of line-elements that need closing to \ form triangles :\ \>", "Text"], Cell["\<\ euler[t_]:=Module[{n=1},(((Flatten[{1,t,0}]/. 0:>{0} )//.eulerrule)[[2]])/. {0} :>n++]\ \>", "Input", InitializationCell->True], Cell["\<\ nice[a_]:=If[a==={},a,Partition[Join[a, Table[Graphics[Point[{0, 0}]], { Mod[Ceiling[Sqrt[#]]-Mod[#,(Ceiling[Sqrt[#]] )],Ceiling[Sqrt[#]]] & @ \ Length[a]}]], Ceiling[Sqrt[Length[a]]] ] ]\ \>", "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Definitions about Trees", "Subsection"], Cell["\<\ A tree is defined here as a structure that can generate a two-dimensional \ graph of a binary planted tree. The structure is given by a binary list, as many \"1\" as \"0\", such that, \ starting with a \"1\", so that there are at least as many preceding \"1\" \ than \"0\". A slightly different definition has the root represented as an initial \"1\", \ and the final leaf with a \"0\" ; in that case the number of ones is \ everywhere greater than the number of zero's, except at the last position \ (where they are equal). A simple graphic of a \"mountain range\" is obtained by replacing the \"0\" \ with \"-1\" : the cumulative sum along the series is nowhere negative, and at \ the start and end-point, it is exactly zero.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ test={1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0,0}; Length[test] t1=FoldList[Plus,0,2 test-1] Show[mountainplot[b2d@test],disp]\ \>", "Input"], Cell[BoxData[ \(20\)], "Output"], Cell[BoxData[ \({0, 1, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 2, 1, 2, 1, 2, 1, 0}\)], "Output"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 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 0.0453515 0 0.0618034 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] 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Note that these trees have (7+1) = 8 leaves plus a root !!\ \>", "Text"], Cell[CellGroupData[{ Cell["Show[mytreeplot[b2d@test],disp]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 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 0.404762 0.380952 0.537918 0.0654004 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .40476 .60332 m .40476 .53792 L .02381 .47252 L .40476 .53792 L .78571 .47252 L .59524 .40712 L .5 .34172 L .45238 .27632 L .42857 .21092 L .45238 .27632 L .47619 .21092 L .46429 .14552 L .47619 .21092 L .4881 .14552 L .48214 .08012 L .47917 .01472 L .48214 .08012 L .48512 .01472 L .48214 .08012 L .4881 .14552 L .49405 .08012 L .4881 .14552 L .47619 .21092 L .45238 .27632 L .5 .34172 L .54762 .27632 L .5 .34172 L .59524 .40712 L .69048 .34172 L .64286 .27632 L .69048 .34172 L .7381 .27632 L .71429 .21092 L .7381 .27632 L .7619 .21092 L .7381 .27632 L .69048 .34172 L .59524 .40712 L .78571 .47252 L .97619 .40712 L .78571 .47252 L .40476 .53792 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0?ooo`<000004`3oool100000040oooo0000 0`000000oooo0?ooo`2g0?ooo`<00000B03oool3000001H0oooo0@0000010?ooo`0000<000000?oo o`3oool0^P3oool300000480oooo0`00000I0?ooo`4000000@3oool000030000003oool0oooo0;d0 oooo0`00000l0?ooo`<00000703oool100000040oooo00000`000000oooo0?ooo`300?ooo`<00000 =P3oool3000001l0oooo0@0000010?ooo`0000<000000?ooo`3oool0``3oool300000340oooo0P00 000R0?ooo`4000000@3oool000030000003oool0oooo0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.06576, -8.22504, \ 0.00916904, 0.0534089}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Table[{i,cat[i]},{i,12}]", "Input"], Cell[BoxData[ \({{1, 1}, {2, 2}, {3, 5}, {4, 14}, {5, 42}, {6, 132}, {7, 429}, {8, 1430}, {9, 4862}, {10, 16796}, {11, 58786}, {12, 208012}}\)], "Output"] }, Open ]], Cell["The five trees of length 3 are:", "Text"], Cell[CellGroupData[{ Cell["d2b/@wood[3]//ColumnForm", "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({1, 0, 1, 0, 1, 0}\)}, {\({1, 0, 1, 1, 0, 0}\)}, {\({1, 1, 0, 0, 1, 0}\)}, {\({1, 1, 0, 1, 0, 0}\)}, {\({1, 1, 1, 0, 0, 0}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ {{1, 0, 1, 0, 1, 0}, {1, 0, 1, 1, 0, 0}, {1, 1, 0, 0, 1, 0}, {1, 1, 0, 1, 0, 0}, {1, 1, 1, 0, 0, 0}}], Editable->False]], "Output"] }, Open ]], Cell["\<\ The set of all trees of length n is here defined as \"wood[n]\". The trees are ordered reversed lexicographicaly. For compactness, every tree \ is generated by the integer obtained by interpreting the list as a binary \ number : \"1100\" becomes 12 in decimal notation. This function is defined here as \ \"b2d\", short for \"Binary to Decimal\". Its inverse is \"d2b\". In this notation, wood[3] looks like:\ \>", "Text"], Cell[CellGroupData[{ Cell["w3=wood[3]", "Input"], Cell[BoxData[ \({42, 44, 50, 52, 56}\)], "Output"] }, Open ]], Cell["\<\ The \"classic\" planar plot of a binary tree is generated by the following \ trick (J. Lukasiewicz ) : Create a text string using the letters \"N\", \"S\", \"E\" and \"W\" (north, \ south, east & west), starting from a string \"C\" (Choose); using the binary representation as a guide (a kind of DNA that dictates \ protein structure). First, add a terminator \"0\" to the end of the binary list. (The starter \"1\ \" from the root is thought to generate the starter \"C\".) Then : for every \"1\" in the binary list, replace the first \"C\" by \"S C N \ E C W\", and for every zero, delete the first \"C\".\ \>", "Text"], Cell["\<\ This function translates a tree into a set of graphical directives: South, \ North, East, West :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(graf[test]\)], "Input"], Cell[BoxData[ \("SNESSSSNESNESSNEWNEWWWNEWNESNESNEWWWNEWW"\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Plotting a tree.", "Subsection"], Cell["\<\ Just for fun, lets plot a tree by manipulating vectors in the complex plane. \ There are easier ways to do it, but it has a certain charm to stick to \ stricktly (complex) numbers instead of using Hewlett Packard plot language \ :\ \>", "Text"], Cell[CellGroupData[{ Cell["dna=test", "Input"], Cell[BoxData[ \({1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["graf[dna]", "Input"], Cell[BoxData[ \("SNESSSSNESNESSNEWNEWWWNEWNESNESNEWWWNEWW"\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["rna=Characters[%]", "Input"], Cell[BoxData[ \({"S", "N", "E", "S", "S", "S", "S", "N", "E", "S", "N", "E", "S", "S", "N", "E", "W", "N", "E", "W", "W", "W", "N", "E", "W", "N", "E", "S", "N", "E", "S", "N", "E", "W", "W", "W", "N", "E", "W", "W"}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ vect=rna/.Thread[Rule[{\"S\",\"N\",\"E\",\"W\"},{-1-I,1+I,1-I,-1+I}]]\ \>", "Input"], Cell[BoxData[ \({\(-1\) - 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E. Shannon, the information content of a normed set of \ probabilities ", Cell[BoxData[ \(p\_i\)]], " is -", Cell[BoxData[ \(\[Sum]p\_i\ Log[2, p\_i]\)]], ".\nThe probabilities can be defined as the chance ( the density, the \ frequency ) of \"1\" in a given position over the complete set of trees of \ order n :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[countit = Table[Plus @@ \(d2b /@ wood[i]\), {i, 9}]]\)], "Input"], Cell[BoxData[ \({0.796`\ Second, {{1, 0}, {2, 1, 1, 0}, {5, 3, 3, 2, 2, 0}, {14, 9, 9, 7, 7, 5, 5, 0}, {42, 28, 28, 23, 23, 19, 19, 14, 14, 0}, {132, 90, 90, 76, 76, 66, 66, 56, 56, 42, 42, 0}, {429, 297, 297, 255, 255, 227, 227, 202, 202, 174, 174, 132, 132, 0}, {1430, 1001, 1001, 869, 869, 785, 785, 715, 715, 645, 645, 561, 561, 429, 429, 0}, {4862, 3432, 3432, 3003, 3003, 2739, 2739, 2529, 2529, 2333, 2333, 2123, 2123, 1859, 1859, 1430, 1430, 0}}}\)], "Output"] }, Open ]], Cell["This table can be mimicked by:", "Text"], Cell[CellGroupData[{ Cell["\<\ 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{1, 4, 9, 14, 14}, {1, 5, 14, 28, 42, 42}}\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Flipping over (Reflecting) and Reversing trees.", "Subsection"], Cell["\<\ The function \"f\" (flip) operates on the decimal representation of a tree, \ to produce the flipped (mirror image) plot of the tree. The function \"r\" (reverse) reverses the \"mountain range graphic\" of the \ tree. 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oooo00@0oooo00<000000?ooo`3oool09P3oool00`000000oooo0?ooo`0C0?ooo`030000003oool0 oooo00H0oooo00070?ooo`030000003oool0oooo01L0oooo00<000000?ooo`3oool08P3oool00`00 0000oooo0?ooo`040?ooo`030000003oool0oooo0280oooo00<000000?ooo`3oool05`3oool00`00 0000oooo0?ooo`030?ooo`030000003oool0oooo01L0oooo00<000000?ooo`3oool08P3oool00`00 0000oooo0?ooo`040?ooo`030000003oool0oooo02H0oooo00<000000?ooo`3oool04`3oool00`00 0000oooo0?ooo`060?ooo`001`3ooom0000000H0oooo@00000050?oood0000001P3ooom0000000P0 oooo003o0?ooob40oooo003o0?ooob40oooo003o0?ooob40oooo0000\ \>"], ImageRangeCache->{{{0, 287}, {131.438, 0}} -> {-0.109007, -0.0460031, \ 0.0146969, 0.0146969}, {{7.375, 70.6875}, {128.25, 89.1875}} -> {-2.02684, \ -1.17975, 0.0519287, 0.0519287}, {{77, 140.313}, {128.25, 89.1875}} -> \ {-5.64237, -1.17975, 0.0519287, 0.0519287}, {{146.625, 209.938}, {128.25, \ 89.1875}} -> {-9.25791, -1.17975, 0.0519287, 0.0519287}, {{216.25, 279.563}, \ {128.25, 89.1875}} -> {-12.8734, -1.17975, 0.0519287, 0.0519287}, {{7.375, \ 70.6875}, {85.25, 46.125}} -> {-3.14753, -11.8701, 0.349656, 0.256579}, {{77, \ 140.313}, {85.25, 46.125}} -> {-27.4924, -11.8701, 0.349656, 0.256579}, \ {{146.625, 209.938}, {85.25, 46.125}} -> {-51.8372, -11.8701, 0.349656, \ 0.256579}, {{216.25, 279.563}, {85.25, 46.125}} -> {-76.182, -11.8701, \ 0.349656, 0.256579}, {{7.375, 70.6875}, {42.1875, 3.125}} -> {-1.37471, \ -29.8759, 0.0416258, 0.242467}, {{77, 140.313}, {42.1875, 3.125}} -> \ {-4.77291, -29.8759, 0.0416258, 0.242467}, {{146.625, 209.938}, {42.1875, \ 3.125}} -> {-7.1711, -29.8759, 0.0416258, 0.242467}, {{216.25, 279.563}, \ {42.1875, 3.125}} -> {-9.66237, -29.8759, 0.0374632, 0.242467}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] GraphicsArray \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["The permutations R , F and RF", "Subsection"], Cell["\<\ We can apply operation \"f\" or \"r\" on all trees in a wood at once. Since \ each tree has a flipped and reversed version, and since f^2==r^2==Identity, \ and the operations f and r permute the trees in a wood. Since the decimal values for the trees stand in increasing order, a \ \"Ordering[ ]\" produces the standard form of the inverse permutation. The \ permutations \"R\" and \"F\" are lists of integers from 1 to cat[n], each \ integer representing a tree by its index : the position of that tree in \ wood[n]. Both F and R are involutions (self inverse permutations) \"over\" the wood. \ \>", "Text"], Cell[CellGroupData[{ Cell["?Ordering", "Input"], Cell[BoxData[ \("Ordering[list] gives the permutation that puts the elements of list in \ order."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell["?RandomPermutation", "Input"], Cell[BoxData[ \("RandomPermutation[n] gives a random permutation of n elements."\)], \ "Print"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Table[(wou=RandomPermutation[10])==Ordering[Ordering[wou]],{10}]\ \>", "Input"], Cell[BoxData[ \({True, True, True, True, True, True, True, True, True, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["w4=wood[4]", "Input"], Cell[BoxData[ \({170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240}\)], "Output"] }, Open ]], Cell["\<\ For any n, \"in\" is the identity permutation : it is simply Range[cat[n]].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(i4 = Ordering[\ w4\ ]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f4 = Ordering[\ f /@ \ w4\ ]\)], "Input"], Cell[BoxData[ \({14, 13, 12, 10, 9, 11, 8, 7, 5, 4, 6, 3, 2, 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(f/@ w4 ) == w4 [[f4]]", "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(r4 = Ordering[\ r /@ \ w4\ ]\)], "Input"], Cell[BoxData[ \({1, 6, 3, 8, 11, 2, 7, 4, 9, 12, 5, 10, 13, 14}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(r/@ w4 ) == w4 [[r4]]", "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["f4 [[f4]] == i4 == r4 [[r4]]", "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ Note that (RF) is the permutation that has an equivalent effect on wood[n] as \ r/@f/@ wood[n]. The operation Ordering[ ] does two things: first, it converts the decimal tree-numbers to integer indices, second, it inverses the permutation. Twice using Ordering[ ] puts it back in order. For an involution, the \ Ordering[ ] has no effect. So, in the definition of R and F, one ordering \ suffices. The product operation (RF) is however not self-inverse ! This implies that the definition r/@f/@ wood == wood [[ (RF) ]] means that \ the operation wood [[ (RF) ]] == r/@ ( f/@ wood ) == r/@ (wood [[F]] ) == (r/@ wood ) \ [[F]] == wood [[R]] [[F]] It is faster to calculate the effect of r/@f/@wood as a product of \ permutations R and F then as two successive operations on all cat[n] trees. \ \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ it = r /@ f /@ w4 rf4 = Ordering[Ordering[it]] it == r /@ w4[[f4]] == (r /@ w4)[[f4]] == w4[[r4]][[f4]] it == w4[[r4[[f4]]]] == w4[[rf4]] rf4 == r4[[f4]]\ \>", "Input", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \({240, 232, 216, 228, 212, 184, 180, 204, 226, 210, 172, 178, 202, 170}\)], "Output"], Cell[BoxData[ \({14, 13, 10, 12, 9, 5, 4, 7, 11, 8, 2, 3, 6, 1}\)], "Output"], Cell[BoxData[ \(True\)], "Output"], Cell[BoxData[ \(True\)], "Output"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["Note that f and r do not commute : F R =!= R F", "Text"], Cell[CellGroupData[{ Cell["f4 [[r4]] =!= r4 [[f4]] == Ordering[ f4 [[r4]] ]", "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Decomposition into cycles of RF.", "Section"], Cell[CellGroupData[{ Cell["Silva Octifolia.", "Subsection"], Cell["\<\ Lets calculate the permutations for wood[7] (the wood of eight-leafed-trees). \ It serves as an example of sufficient length.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ num = 7 cat[num] Timing[wn = wood[num]; ] Timing[rn = Ordering[r /@ wn]; ] Timing[fn = Ordering[f /@ wn]; ] Timing[rfn = rn[[fn]]; ] Timing[frn = fn[[rn]]; ]\ \>", "Input"], Cell[BoxData[ \(7\)], "Output"], Cell[BoxData[ \(429\)], "Output"], Cell[BoxData[ \({0.04600000000000026`\ Second, Null}\)], "Output"], Cell[BoxData[ \({0.04699999999999971`\ Second, Null}\)], "Output"], Cell[BoxData[ \({0.125`\ Second, Null}\)], "Output"], Cell[BoxData[ \({0.`\ Second, Null}\)], "Output"], Cell[BoxData[ \({0.`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ Now we can look at the cycle structure of the permutations RF and FR :\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["ToCycles.", "Subsection"], Cell["\<\ The built-in ToCycles (author S. Skiena, DiscreteMath`Permutations`) needs \ 1328 seconds (20 min.) to calculate the 1430 element cyclic decomposition of \ rf8 on a Windows NT 3.51 serv. pack 4, at 75 Mhz and 24 Mb RAM. On a Windows \ 95 at 100 MHz and 32 Mb RAM, it takes 1144 seconds under Mma 3.0. It takes 136 seconds on a Acer 1500 MHz, 256 Mb RAM.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Information["\", LongForm \[Rule] True]\)], "Input"], Cell[BoxData[ \("ToCycles[p] gives the cycle structure of permutation p, as a list of \ cyclic permutations."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(ToCycles[ DiscreteMath`Permutations`Private`perm_List] := \ \((Take[#1, \(Position[Rest[#1], First[#1]]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]] &)\) /@ Last[FoldList[ If[MemberQ[Flatten[#1], #2], #1, Append[#1, NestList[ DiscreteMath`Permutations`Private`perm\ \[LeftDoubleBracket]#1\[RightDoubleBracket] &, #2, Length[ DiscreteMath`Permutations`Private`perm]]]]\ &, {}, DiscreteMath`Permutations`Private`perm]]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ ToCycles], Editable->False]], "Print"] }, Open ]], Cell[TextData[{ "ToCycles[p] is a relatively slow operation, and can be improved for the \ kind of permutation we are dealing with.\n ", StyleBox["I prefer to have the cycles begin with their lowest element : \ therefore the \"RotateRight\".", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[crfn = RotateRight /@ ToCycles[rfn]]\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(crfn\)\" is similar to \ existing symbol \"\!\(rfn\)\"."\)], "Message"], Cell[BoxData[ \({3.8910000000000005`\ Second, {{1, 429}, {2, 428, 133, 132, 197, 423}, {3, 422, 175, 174, 65, 424}, {4, 427, 43, 264, 196, 404}, {5, 421, 265, 42, 328, 403}, {6, 402, 189, 278, 23, 425}, {7, 401, 279, 56, 238, 405}, {8, 416, 57, 306, 64, 407}, {9, 426, 15, 354, 194, 359}, {10, 420, 147, 84, 327, 357}, {11, 396, 307, 146, 106, 406}, {12, 400, 217, 188, 155, 128, 200, 397, 176, 173, 93, 260, 199, 384, 136, 130, 225, 259, 195, 358}, {13, 415, 85, 216, 326, 356}, {14, 395, 355}, {16, 353, 284, 89, 210, 408}, {17, 348, 312, 151, 78, 409}, {18, 352, 222, 221, 183, 171, 68, 398, 177, 169, 102, 303, 67, 387, 140, 121, 234, 302, 63, 367, 31, 343, 231, 283, 51, 261, 205, 336, 178, 172, 74, 350, 191, 274, 35, 380, 71, 320, 154, 117, 209, 378, 58, 305, 92, 249, 208, 370, 46, 262, 224, 246, 192, 276, 26, 399, 180, 160, 105, 366, 25, 391, 152, 98, 237, 365, 21, 381, 90, 230, 236, 361}, {19, 347, 360}, {20, 382, 62, 368, 22, 411}, {24, 419, 138, 112, 324, 275}, {27, 414, 48, 244, 323, 271}, {28, 394, 270}, {29, 334, 321, 269, 36, 410}, {30, 333, 369, 47, 257, 362}, {32, 351, 203, 311, 141, 129, 206, 349, 190, 277}, {33, 346, 293, 61, 300, 115, 207, 344, 280, 55, 252, 247, 204, 310, 144, 110, 338, 185, 156, 123, 290, 108, 316, 166, 94, 255, 289, 100, 291, 118, 226, 254, 285, 88, 214, 340, 218, 187, 164, 81, 331, 309, 142, 124, 296, 80, 325, 272}, {34, 377, 99, 292, 104, 364}, {37, 390, 161, 70, 393, 267}, {38, 329, 383, 137, 125, 363}, {39, 332, 335, 179, 168, 116, 213, 330, 308, 145, 120, 248, 212, 322, 268, 40, 342, 245, 193, 273}, {41, 376, 113, 202, 392, 266}, {44, 263, 286, 114, 201, 412}, {45, 258, 314, 162, 69, 413}, {49, 243, 371}, {50, 253, 233, 294, 60, 304, 73, 339, 182, 165, 77, 379, 59, 301, 101, 297, 76, 373}, {52, 256, 295, 72, 319, 163, 75, 345, 281}, {53, 239, 385, 139, 111, 372}, {54, 242, 337, 181, 159, 119, 228, 240, 313, 150, 82, 341, 219, 184, 167, 96, 241, 317, 153, 97, 251, 227, 232, 282}, {66, 418, 135, 126, 315, 170}, {79, 389, 149}, {83, 375, 87, 211, 388, 148}, {86, 215, 374}, {91, 229, 250, 223, 220, 186, 158, 109, 318, 157, 103, 299, 95, 235, 298}, {107, 386, 143}, {122, 288}, {127, 198, 417, 134, 131, 287}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Length /@ crfn\)], "Input"], Cell[BoxData[ \({2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 20, 6, 3, 6, 6, 72, 3, 6, 6, 6, 3, 6, 6, 10, 48, 6, 6, 6, 20, 6, 6, 6, 3, 18, 9, 6, 24, 6, 3, 6, 3, 15, 3, 2, 6}\)], "Output"] }, Open ]], Cell[TextData[{ "Cycles of length 6 seem very abundant. As it seems to turn out, for a wood \ of even n, all cycle lengths are even. For a wood of odd n, some cycles ", StyleBox["can", FontWeight->"Bold"], " be odd. We end up with 46 different cycles :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[crfn]\)], "Input"], Cell[BoxData[ \(46\)], "Output"] }, Open ]], Cell["take the second cycle as a test case:", "Text"], Cell[CellGroupData[{ Cell["crfn[[2]]", "Input"], Cell[BoxData[ \({2, 428, 133, 132, 197, 423}\)], "Output"] }, Open ]], Cell["the trees are:", "Text"], Cell[CellGroupData[{ Cell["it=wn[[%]]", "Input"], Cell[BoxData[ \({10924, 16192, 12970, 12224, 13652, 16130}\)], "Output"] }, Open ]], Cell["we can test the validity of this cycle:", "Text"], Cell[CellGroupData[{ Cell["NestList[r[ f[#] ]&, it[[1]], 6 ]", "Input"], Cell[BoxData[ \({10924, 16192, 12970, 12224, 13652, 16130, 10924}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["NestList[ rfn[[#]]&, crfn[[2]][[1]], 6 ]", "Input"], Cell[BoxData[ \({2, 428, 133, 132, 197, 423, 2}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["A faster routine. ", "Subsection"], Cell["\<\ This routine does rf8 on the same machine in 20.3 seconds, and in 50.8 \ seconds on a 100 MHz, 32 Mb machine running Windows 95 ; after installing the \"addtwo\" \ fix, it takes only 22.5 seconds. This last machine does it in 15.6 seconds under \ Mma 3.0. In 1.0 sec on the 1500 MHz 256Mb machine.\ \>", "Text"], Cell["it comes to the same result, check :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[mytocycles[rfn] == crfn]\)], "Input"], Cell[BoxData[ \({0.09299999999999997`\ Second, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[mytocyclesshort[rfn] == crfn]\)], "Input"], Cell[BoxData[ \({0.03200000000000003`\ Second, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[mytocyclesmixed[rfn] == crfn]\)], "Input"], Cell[BoxData[ \({0.030999999999999694`\ Second, True}\)], "Output"] }, Open ]], Cell["\<\ We chose for size 7, so we will call them by their name from here on :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(crf7 = crfn; w7 = wn; f7 = fn; r7 = rn; rf7 = rfn; Share[]\)], "Input"], Cell[BoxData[ \(349976\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ An even faster one that saves intermediates, and can be stopped and then \ continued at a later time.\ \>", "Subsubsection", FontColor->GrayLevel[0.333333]], Cell[BoxData[ \( (*\ \ it = << \(\*"\""\)\(c : \(\(/\)\(_Wouter\)\)/\(textboompjes/ rf12 . txt\) \*"\"\<; \>"\)*) \)], "Input", FontColor->GrayLevel[0.333333]], Cell[BoxData[ \( (*\ \ \(rfn = it\[LeftDoubleBracket]2\[RightDoubleBracket];\)\ *) \)], "Input", FontColor->GrayLevel[0.333333]], Cell["\<\ it is a statement, not a function. This allows more user control. It does its \ thing in 15.5 seconds on the NT, an in 40.5 seconds on the Win'95 before, \ and 15.4 sec. after the fix..\ \>", "Text", FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(c = {}; from = 1; upto = cat[7]\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(Timing[ Do[If[FreeQ[c, i], predi = \(Position[rfn, i]\)\[LeftDoubleBracket]1, 1 \[RightDoubleBracket]; \(c = Join[ c, {Drop[ FixedPointList[ my[rfn\[LeftDoubleBracket]#1\[RightDoubleBracket], i, predi]&, i], \(-1\)]}]; \) (*\ Last[c] >>> \(\*"\""c : \(/ _wouter\)/c12.txt \*"\""\)*) , \n \t\t\t\tContinue[]], {i, from, upto}]]\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(c == cn\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell["\<\ This method comes to its advantage for wood[11] and higher : in the last stages, most cycles have been found, with most values of i still \ to go. Then it is advantagous to search for the values of i that are not yet part of \ a cycle found so far.\ \>", "Text", FontColor->GrayLevel[0.333333]], Cell["\<\ To read c12.txt, use ReadList : it was saved record by record : (!! set to \ non-evaluatable !!)\ \>", "Text", FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(\(c = ReadList["\"]; \)\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(Max[First/@c]\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(\(theRest = Complement[Range[cat[12]], Flatten[c]]; \)\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(theRest\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(Length[theRest]\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]], Cell["\<\ Now, the Do-loop should go over the relatively small number of elements in \ theRest, as with\ \>", "Text", FontColor->GrayLevel[0.333333]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ Do[i = theRest\[LeftDoubleBracket]j\[RightDoubleBracket]; If[FreeQ[c, i], predi = \(Position[rfn, i]\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]; \(c = Join[c, {Drop[ FixedPointList[ my[rfn\[LeftDoubleBracket]#1\[RightDoubleBracket], i, predi] &, i], \(-1\)]}];\) (*\ Last[c] >>> \(\*"\""c : \(\(/\)\(_wouter\)\)/ c12 . txt \*"\"\< \>"\)*) , \n\t\t\tContinue[]], {j, Length[theRest]}]]\)], "Input", FontColor->GrayLevel[0.333333]], Cell[BoxData[ \({0.`\ Second, Null}\)], "Output"] }, Open ]], Cell["This speeds up things quite a bit.", "Text", FontColor->GrayLevel[0.333333]], Cell["\<\ (Now we save the entire list in one go, so it should from here onwards be \ read with Read in stead of ReadList.)\ \>", "Text", FontColor->GrayLevel[0.333333]], Cell[BoxData[ \(c >> "\"\)], "Input", Evaluatable->False, FontColor->GrayLevel[0.333333]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Description of the cycles of the permutation RF ", "Subsection"], Cell["The number of cycles is :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[crf7]\)], "Input"], Cell[BoxData[ \(46\)], "Output"] }, Open ]], Cell["\<\ Each cycle starts with its lowest element, call those the \"starters\" of \ those cycles :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(starters = First /@ crf7\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 24, 27, 28, 29, 30, 32, 33, 34, 37, 38, 39, 41, 44, 45, 49, 50, 52, 53, 54, 66, 79, 83, 86, 91, 107, 122, 127}\)], "Output"] }, Open ]], Cell["\<\ The lengths of the cycles have a more-than-random structure, note the \ abundance of length 6 :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lencyc = Length /@ crf7\)], "Input"], Cell[BoxData[ \({2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 20, 6, 3, 6, 6, 72, 3, 6, 6, 6, 3, 6, 6, 10, 48, 6, 6, 6, 20, 6, 6, 6, 3, 18, 9, 6, 24, 6, 3, 6, 3, 15, 3, 2, 6}\)], "Output"] }, Open ]], Cell["\<\ The sum of cycle lengths must of course be the size of the wood :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Plus @@ lencyc\), "\n", \(cat[7]\)}], "Input"], Cell[BoxData[ \(429\)], "Output"], Cell[BoxData[ \(429\)], "Output"] }, Open ]], Cell["\<\ the length 6 is very common, the list of different occuring lengths is :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Union[lencyc]\)], "Input"], Cell[BoxData[ \({2, 3, 6, 9, 10, 15, 18, 20, 24, 48, 72}\)], "Output"] }, Open ]], Cell["their frequency is:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((Count[lencyc, #1] &)\) /@ Union[lencyc]\)], "Input"], Cell[BoxData[ \({2, 7, 28, 1, 1, 1, 1, 2, 1, 1, 1}\)], "Output"] }, Open ]], Cell["presented as pairs {length, frequency} :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[{Union[lencyc], \((Count[lencyc, #1] &)\) /@ Union[lencyc]}]\)], "Input"], Cell[BoxData[ \({{2, 2}, {3, 7}, {6, 28}, {9, 1}, {10, 1}, {15, 1}, {18, 1}, {20, 2}, {24, 1}, {48, 1}, {72, 1}}\)], "Output"] }, Open ]], Cell["The cycle lengths have surprisingly small prime factors :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FactorInteger[LCM @@ Union[lencyc]]\)], "Input"], Cell[BoxData[ \({{2, 4}, {3, 2}, {5, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["The cycle structure of the inverse permutation FR .", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[\(cfr7 = mytocyclesmixed[frn];\)]\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(cfr7\)\" is similar to \ existing symbol \"\!\(crf7\)\"."\)], "Message"], Cell[BoxData[ \({0.030999999999999694`\ Second, Null}\)], "Output"] }, Open ]], Cell["The number of cycles is :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[crf7]\)], "Input"], Cell[BoxData[ \(46\)], "Output"] }, Open ]], Cell["\<\ the starters (lowest elements) of the cycles of crf7 and cfr7 are the same:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(starters == First /@ crf7\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ the difference between both permutations is that they have their cycles \ \"reversed\" :\ \>", "Text"], Cell[CellGroupData[{ Cell["cfr7== (Reverse[RotateLeft[#]]& /@ crf7)", "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Permutation F on the starters", "Subsection"], Cell["\<\ The operation RF carries any tree one place further in its cycle. What does \ the 'half operation' F carry it into ? To find out, we first apply F to the \ starters, and then give their positions in cn. This will have the form {cycle \ number, position in cycle} :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f7\[LeftDoubleBracket]starters\[RightDoubleBracket]\)], "Input"], Cell[BoxData[ \({429, 428, 427, 422, 421, 426, 420, 416, 402, 401, 415, 400, 396, 395, 419, 414, 399, 394, 411, 353, 348, 347, 410, 390, 351, 346, 377, 333, 376, 342, 329, 418, 413, 389, 379, 345, 375, 341, 263, 243, 239, 386, 318, 215, 288, 287}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(toCyclePhasefr = Flatten[\((Position[cfr7, #1] &)\) /@ f7\[LeftDoubleBracket]starters\[RightDoubleBracket], 1]\), "\n", \(toCyclePhaserf = Flatten[\((Position[crf7, #1] &)\) /@ f7\[LeftDoubleBracket]starters\[RightDoubleBracket], 1]\)}], "Input"], Cell[BoxData[ \({{1, 2}, {2, 6}, {4, 6}, {3, 6}, {5, 6}, {9, 6}, {10, 6}, {8, 6}, {6, 6}, {7, 6}, {13, 6}, {12, 20}, {11, 6}, {14, 3}, {20, 6}, {21, 6}, {17, 18}, {22, 3}, {19, 2}, {15, 6}, {16, 6}, {18, 3}, {23, 2}, {28, 6}, {25, 10}, {26, 48}, {27, 6}, {24, 6}, {31, 6}, {30, 5}, {29, 6}, {39, 6}, {33, 2}, {40, 3}, {35, 8}, {36, 3}, {41, 6}, {38, 14}, {32, 6}, {34, 3}, {37, 6}, {44, 3}, {43, 8}, {42, 3}, {45, 2}, {46, 2}}\)], "Output"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(toCyclePhaserf\)\" is \ similar to existing symbol \"\!\(toCyclePhasefr\)\"."\)], "Message"], Cell[BoxData[ \({{1, 2}, {2, 2}, {4, 2}, {3, 2}, {5, 2}, {9, 2}, {10, 2}, {8, 2}, {6, 2}, {7, 2}, {13, 2}, {12, 2}, {11, 2}, {14, 2}, {20, 2}, {21, 2}, {17, 56}, {22, 2}, {19, 6}, {15, 2}, {16, 2}, {18, 2}, {23, 6}, {28, 2}, {25, 2}, {26, 2}, {27, 2}, {24, 2}, {31, 2}, {30, 17}, {29, 2}, {39, 2}, {33, 6}, {40, 2}, {35, 12}, {36, 8}, {41, 2}, {38, 12}, {32, 2}, {34, 2}, {37, 2}, {44, 2}, {43, 9}, {42, 2}, {45, 2}, {46, 6}}\)], "Output"] }, Open ]], Cell["\<\ So, the operation F can either leave an element in its cycle, and push it \ further, or switch to an other cycle of the same length. The first three elements of \"toCyclePhasefr\" contain {1,2}, {2,6} and {4,6} \ ; this means that the starter of cycle 1 gets carried (by operation F) to the \ second place of cycle 1; the first element of cycle 2 gets carried to the \ sixth place in cycle 2, and the first element of cycle 3 gets carried all the \ way to the sixth place in cycle 4. This implies a permutation of cycles:\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ mytocycles[First[Transpose[toCyclePhasefr]]] mytocycles[First[Transpose[toCyclePhaserf]]] % == %%\ \>", "Input"], Cell[BoxData[ \({{1}, {2}, {3, 4}, {5}, {6, 9}, {7, 10}, {8}, {11, 13}, {12}, {14}, {15, 20}, {16, 21}, {17}, {18, 22}, {19}, {23}, {24, 28}, {25}, {26}, {27}, {29, 31}, {30}, {32, 39}, {33}, {34, 40}, {35}, {36}, {37, 41}, {38}, {42, 44}, {43}, {45}, {46}}\)], "Output"], Cell[BoxData[ \({{1}, {2}, {3, 4}, {5}, {6, 9}, {7, 10}, {8}, {11, 13}, {12}, {14}, {15, 20}, {16, 21}, {17}, {18, 22}, {19}, {23}, {24, 28}, {25}, {26}, {27}, {29, 31}, {30}, {32, 39}, {33}, {34, 40}, {35}, {36}, {37, 41}, {38}, {42, 44}, {43}, {45}, {46}}\)], "Output"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ So, there are two kinds of cycle in c7, the \"simple\" ones like cycles 1, 2 \ and 5, and those that occur in pairs like cycles 6 and 9. The pairs are those cycles between which a member-tree gets carried by \ alternating F and R oprations. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Starting with F or R ?", "Subsection"], Cell["\<\ Now we can compare (RF)^n with (FR)^n. We can also compare the effect of F (RF)^n and (RF)^n R , say \ post-Flipping or pre-Reversing . Mutatis mutandis, R (FR)^n and (FR)^n F, say post-reversing and \ pre-flipping. First, it is obvious that F (RF)^n == (FR)^n F and (RF)^n R == R (FR)^n. Next, if we denote the cycle-length by 'c', then : (RF)^c == Identity (RF)^(c+n) == (RF)^n and, since for any m : (RF)^m (FR)^m == Identity, (RF)^(c-n) == (FR)^n So, it follows that (RF)^n R == R (RF)^(c-n) == F (RF)^(c-n-1) == R (FR)^n.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Invariants of R and F", "Subsection"], Cell["\<\ What is the behaviour for cycles containing invariants of F and/or R ? These are:\ \>", "Text"], Cell[CellGroupData[{ Cell["r7//Short", "Input"], Cell[BoxData[ TagBox[\(\[LeftSkeleton]1\[RightSkeleton]\), Short]], "Output"] }, Open ]], Cell["The invariants of R are :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fixr = Cases[Range[cat[7]], a_ /; r7\[LeftDoubleBracket]a\[RightDoubleBracket] === a]\)], "Input"], Cell[BoxData[ \({1, 20, 29, 45, 64, 73, 95, 104, 122, 127, 134, 153, 162, 178, 197, 206, 228, 237, 255, 260, 269, 288, 297, 319, 328, 346, 351, 368, 377, 395, 400, 416, 421, 428, 429}\)], "Output"] }, Open ]], Cell["and their number is :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[fixr]\)], "Input"], Cell[BoxData[ \(35\)], "Output"] }, Open ]], Cell["Those of F are :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fixf = Cases[Range[cat[7]], a_ /; f7\[LeftDoubleBracket]a\[RightDoubleBracket] === a]\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(fixf\)\" is similar to \ existing symbol \"\!\(fixr\)\"."\)], "Message"], Cell[BoxData[ \({193, 220, 281, 308, 355}\)], "Output"] }, Open ]], Cell["and there are fewer of them :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[fixf]\)], "Input"], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell["\<\ The following table gives the number of fixed points for F and R as a \ function of n :\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Table[{n, If[EvenQ[n],0,cat[(n-1)/2]], If[EvenQ[n],2 (n-1) cat[n/2-1],n cat[(n-1)/2]] },{n,12}]//TableForm\ \>", "Input"], Cell[BoxData[ TagBox[GridBox[{ {"1", "1", "1"}, {"2", "0", "2"}, {"3", "1", "3"}, {"4", "0", "6"}, {"5", "2", "10"}, {"6", "0", "20"}, {"7", "5", "35"}, {"8", "0", "70"}, {"9", "14", "126"}, {"10", "0", "252"}, {"11", "42", "462"}, {"12", "0", "924"} }, RowSpacings->0, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]], Cell["\<\ The positions of the fixed points of R in the cycles are: {fixed point element, {cyle number, position in that cycle}, length of that \ cycle }\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ pofixr = ({#1, Flatten[Position[cfr7, #1]], \ Length[cfr7[[Flatten[Position[cfr7, #1]][[1]]] ] ]} & ) /@ fixr\ \>", "Input"], Cell[BoxData[ \({{1, {1, 1}, 2}, {20, {19, 1}, 6}, {29, {23, 1}, 6}, {45, {33, 1}, 6}, {64, {8, 3}, 6}, {73, {35, 13}, 18}, {95, {43, 4}, 15}, {104, {27, 3}, 6}, {122, {45, 1}, 2}, {127, {46, 1}, 6}, {134, {46, 4}, 6}, {153, {38, 7}, 24}, {162, {33, 4}, 6}, {178, {17, 45}, 72}, {197, {2, 3}, 6}, {206, {25, 5}, 10}, {228, {38, 19}, 24}, {237, {17, 9}, 72}, {255, {26, 24}, 48}, {260, {12, 10}, 20}, {269, {23, 4}, 6}, {288, {45, 2}, 2}, {297, {35, 4}, 18}, {319, {36, 6}, 9}, {328, {5, 3}, 6}, {346, {26, 48}, 48}, {351, {25, 10}, 10}, {368, {19, 4}, 6}, {377, {27, 6}, 6}, {395, {14, 3}, 3}, {400, {12, 20}, 20}, {416, {8, 6}, 6}, {421, {5, 6}, 6}, {428, {2, 6}, 6}, {429, {1, 2}, 2}}\)], "Output"] }, Open ]], Cell["\<\ ordered per cycle number, they appear in pairs if the cycle lengts are even :\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RotateRight /@ Sort[RotateLeft /@ pofixr]\)], "Input"], Cell[BoxData[ \({{1, {1, 1}, 2}, {429, {1, 2}, 2}, {197, {2, 3}, 6}, {428, {2, 6}, 6}, {328, {5, 3}, 6}, {421, {5, 6}, 6}, {64, {8, 3}, 6}, {416, {8, 6}, 6}, {260, {12, 10}, 20}, {400, {12, 20}, 20}, {395, {14, 3}, 3}, {237, {17, 9}, 72}, {178, {17, 45}, 72}, {20, {19, 1}, 6}, {368, {19, 4}, 6}, {29, {23, 1}, 6}, {269, {23, 4}, 6}, {206, {25, 5}, 10}, {351, {25, 10}, 10}, {255, {26, 24}, 48}, {346, {26, 48}, 48}, {104, {27, 3}, 6}, {377, {27, 6}, 6}, {45, {33, 1}, 6}, {162, {33, 4}, 6}, {297, {35, 4}, 18}, {73, {35, 13}, 18}, {319, {36, 6}, 9}, {153, {38, 7}, 24}, {228, {38, 19}, 24}, {95, {43, 4}, 15}, {122, {45, 1}, 2}, {288, {45, 2}, 2}, {127, {46, 1}, 6}, {134, {46, 4}, 6}}\)], "Output"] }, Open ]], Cell["\<\ the position of the second seems to be half a cyclength c/2 further down than \ the position of the first. So they sit on opposite sides of these even-length cycles. What about the odd cycles containing fixed points of R? \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DeleteCases[pofixr, {a_, {b_, c_}, d_} /; EvenQ[d]]\)], "Input"], Cell[BoxData[ \({{95, {43, 4}, 15}, {319, {36, 6}, 9}, {395, {14, 3}, 3}}\)], "Output"] }, Open ]], Cell["The positions of the fixed points of F are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pofixf = \(({#1, Flatten[Position[cfr7, #1]], Length[cfr7\[LeftDoubleBracket]\(Flatten[ Position[ cfr7, #1]]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ \[RightDoubleBracket]]} &)\) /@ fixf\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(pofixf\)\" is similar \ to existing symbol \"\!\(pofixr\)\"."\)], "Message"], Cell[BoxData[ \({{193, {30, 3}, 20}, {220, {43, 12}, 15}, {281, {36, 2}, 9}, {308, {30, 13}, 20}, {355, {14, 2}, 3}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RotateRight /@ Sort[RotateLeft /@ pofixf]\)], "Input"], Cell[BoxData[ \({{355, {14, 2}, 3}, {193, {30, 3}, 20}, {308, {30, 13}, 20}, {281, {36, 2}, 9}, {220, {43, 12}, 15}}\)], "Output"] }, Open ]], Cell["\<\ So, the odd cycles containing fixed points of R contain fixed points of F \ too. The fixed point of F sitting (c+1)/2 further than that of R. And, in the even cycles, fixed points of F occur in pairs, sitting on \ opposite sides.\ \>", "Text"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Rotation : uprooting and replanting a tree.", "Section"], Cell["\<\ The standard plot of a tree can be traversed from root (at the top) towards \ the first (leftmost) leaf. If we now consider the tree as a rootless graph, than we can define a \ \"rotated tree\" by making the first leaf its new root. The \"old\" root now becomes its last (rightmost) leaf. To a tree with n leaves and a root thus correspond at most n+1 different \ rotated forms, each a valid tree in the wood[n]. If a tree has less than n+1 rotated forms, say only p, then it can be said to \ have a p-fold rotation symmetry. Even then, of course, it must com back to its start position after n+1 turns; \ so p must be a divisor of n+1.\ \>", "Text"], Cell[CellGroupData[{ Cell["A function for Turning (roTating) a tree", "Subsection"], Cell["\<\ This function t produces a \"turned\" tree; it takes the integer \ representation as input & output :\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ test d2b @ t[b2d @ test]\ \>", "Input"], Cell[BoxData[ \({1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0}\)], "Output"], Cell[BoxData[ \({1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ Lets check this by rotating all the trees in wood[4] through a full cyclus : \ all 4+2=6 different positions, plus once more to the starting position again \ :\ \>", "Text"], Cell[CellGroupData[{ Cell["w4=wood[4]", "Input"], Cell[BoxData[ \({170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["NestList[t/@#&, w4 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L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.299499 0.0296238 0.424812 0.107072 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .59524 .60332 m .59524 .4856 L .21429 .36788 L .02381 .25016 L .21429 .36788 L .40476 .25016 L .30952 .13244 L .40476 .25016 L .5 .13244 L .45238 .01472 L .5 .13244 L .54762 .01472 L .5 .13244 L .40476 .25016 L .21429 .36788 L .59524 .4856 L .97619 .36788 L .59524 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.437343 0.0296238 0.562657 0.107072 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0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .62987 .60332 m .62987 .4856 L .28355 .36788 L .11039 .25016 L .02381 .13244 L .11039 .25016 L .19697 .13244 L .15368 .01472 L .19697 .13244 L .24026 .01472 L .19697 .13244 L .11039 .25016 L .28355 .36788 L .45671 .25016 L .28355 .36788 L .62987 .4856 L .97619 .36788 L .62987 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.161654 0.114816 0.286967 0.192264 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.46337 0.29304 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .46337 .60332 m .46337 .45617 L 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.36788 L .62987 .4856 L .97619 .36788 L .62987 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.575188 0.114816 0.700501 0.192264 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.46337 0.29304 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .46337 .60332 m .46337 .45617 L .17033 .30902 L .02381 .16187 L .17033 .30902 L .31685 .16187 L .17033 .30902 L .46337 .45617 L .75641 .30902 L .60989 .16187 L .75641 .30902 L .90293 .16187 L .82967 .01472 L .90293 .16187 L .97619 .01472 L .90293 .16187 L .75641 .30902 L .46337 .45617 L s MathSubEnd P % End of sub-graphic % Start 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0.34632 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .62987 .60332 m .62987 .45617 L .28355 .30902 L .11039 .16187 L .02381 .01472 L .11039 .16187 L .19697 .01472 L .11039 .16187 L .28355 .30902 L .45671 .16187 L .37013 .01472 L .45671 .16187 L .54329 .01472 L .45671 .16187 L .28355 .30902 L .62987 .45617 L .97619 .30902 L .62987 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.161654 0.200009 0.286967 0.277457 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.37013 0.34632 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 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L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .37013 .60332 m .37013 .45617 L .02381 .30902 L .37013 .45617 L .71645 .30902 L .54329 .16187 L .45671 .01472 L .54329 .16187 L .62987 .01472 L .54329 .16187 L .71645 .30902 L .88961 .16187 L .80303 .01472 L .88961 .16187 L .97619 .01472 L .88961 .16187 L .71645 .30902 L .37013 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.575188 0.200009 0.700501 0.277457 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.62987 0.34632 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .62987 .60332 m .62987 .45617 L .28355 .30902 L .11039 .16187 L .02381 .01472 L .11039 .16187 L .19697 .01472 L .11039 .16187 L .28355 .30902 L .45671 .16187 L .37013 .01472 L .45671 .16187 L .54329 .01472 L .45671 .16187 L .28355 .30902 L .62987 .45617 L .97619 .30902 L .62987 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.713033 0.200009 0.838346 0.277457 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.37013 0.34632 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .37013 .60332 m .37013 .45617 L .02381 .30902 L .37013 .45617 L .71645 .30902 L .54329 .16187 L .45671 .01472 L .54329 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0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .53663 .60332 m .53663 .45617 L .24359 .30902 L .09707 .16187 L .02381 .01472 L .09707 .16187 L .17033 .01472 L .09707 .16187 L .24359 .30902 L .39011 .16187 L .24359 .30902 L .53663 .45617 L .82967 .30902 L .68315 .16187 L .82967 .30902 L .97619 .16187 L .82967 .30902 L .53663 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.0238095 0.370394 0.149123 0.447842 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .59524 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.88961 .25016 L .97619 .13244 L .88961 .25016 L .71645 .36788 L .37013 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.437343 0.370394 0.562657 0.447842 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .59524 .60332 m .59524 .4856 L .21429 .36788 L .02381 .25016 L .21429 .36788 L .40476 .25016 L .30952 .13244 L .2619 .01472 L .30952 .13244 L .35714 .01472 L .30952 .13244 L .40476 .25016 L .5 .13244 L .40476 .25016 L .21429 .36788 L .59524 .4856 L .97619 .36788 L .59524 .4856 L s MathSubEnd P % End of 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Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .59524 .60332 m .59524 .4856 L .21429 .36788 L .02381 .25016 L .21429 .36788 L .40476 .25016 L .30952 .13244 L .2619 .01472 L .30952 .13244 L .35714 .01472 L .30952 .13244 L .40476 .25016 L .5 .13244 L .40476 .25016 L .21429 .36788 L .59524 .4856 L .97619 .36788 L .59524 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.0238095 0.455587 0.149123 0.533035 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 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1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .5 .60332 m .5 .45617 L .18254 .30902 L .02381 .16187 L .18254 .30902 L .34127 .16187 L .18254 .30902 L .5 .45617 L .81746 .30902 L .65873 .16187 L .57937 .01472 L .65873 .16187 L .7381 .01472 L .65873 .16187 L .81746 .30902 L .97619 .16187 L .81746 .30902 L .5 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.437343 0.455587 0.562657 0.533035 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.404762 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .40476 .60332 m .40476 .4856 L .02381 .36788 L 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L .21429 .36788 L .02381 .25016 L .21429 .36788 L .40476 .25016 L .30952 .13244 L .40476 .25016 L .5 .13244 L .45238 .01472 L .5 .13244 L .54762 .01472 L .5 .13244 L .40476 .25016 L .21429 .36788 L .59524 .4856 L .97619 .36788 L .59524 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.850877 0.54078 0.97619 0.618228 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.31746 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .5 .60332 m .5 .45617 L .18254 .30902 L .02381 .16187 L .18254 .30902 L .34127 .16187 L .2619 .01472 L .34127 .16187 L .42063 .01472 L .34127 .16187 L .18254 .30902 L .5 .45617 L .81746 .30902 L .65873 .16187 L .81746 .30902 L .97619 .16187 L .81746 .30902 L .5 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.0238095 0.625972 0.149123 0.70342 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.31746 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .5 .60332 m .5 .45617 L .18254 .30902 L .02381 .16187 L .18254 .30902 L .34127 .16187 L .18254 .30902 L .5 .45617 L .81746 .30902 L .65873 .16187 L .57937 .01472 L .65873 .16187 L .7381 .01472 L .65873 .16187 L .81746 .30902 L .97619 .16187 L .81746 .30902 L .5 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.161654 0.625972 0.286967 0.70342 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.404762 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .40476 .60332 m .40476 .4856 L .02381 .36788 L .40476 .4856 L .78571 .36788 L .59524 .25016 L .5 .13244 L .45238 .01472 L .5 .13244 L .54762 .01472 L .5 .13244 L .59524 .25016 L .69048 .13244 L .59524 .25016 L .78571 .36788 L .97619 .25016 L .78571 .36788 L .40476 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.299499 0.625972 0.424812 0.70342 MathSubStart 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L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .5 .60332 m .5 .45617 L .18254 .30902 L .02381 .16187 L .18254 .30902 L .34127 .16187 L .18254 .30902 L .5 .45617 L .81746 .30902 L .65873 .16187 L .57937 .01472 L .65873 .16187 L .7381 .01472 L .65873 .16187 L .81746 .30902 L .97619 .16187 L .81746 .30902 L .5 .45617 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.0238095 0.711165 0.149123 0.788613 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.46337 0.29304 0.456168 0.147151 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .46337 .60332 m .46337 .45617 L .17033 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L .64286 .01472 L .69048 .13244 L .7381 .01472 L .69048 .13244 L .59524 .25016 L .78571 .36788 L .97619 .25016 L .78571 .36788 L .40476 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.299499 0.711165 0.424812 0.788613 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.62987 0.34632 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .62987 .60332 m .62987 .4856 L .28355 .36788 L .11039 .25016 L .02381 .13244 L .11039 .25016 L .19697 .13244 L .15368 .01472 L .19697 .13244 L .24026 .01472 L .19697 .13244 L .11039 .25016 L .28355 .36788 L .45671 .25016 L .28355 .36788 L 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.61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .37013 .60332 m .37013 .4856 L .02381 .36788 L .37013 .4856 L .71645 .36788 L .54329 .25016 L .71645 .36788 L .88961 .25016 L .80303 .13244 L .75974 .01472 L .80303 .13244 L .84632 .01472 L .80303 .13244 L .88961 .25016 L .97619 .13244 L .88961 .25016 L .71645 .36788 L .37013 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.161654 1.05194 0.286967 1.12938 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath 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.01472 L .80303 .13244 L .88961 .25016 L .97619 .13244 L .88961 .25016 L .71645 .36788 L .37013 .4856 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.575188 1.05194 0.700501 1.12938 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.380952 0.485598 0.117721 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 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 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .59524 .60332 m .59524 .4856 L .21429 .36788 L .02381 .25016 L .21429 .36788 L .40476 .25016 L .30952 .13244 L .2619 .01472 L .30952 .13244 L .35714 .01472 L .30952 .13244 L .40476 .25016 L .5 .13244 L .40476 .25016 L .21429 .36788 L .59524 .4856 L .97619 .36788 L .59524 .4856 L s 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161.375}, {83.625, 65.8125}} -> {-15.1972, -64.478, 0.100874, \ 0.29676}, {{164.25, 193.188}, {83.625, 65.8125}} -> {-21.1346, -51.2727, \ 0.118956, 0.236891}, {{196.063, 224.938}, {83.625, 65.8125}} -> {-19.0537, \ -64.478, 0.0917041, 0.29676}, {{5.4375, 34.3125}, {64, 46.1875}} -> \ {-1.62988, -56.0416, 0.100874, 0.237408}, {{37.25, 66.125}, {64, 46.1875}} -> \ {-5.58895, -56.0416, 0.100874, 0.237408}, {{69, 97.875}, {64, 46.1875}} -> \ {-8.04171, -56.0416, 0.100874, 0.237408}, {{100.75, 129.625}, {64, 46.1875}} -> \ {-11.9945, -56.0416, 0.100874, 0.237408}, {{132.5, 161.375}, {64, 46.1875}} -> \ {-14.4472, -56.0416, 0.100874, 0.237408}, {{164.25, 193.188}, {64, 46.1875}} -> \ {-18.3639, -55.9217, 0.100655, 0.236891}, {{196.063, 224.938}, {64, 46.1875}} -> \ {-20.8591, -56.0416, 0.100874, 0.237408}, {{5.4375, 34.3125}, {44.375, \ 26.5625}} -> {-1.62988, -76.1258, 0.100874, 0.29676}, {{37.25, 66.125}, \ {44.375, 26.5625}} -> {-4.98995, -76.1258, 0.0917041, 0.29676}, {{69, \ 97.875}, {44.375, 26.5625}} -> {-10.072, -60.7007, 0.119215, 0.237408}, \ {{100.75, 129.625}, {44.375, 26.5625}} -> {-11.2445, -76.1258, 0.100874, \ 0.29676}, {{132.5, 161.375}, {44.375, 26.5625}} -> {-13.7248, -76.1258, \ 0.0917041, 0.29676}, {{164.25, 193.188}, {44.375, 26.5625}} -> {-21.3846, \ -60.5707, 0.118956, 0.236891}, {{196.063, 224.938}, {44.375, 26.5625}} -> \ {-20.8591, -76.1258, 0.100874, 0.29676}, {{5.4375, 34.3125}, {24.75, 6.9375}} -> \ {-1.65851, -81.9497, 0.10546, 0.29676}, {{37.25, 66.125}, {24.75, 6.9375}} -> \ {-4.98995, -81.9497, 0.0917041, 0.29676}, {{69, 97.875}, {24.75, 6.9375}} -> \ {-9.18187, -65.3599, 0.110045, 0.237408}, {{100.75, 129.625}, {24.75, \ 6.9375}} -> {-12.6758, -65.3599, 0.110045, 0.237408}, {{132.5, 161.375}, \ {24.75, 6.9375}} -> {-13.2248, -81.9497, 0.0917041, 0.29676}, {{164.25, \ 193.188}, {24.75, 6.9375}} -> {-19.244, -81.7744, 0.10523, 0.296113}, \ {{196.063, 224.938}, {24.75, 6.9375}} -> {-21.7617, -81.9497, 0.10546, \ 0.29676}}], Cell[BoxData[ 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0000003oool0oooo02`0oooo00<000000?ooo`3oool01`3oool000P0oooo00<000000?ooo`3oool0 ;03oool00`000000oooo0?ooo`1>0?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool0 CP3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo00L0oooo00080?ooo`030000003o ool0oooo02`0oooo00<000000?ooo`3oool0CP3oool00`000000oooo0?ooo`0;0?ooo`030000003o ool0oooo04h0oooo00<000000?ooo`3oool0;03oool00`000000oooo0?ooo`070?ooo`00203oool0 0`000000oooo0?ooo`0/0?ooo`030000003oool0oooo04h0oooo00<000000?ooo`3oool02`3oool0 0`000000oooo0?ooo`1>0?ooo`030000003oool0oooo02`0oooo00<000000?ooo`3oool01`3oool0 00P0oooo00<000000?ooo`3oool0;03oool00`000000oooo0?ooo`1>0?ooo`030000003oool0oooo 00/0oooo00<000000?ooo`3oool0CP3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo 00L0oooo00080?oooh4000003@3ooon1000000T0oooo003o0?ooob40oooo003o0?ooob40oooo0000 \ \>"], ImageRangeCache->{{{0, 287}, {83.75, 0}} -> {-0.0589635, -0.0147182, \ 0.00737954, 0.00737954}, {{7.9375, 137}, {81.75, 1.9375}} -> {-1.2495, \ -3.21088, 0.0224165, 0.0527574}, {{149.938, 279}, {81.75, 1.9375}} -> \ {-5.18265, -3.21088, 0.0224165, 0.0527574}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] GraphicsArray \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["below are the cycles of woods 5 and 6:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mytocyclesmixed[Ordering[Ordering[t /@ wood[5]]]]\)], "Input"], Cell[BoxData[ \({{1, 23, 22, 21, 19, 14, 42}, {2, 24, 25, 30, 5, 28, 38}, {3, 26, 31, 7, 33, 11, 39}, {4, 27, 34, 16, 10, 37, 29}, {6, 32, 8, 35, 17, 12, 40}, {9, 36, 20, 18, 13, 41, 15}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((Length /@ #1 &)\)[ mytocyclesmixed[Ordering[Ordering[t /@ wood[6]]]]]\)], "Input"], Cell[BoxData[ \({8, 8, 8, 8, 4, 8, 8, 8, 8, 4, 8, 8, 8, 4, 8, 8, 8, 4, 4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Share[]", "Input"], Cell[BoxData[ \(289640\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Divisibility", "Subsubsection"], Cell["\<\ In most cases, the rotations of a tree of order 7 have a cyclicity of 7+2 = \ 9 : 7 counted leaves + the uncounted terminal leaf + the root. This means that cat[n-2] must be divisible by n if n is prime :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{nn = Prime[p], Mod[cat[nn - 2], nn]}, {p, 3, 16}]\)], "Input"], Cell[BoxData[ \({{5, 0}, {7, 0}, {11, 0}, {13, 0}, {17, 0}, {19, 0}, {23, 0}, {29, 0}, {31, 0}, {37, 0}, {41, 0}, {43, 0}, {47, 0}, {53, 0}}\)], "Output"] }, Open ]], Cell["The number of cycles, each of prime length, is predictable :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{nn = Prime[p], cat[nn - 2]/nn}, {p, 3, 16}]\)], "Input"], Cell[BoxData[ \({{5, 1}, {7, 6}, {11, 442}, {13, 4522}, {17, 570285}, {19, 6823410}, {23, 1063750740}, {29, 2397708652276}, {31, 32330394085528}, {37, 84223932294791926}, {41, 16595740773901848790}, {43, 235207409107858096140}, {47, 48023784129303150494760}, {53, 144995956047439931796807852}}\)], "Output"] }, Open ]], Cell["\<\ For n non-prime, cat[n-2] must either be divisible by n, or Mod[ cat[n-2] , n \ ] must divide n. Three series exist : if n is a power of 2, then we get Mod[ cat[n-2] , n ] = \ n/2 ; a second series where Mod[ cat[n-2] , n ] = n/3 ; a third series where Mod[ cat[n-2] , n ] = 2n/3 ; Even for *very* large numbers n, there exists a fast routine to decide to \ which series cat[n-2] belongs. (Not given here).\ \>", "Text"], Cell[BoxData[ \(\(\(({#1, Mod[cat[#1 - 2], #1]} &)\) /@ DeleteCases[Range[3, 259], p_?PrimeQ, 1];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(DeleteCases[%, {_, 0}]\)], "Input"], Cell[BoxData[ \({{4, 2}, {6, 2}, {8, 4}, {9, 6}, {12, 8}, {15, 10}, {16, 8}, {27, 18}, {30, 20}, {32, 16}, {33, 22}, {36, 12}, {39, 13}, {42, 14}, {64, 32}, {81, 54}, {84, 56}, {87, 58}, {90, 30}, {93, 31}, {96, 32}, {108, 36}, {111, 37}, {114, 38}, {117, 78}, {120, 80}, {123, 82}, {128, 64}, {243, 162}, {246, 164}, {249, 166}, {252, 84}, {255, 85}, {256, 128}, {258, 86}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Apply[{#1, \(6\ #2\)\/#1} &, %, 1]\)], "Input"], Cell[BoxData[ \({{4, 3}, {6, 2}, {8, 3}, {9, 4}, {12, 4}, {15, 4}, {16, 3}, {27, 4}, {30, 4}, {32, 3}, {33, 4}, {36, 2}, {39, 2}, {42, 2}, {64, 3}, {81, 4}, {84, 4}, {87, 4}, {90, 2}, {93, 2}, {96, 2}, {108, 2}, {111, 2}, {114, 2}, {117, 4}, {120, 4}, {123, 4}, {128, 3}, {243, 4}, {246, 4}, {249, 4}, {252, 2}, {255, 2}, {256, 3}, {258, 2}}\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Turning the Silva Octifolia.", "Subsection"], Cell["\<\ Now, in the case of wood[7], there are 7+2 = 9 rotated positions per tree. \ The cycle structure of the permutation \"ct7\" contains just two cycles of \ length 3 :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[\(t7 = Ordering[Ordering[t /@ \ w7\ ]];\)]\)], "Input"], Cell[BoxData[ \({0.1719999999999997`\ Second, Null}\)], "Output"] }, Open ]], Cell["w7=wn;", "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[\ \(ct7 = mytocyclesmixed[\ t7\ ];\)\ ]\)], "Input"], Cell[BoxData[ \({0.016000000000000014`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Length /@ ct7\)], "Input"], Cell[BoxData[ \({9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9}\)], "Output"] }, Open ]], Cell["there are 49 cycles :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length@ct7\)], "Input"], Cell[BoxData[ \(49\)], "Output"] }, Open ]], Cell["2 of length 3 and 47 of length 9 :", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[{Union[it = Length /@ ct7], \((Count[it, #1] &)\) /@ Union[it]}]\)], "Input"], Cell[BoxData[ \({{3, 2}, {9, 47}}\)], "Output"] }, Open ]], Cell["\<\ here we generate a plot of the trees with threefold rotation symmetry :\ \>", "Text"], Cell[CellGroupData[{ Cell["Select[ct7,Length[#]===3&]", "Input"], Cell[BoxData[ \({{30, 249, 365}, {102, 390, 152}}\)], "Output"] }, Open ]], Cell["replace the index of a tree by tree itself :", "Text"], Cell[CellGroupData[{ Cell["%/.a_Integer:>w7[[a]]", "Input"], Cell[BoxData[ \({{11148, 14104, 15458}, {11924, 15656, 13130}}\)], "Output"] }, Open ]], Cell["\<\ check the \"rotate\" operator on the stater tree, and compare with the \ permutation result: \ \>", "Text"], Cell[CellGroupData[{ Cell["NestList[t,#,3]& /@ 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\ {-10.247, -10.219, 0.0564747, 0.135374}, {{215.938, 278.938}, {40.9375, 2}} -> \ {-11.4662, -12.4628, 0.048108, 0.16245}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] GraphicsArray \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Combining T and F", "Subsection"], Cell["\<\ We can ask ourselves if all trees in a cycle of ct7 get carried into an other \ cycle of the same length by the mirror symmetry operator F (\"flipping \ over\"). The starters of ct7 can be called \"startct7\":\ \>", "Text"], Cell[CellGroupData[{ Cell["startct7=First/@ct7", "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 34, 35, 36, 40, 41, 43, 45, 46, 48, 50, 51, 55, 57, 60, 62, 65, 69, 77, 78, 102, 106}\)], "Output"] }, Open ]], Cell["Operating with F on them produces:", "Text"], Cell["f7=fn;", "Input"], Cell[CellGroupData[{ Cell["flipstart=f7[[startct7]]", "Input"], Cell[BoxData[ \({429, 428, 427, 422, 421, 426, 420, 416, 402, 401, 415, 400, 396, 425, 419, 414, 399, 411, 391, 382, 354, 353, 381, 352, 348, 390, 380, 351, 377, 343, 334, 332, 329, 424, 413, 398, 409, 379, 350, 331, 407, 339, 368, 264, 258, 253, 244, 249, 216}\)], "Output"] }, Open ]], Cell["\<\ In which cycles of ct7 are these to be found, and at what position? To compare the positions , the following table was built : { cycle #, cycle containing the mirror starter-tree, position in that cycle, \ length of cycle } :\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ it=Table[{i,Position[ct7,flipstart[[i]] ], \t\tLength[ct7[[i]] ]}//Flatten, \t{i,Length[flipstart]}]\t\t\ \>", "Input"], Cell[BoxData[ \({{1, 1, 9, 9}, {2, 44, 8, 9}, {3, 34, 9, 9}, {4, 21, 8, 9}, {5, 49, 7, 9}, {6, 14, 9, 9}, {7, 43, 6, 9}, {8, 20, 7, 9}, {9, 9, 8, 9}, {10, 47, 7, 9}, {11, 41, 9, 9}, {12, 39, 8, 9}, {13, 31, 7, 9}, {14, 6, 9, 9}, {15, 42, 5, 9}, {16, 37, 9, 9}, {17, 28, 8, 9}, {18, 19, 6, 9}, {19, 18, 6, 9}, {20, 8, 7, 9}, {21, 4, 8, 9}, {22, 45, 7, 9}, {23, 46, 6, 9}, {24, 36, 8, 9}, {25, 25, 7, 9}, {26, 48, 2, 3}, {27, 38, 7, 9}, {28, 17, 8, 9}, {29, 30, 6, 9}, {30, 29, 6, 9}, {31, 13, 7, 9}, {32, 40, 7, 9}, {33, 33, 6, 9}, {34, 3, 9, 9}, {35, 35, 9, 9}, {36, 24, 8, 9}, {37, 16, 9, 9}, {38, 27, 7, 9}, {39, 12, 8, 9}, {40, 32, 7, 9}, {41, 11, 9, 9}, {42, 15, 5, 9}, {43, 7, 6, 9}, {44, 2, 8, 9}, {45, 22, 7, 9}, {46, 23, 6, 9}, {47, 10, 7, 9}, {48, 26, 2, 3}, {49, 5, 7, 9}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Transpose[%]", "Input"], Cell[BoxData[ \({{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49}, {1, 44, 34, 21, 49, 14, 43, 20, 9, 47, 41, 39, 31, 6, 42, 37, 28, 19, 18, 8, 4, 45, 46, 36, 25, 48, 38, 17, 30, 29, 13, 40, 33, 3, 35, 24, 16, 27, 12, 32, 11, 15, 7, 2, 22, 23, 10, 26, 5}, {9, 8, 9, 8, 7, 9, 6, 7, 8, 7, 9, 8, 7, 9, 5, 9, 8, 6, 6, 7, 8, 7, 6, 8, 7, 2, 7, 8, 6, 6, 7, 7, 6, 9, 9, 8, 9, 7, 8, 7, 9, 5, 6, 8, 7, 6, 7, 2, 7}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9}}\)], "Output"] }, Open ]], Cell["\<\ it apears that the rotation-cycles are paired by the mirroring : cycle 2 \ mirrors into cycle 44, etc. some cycles contain their own mirror image (fixed points) : 1, 9, 25, 33 and \ 35. Five of them in ct7. This implies trees that are rotated into their own \ mirror image by a number of rotations that is a divisor of the cycle length \ n+2.\ \>", "Text"], Cell[CellGroupData[{ Cell["mytocycles[Transpose[it][[2]]]", "Input"], Cell[BoxData[ \({{1}, {2, 44}, {3, 34}, {4, 21}, {5, 49}, {6, 14}, {7, 43}, {8, 20}, {9}, {10, 47}, {11, 41}, {12, 39}, {13, 31}, {15, 42}, {16, 37}, {17, 28}, {18, 19}, {22, 45}, {23, 46}, {24, 36}, {25}, {26, 48}, {27, 38}, {29, 30}, {32, 40}, {33}, {35}}\)], "Output"] }, Open ]], Cell[TextData[{ "So, we can pick out the lowest cycles ", StyleBox["excluding mirroring :", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell["First/@mytocycles[Transpose[it][[2]]]", "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 29, 32, 33, 35}\)], "Output"] }, Open ]], Cell["the starter-trees of these cycles have indices:", "Text"], Cell[CellGroupData[{ Cell["startct7[[%]]", "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 20, 24, 25, 26, 27, 30, 31, 34, 40, 41, 45}\)], "Output"] }, Open ]], Cell["and the trees themselves are:", "Text"], Cell[CellGroupData[{ Cell["noTnoF7=w7[[%]]", "Input"], Cell[BoxData[ \({10922, 10924, 10930, 10932, 10936, 10954, 10956, 10962, 10964, 10968, 10978, 10980, 10984, 11052, 11058, 11060, 11082, 11096, 11106, 11108, 11112, 11148, 11154, 11170, 11208, 11216, 11442}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[noTnoF7]", "Input"], Cell[BoxData[ \(27\)], "Output"] }, Open ]], Cell["Their positions make no sense:", "Text"], Cell[CellGroupData[{ Cell["{#[[3]],#[[4]]}&/@it", "Input"], Cell[BoxData[ \({{9, 9}, {8, 9}, {9, 9}, {8, 9}, {7, 9}, {9, 9}, {6, 9}, {7, 9}, {8, 9}, {7, 9}, {9, 9}, {8, 9}, {7, 9}, {9, 9}, {5, 9}, {9, 9}, {8, 9}, {6, 9}, {6, 9}, {7, 9}, {8, 9}, {7, 9}, {6, 9}, {8, 9}, {7, 9}, {2, 3}, {7, 9}, {8, 9}, {6, 9}, {6, 9}, {7, 9}, {7, 9}, {6, 9}, {9, 9}, {9, 9}, {8, 9}, {9, 9}, {7, 9}, {8, 9}, {7, 9}, {9, 9}, {5, 9}, {6, 9}, {8, 9}, {7, 9}, {6, 9}, {7, 9}, {2, 3}, {7, 9}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["{#,Count[%,#]}&/@Union[%]", "Input"], Cell[BoxData[ \({{{2, 3}, 2}, {{5, 9}, 2}, {{6, 9}, 9}, {{7, 9}, 15}, {{8, 9}, 11}, {{9, 9}, 10}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Combining T and RF", "Subsection"], Cell["\<\ To compare the positions of the cat[n] different trees in both the RF'ed \ wood and the Rotated wood, following table was built : { i, cycle containing i in RF'ed wood, cycle containing i in the Turned wood \ } :\ \>", "Text"], Cell["crfn=mytocycles[Ordering[Ordering[r/@f/@wood[n]]]]", "Text"], Cell["ctn=ct7;", "Input"], Cell[CellGroupData[{ Cell["rfn=rn[[fn]]", "Input"], Cell[BoxData[ \({429, 428, 422, 427, 421, 402, 401, 416, 426, 420, 396, 400, 415, 395, 354, 353, 348, 352, 347, 382, 381, 411, 425, 419, 391, 399, 414, 394, 334, 333, 343, 351, 346, 377, 380, 410, 390, 329, 332, 342, 376, 328, 264, 263, 258, 262, 257, 244, 243, 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Cell[BoxData[ \({0.01499999999999968`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[crfn]", "Input"], Cell[BoxData[ \(46\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length/@crfn", "Input"], Cell[BoxData[ \({2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 20, 6, 3, 6, 6, 72, 3, 6, 6, 6, 3, 6, 6, 10, 48, 6, 6, 6, 20, 6, 6, 6, 3, 18, 9, 6, 24, 6, 3, 6, 3, 15, 3, 2, 6}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Union[Length/@crfn]", "Input"], Cell[BoxData[ \({2, 3, 6, 9, 10, 15, 18, 20, 24, 48, 72}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["LCM@@%", "Input"], Cell[BoxData[ \(720\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[ctn]", "Input"], Cell[BoxData[ \(49\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length/@ctn", "Input"], Cell[BoxData[ \({9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9}\)], "Output"] }, Open ]], 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.61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .0873 0 m .15079 .08829 L .21429 .17658 L .27778 .08829 L .34127 .17658 L .40476 .26487 L .46825 .17658 L .53175 .08829 L .59524 0 L .65873 .08829 L .72222 .17658 L .78571 .08829 L .84921 0 L .9127 .08829 L .97619 0 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.754707 0.50331 0.97619 0.721429 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.988095 0.47619 [ [ 0 0 0 0 ] [ 1 1.5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1.5 L 0 1.5 L closepath clip newpath 1 0 0 r .008 w .02381 .9881 Mdot .5 1.46429 Mdot gsave .97619 .03571 -322.438 -12.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} 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%%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (,) show 160.938 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (0) show 167.938 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (,) show 179.125 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (0) show 186.125 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (,) show 197.313 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (0) show 204.313 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (,) show 215.500 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (1) show 222.500 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (,) show 233.688 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (1) show 240.688 17.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 12.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 1.000 0.000 0.000 setrgbcolor (,) 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