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Here ", StyleBox["t ", FontSlant->"Italic"], "is the variable which changes from -\[Infinity] to +\[Infinity] as we move \ along the spiral, and ", StyleBox["c", FontSlant->"Italic"], " is a real constant. Since all our considerations will be preserved by \ isometries of hyperbolic 3-space, and these include ", StyleBox["z\[Rule]", FontSlant->"Italic"], "1", StyleBox["/z ", FontSlant->"Italic"], "and ", StyleBox["z\[Rule]", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`z\&_\)]], ", we may as well assume that ", StyleBox["c>", FontSlant->"Italic"], "0. When analyzing the hyperbolic convex hull of the logarithmic spiral, \ and the complement of the logarithmic spiral in the complex plane, the \ important thing to realize is that there is a 1-parameter group preserving \ the subset.\nIn fact, the group of orientation preserving isometries of \ hyperbolic 3-space preserving the logarithmic spiral is isomorphic to the \ group of isometries of the real line. An orientation-preserving isometry of \ hyperbolic 3-space preserving the logarithmic spiral has either \n1) the \ form ", StyleBox["z\[Rule] exp", FontSlant->"Italic"], StyleBox[" (", FontVariations->{"CompatibilityType"->0}], StyleBox["2\[Pi] \[Tau] ", FontSlant->"Italic"], StyleBox["(", FontVariations->{"CompatibilityType"->0}], StyleBox["i + c", FontSlant->"Italic"], StyleBox[")) ", FontVariations->{"CompatibilityType"->0}], StyleBox["z", FontSlant->"Italic"], ", where ", StyleBox["\[Tau]", FontSlant->"Italic"], " is a real constant, or\n2) the form ", StyleBox["z\[Rule] ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ StyleBox["exp", FontSlant->"Italic"], " ", \((2 \[Pi]\ \[Tau]\ \((i\ + \ c)\))\), " "}], "z"], FontSize->16], TraditionalForm]]], ".\nIn terms of isometries of the real line, these correspond to ", StyleBox["t\[Rule]t+\[Tau] ", FontSlant->"Italic"], "and to ", StyleBox["t\[Rule]\[Tau]-t.", FontSlant->"Italic"], "\n\nThe second of these will be of most use to us. Note that it is an \ elliptic involution, with fixed points at ", StyleBox["z\[Rule] \[PlusMinus]exp", FontSlant->"Italic"], " (", StyleBox["\[Pi] \[Tau]", FontSlant->"Italic"], " (", StyleBox["i + c", FontSlant->"Italic"], "))", StyleBox[" z", FontSlant->"Italic"], ". Let \[Alpha](\[Tau],", StyleBox["c", FontSlant->"Italic"], ") be the geodesic in upper half 3-space joining these two points.\n\nThe \ original group acts in a loxodromic manner. There are two hyperbolic planes \ which are relevant. One is the complement \[CapitalOmega] of the logarithmic \ spiral, endowed with the hyperbolic metric coming from the Riemann mapping. \ The other is the convex hull boundary of the logarithmic spiral. In each of \ these the group acts with a fixed geodesic as the axis. These geodesics \ travel from one end of the spiral to the other, either inside \ \[CapitalOmega], or on the convex hull boundary. The two geodesics correspond \ under the nearest point retraction from the floor \[CapitalOmega] to the \ convex hull boundary. The convex hull boundary has the shape of half a snail \ shell. We denote by \[Gamma] the geodesic on the snail shell.\n\nFrom the \ fact that the situation is invariant under a group, we see that the bending \ lines all make the same fixed angle \[Theta](", StyleBox["c", FontSlant->"Italic"], ") with \[Gamma]. Then \[Theta](", StyleBox["c", FontSlant->"Italic"], ") varies from \[Pi]/2 to 0 as ", StyleBox["c", FontSlant->"Italic"], " varies from 0 to \[Infinity]. It would be easy to plot the curve of \ \[Theta](", StyleBox["c", FontSlant->"Italic"], ") against ", StyleBox["c", FontSlant->"Italic"], " but I haven't done so. The bending measure for the logarithmic spiral \ induces a multiple of Lebesgue measure on \[Gamma]. Any smaller measure with \ the same foliation by geodesics at angle \[Theta](", StyleBox["c", FontSlant->"Italic"], ") gives rise to a spiral region whose complement is a second spiral \ region. Any larger measure will give a pleated surface which is not embedded.\ \n\n", "The orbit of any point in the domain \[CapitalOmega] is a spiral. In the \ hyperbolic plane these become lines of constant curvature, at a fixed \ distance from the geodesic \[Gamma].\n\nFinding the bending measure (some \ constant multiple of the Lebesgue measure) along \[Gamma] will enable us to \ compute any type of measurement of bending which interests us. In order to \ find \[Gamma] explicitly, we proceed as follows. The key observation is that \ if an elliptic involution preserves the spiral, then its set of fixed points \ forms a geodesic \[Alpha] which meets \[Gamma] on the snail shell. One \ endpoint of \[Alpha] lies in \[CapitalOmega] on the spiral which is a \ geodesic in the hyperbolic metric on \[CapitalOmega]. The other end can't lie \ in \[CapitalOmega] because an elliptic has only one fixed point in the \ hyperbolic plane, and so the other end must lie on the main logarithmic \ spiral.\n\nThe general elliptic involution with these properties is ", StyleBox["z\[Rule]", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`e\^\(2 \[Pi]\ k\ \((\ i + c\ )\)\)\)]], StyleBox["/ z.", FontSlant->"Italic"], " From the maximal circle which is tangent at 1 to the logarithmic spiral, \ we obtain a single bending line, with endpoints at ", StyleBox["t=0 ", FontSlant->"Italic"], " and at ", StyleBox["t=time[c]. ", FontSlant->"Italic"], "The involution which interchanges these two endpoints has ", StyleBox["k=time[c].", FontSlant->"Italic"], " This involution interchanges the two endpoints of the bending line, 1 and \ ", Cell[BoxData[ \(TraditionalForm\`e\^\(2 \[Pi]\ k\ \((\ i + c\ )\)\)\)]], ". Its fixed points are \[PlusMinus]", Cell[BoxData[ \(TraditionalForm\`e\^\(\[Pi]\ k\ \((\ i + c\ )\)\)\)]], ". The geodesic in upper half 3-space connecting these two fixed points is \ the set of fixed points of the elliptic involution in 3-space. ", "\n\nWe want to study the bending of the convex hull boundary. This means \ we need to compute the maximal circles in the complement of the logarithmic \ spiral. " }], "Text", TextAlignment->Left] }, Open ]], Cell[CellGroupData[{ Cell["Compute the maximal circles", "Section"], Cell["\<\ Because of the group action, we need only find a single maximal \ circle in the complement of the logarithmic spiral. We will find the circle \ passing through the point 1. We need to compute the other tangent \ point.\ \>", "Text"], Cell[BoxData[{ \(Off[ Remove::rmnsm] (*no\ error\ message\[IndentingNewLine]if\ there\ is\ \ nothing\ to\ remove*) \), "\[IndentingNewLine]", \(\(Remove["\<`*\>"]\ ;\) (*remove\ the\ values\ of\ all\ existing\ \ variables*) \), "\[IndentingNewLine]", \(\(On[Remove::rmnsm];\)\)}], "Input"], Cell["The spiral is defined by the formula", "Text"], Cell[BoxData[ \(\(sp[t_] := Exp[2 Pi\ \((c + I)\)\ t];\)\)], "Input"], Cell["\<\ We will find the maximal circle by looking for the equation of the \ centre, as determined by the unit normal at two different points on the \ spiral. The unit normal is \ \>", "Text"], Cell["n[t_] := -I sp'[t]/Abs[sp'[t]];", "Input"], Cell[TextData[{ "The following clever way of finding our circle was shown to me by Allan \ Hayes.\nIf ", StyleBox["r", FontSlant->"Italic"], " is positive, the following equation says that we have a circle of radius \ ", StyleBox["r ", FontSlant->"Italic"], " which is tangent at the two points on the spiral corresponding to ", StyleBox["t", FontSlant->"Italic"], " = 0 and to some general value of ", StyleBox["t", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell["\<\ equ=sp[0]+r*n[0] == sp[t]-r*n[t]; rsol=Solve[equ, r] \t(*solve for r in terms of the other variables*)\ \>", "Input"], Cell[BoxData[ \({{r \[Rule] \(\(-1\) + \[ExponentialE]\^\(2\ \((\[ImaginaryI] + c)\)\ \ \[Pi]\ t\)\)\/\(\(-\(\(\[ImaginaryI]\ \((\[ImaginaryI] + c)\)\)\/Abs[\ \[ImaginaryI] + c]\)\) - \(\[ImaginaryI]\ \((\[ImaginaryI] + c)\)\ \ \[ExponentialE]\^\(2\ \((\[ImaginaryI] + c)\)\ \[Pi]\ t - 2\ \[Pi]\ Re[\((\ \[ImaginaryI] + c)\)\ t]\)\)\/Abs[\[ImaginaryI] + c]\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The general solution found by ", StyleBox["Mathematica", FontSlant->"Italic"], " is a complex number. We want to find ", StyleBox["t", FontSlant->"Italic"], " such that ", StyleBox["r", FontSlant->"Italic"], " is real. So we direct our attention to the imaginary part of the above \ solution.\n", "ComplexExpand expands an expression assuming that all symbols stand for \ real variables, converting to use of the real and imaginary part functions \ wherever possible. After doing this, we simplify the results." }], "Text"], Cell[CellGroupData[{ Cell["\<\ num=ComplexExpand[Im[r]/.rsol[[1]], \tTargetFunctions->{Re,Im}]//Simplify\ \>", "Input"], Cell[BoxData[ \(\(c\ \((\(-1\) + \[ExponentialE]\^\(2\ c\ \[Pi]\ t\))\) + \((1 + \ \[ExponentialE]\^\(2\ c\ \[Pi]\ t\))\)\ Tan[\[Pi]\ t]\)\/\(2\ \@\(1 + \ c\^2\)\)\)], "Output"] }, Open ]], Cell["\<\ We want this to be zero, so we only need to look at the numerator. \ % means \"the result of the preceding computation\". We convert to \ trigonometric functions and then multiply by various non-zero functions, \ until we get a nice simple condition.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ num1=ExpToTrig[Numerator[num]]; num2=Simplify[num1,c>0 && t>0]; (*simplify the result of the preceding computation under the assumption that \ c>0 and t>0*) num3 = Simplify[num2/((2 Sec[\[Pi]*t])(Cosh[c*\[Pi]*t] + Sinh[c*\[Pi]*t]))]; Expand[num3/(Cosh[c \[Pi] t]*Cos[\[Pi]*t])] Clear[\"num*\"];\ \>", "Input"], Cell[BoxData[ \(Tan[\[Pi]\ t] + c\ Tanh[c\ \[Pi]\ t]\)], "Output"] }, Open ]], Cell[BoxData[ \(tst[t_, c_] := Tan[\[Pi]\ t] + c\ Tanh[c\ \[Pi]\ t]\)], "Input"], Cell[TextData[{ "This is a nice simple formula whose smallest positive solution for ", StyleBox["t ", FontSlant->"Italic"], "gives the second point of contact for the maximal circle tangent at 1.\n\ For c>0, in each t-interval [(n-1/2), (n+1/2)], n an integer, it increases \ from -\[Infinity] to to \[Infinity]; so there is one zero in each interval. \n\ As c\[LongRightArrow] \[Infinity] the zeros move from the middle to the left \ of the intervals." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Do[str=\"c=\"<>ToString[c]; \tPlot[tst[t,c],{t,0,5}, \t\tPlotRange->{-1,1}, Frame->True,PlotLabel->str], {c,0,4,2} ]\ \>", "Input"], Cell[CellGroupData[{ 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 0.190476 0.309017 0.309017 [ [.02381 -0.0125 -3 -9 ] [.02381 -0.0125 3 0 ] [.21429 -0.0125 -3 -9 ] [.21429 -0.0125 3 0 ] [.40476 -0.0125 -3 -9 ] [.40476 -0.0125 3 0 ] [.59524 -0.0125 -3 -9 ] [.59524 -0.0125 3 0 ] [.78571 -0.0125 -3 -9 ] [.78571 -0.0125 3 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [ 0 0 -0.125 0 ] [-0.0125 .07725 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Now we do a careful computation of where the maximal circle tangent at \ ", StyleBox["t", FontSlant->"Italic"], "=0 again touches the spiral when ", StyleBox["c", FontSlant->"Italic"], "=1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(carefulTime[c_]\ := \ t /. FindRoot[ tst[t, c], {t, .501}, \[IndentingNewLine]WorkingPrecision \ \[Rule] 70];\)\), "\[IndentingNewLine]", \(carefulTime[1]\)}], "Input"], Cell[BoxData[ \(0.7528093655709698845373871364101707096158800084792226485643434568144457\ 320592391`70\)], "Output"] }, Open ]], Cell[TextData[{ "\nWe next compute the real part of ", StyleBox["r", FontSlant->"Italic"], ", the radius of the circle we are looking for." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[a, c, d, t]; a = Simplify[ ComplexExpand[Re[r /. rsol[\([1]\)]], TargetFunctions \[Rule] {Re, Im}], {t > 0, c > 0}]\)], "Input"], Cell[BoxData[ \(\(\(-1\) + \[ExponentialE]\^\(2\ c\ \[Pi]\ t\) - c\ \((1 + \ \[ExponentialE]\^\(2\ c\ \[Pi]\ t\))\)\ Tan[\[Pi]\ t]\)\/\(2\ \@\(1 + \ c\^2\)\)\)], "Output"] }, Open ]], Cell[TextData[{ " We substitute for Tan[\[Pi] ", StyleBox["t", FontSlant->"Italic"], "], using ", StyleBox["tst", FontSlant->"Italic"], "[t] == 0." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ a /. Tan[\[Pi]\ t]\ \[Rule] \ \(-c\)\ Tanh[c\ \[Pi]\ t]]\)], "Input"], Cell[BoxData[ \(1\/2\ \@\(1 + c\^2\)\ \((\(-1\) + \[ExponentialE]\^\(2\ c\ \[Pi]\ \ t\))\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["nv", FontSlant->"Italic"], "[", StyleBox["c", FontSlant->"Italic"], "]", StyleBox[" ", FontSlant->"Italic"], "is the normal vector at ", StyleBox["t", FontSlant->"Italic"], "=0", ", with the dependence on ", StyleBox["c", FontSlant->"Italic"], " explicit", ". ", StyleBox["n", FontSlant->"Italic"], "[0] was the same thing conceptually, but it was expressed as a complex \ number, and the dependence on c was implicit. The idea will be to take ", StyleBox["k=time[c]", FontSlant->"Italic"], ", the time of the first tangency to the maximal circle after ", StyleBox["t", FontSlant->"Italic"], "=0, ", "but we don't do this for now. ", StyleBox["r[c,k]", FontSlant->"Italic"], " is the radius of the maximal circle with the dependence on ", StyleBox["c ", FontSlant->"Italic"], "and ", StyleBox["k=time[c]", FontSlant->"Italic"], " made explicit." }], "Text"], Cell[BoxData[{ \(\(Clear[r, c, k, nv];\)\), "\n", \(\(r[c_, k_]\ := \ 1\/2\ \@\(1 + c\^2\)\ \((\(-1\) + Exp[2\ c*\[Pi]\ *k])\);\)\), "\n", \(\(nv[c_]\ = ComplexExpand[{Re[n[0]], Im[n[0]]}, TargetFunctions \[Rule] {Re, Im}];\)\)}], "Input"], Cell[TextData[{ "The next function gives the centre of the maximal circle, provided we put \ ", StyleBox["k=time[c].", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ \(\(\(\(Clear[ centre]\) \)\(;\)\( (*remove\ all\ previous\ values*) \)\(\ \[IndentingNewLine]\)\(\(centre[c_, k_]\ := \ {1, 0} + r[c, k]*nv[c]\) \)\(;\)\(\n\)\)\)], "Input"], Cell["\<\ Now let's draw some pictures and make sure the computation is \ correct. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[str = StringJoin["\", ToString[c]]; (*make\ the\ 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Cell[TextData[{ "Look for the point on the convex hull boundary where the unique geodesic \ \[Gamma] on the convex hull boundary preserved by the 1-parameter group \ meets the geodesic \[Alpha] of fixed points of the\nelliptic involution \ interchanging the two points of contact of the maximal circle. The two points \ of contact are {1,0} and ", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "]", StyleBox[".", FontSlant->"Italic"], " The two fixed points of the involution in the plane are \[PlusMinus]", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,k", FontSlant->"Italic"], "]", StyleBox[". ", FontSlant->"Italic"], "The last line of the next computation is the expression which is zero at \ the intersection of the two intervals connecting these pairs of points. Each \ such interval is the image of the vertical projection of one of the two \ geodesics onto the complex plane." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[c, k, term, vec, equ2, s, t]; term[c_, k_] = ComplexExpand[ Exp[Pi*k*\((I + c)\)], \[IndentingNewLine]TargetFunctions \[Rule] {Re, Im}]; (*the\ second\ tangent\ point\ of\ the\ maximal\ circle, \ \[IndentingNewLine]expressed\ as\ a\ complex\ number*) \), "\n", \(\(vec[c_, k_] = {\(term[c, k]\)[\([1]\)], \(term[c, k]\)[\([2]\)]/ I};\)\n (*the\ second\ tangent\ point, \ expressed\ as\ a\ vector*) \), "\[IndentingNewLine]", \(equ2 = \((1 - t)\) {1, 0} + t*vec[c, 2 k] - s*vec[c, k]\)}], "Input"], Cell[BoxData[ \({1 - t - \[ExponentialE]\^\(c\ k\ \[Pi]\)\ s\ Cos[ k\ \[Pi]] + \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\ t\ Cos[ 2\ k\ \[Pi]], \(-\[ExponentialE]\^\(c\ k\ \[Pi]\)\)\ s\ Sin[ k\ \[Pi]] + \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\ t\ Sin[ 2\ k\ \[Pi]]}\)], "Output"] }, Open ]], Cell[TextData[{ "The above expression has to equal 0 at the intersection of the two \ intervals. We now compute the ", StyleBox["(x,y)-", FontSlant->"Italic"], "coordinates of the intersection \[Alpha]", StyleBox["\[Intersection]", "Title", FontSize->9], StyleBox["\[Gamma]", "Title", FontSize->12], StyleBox[" ", "Title", FontSize->9], "in terms of c and k. " }], "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[c, k, t, s, xycoords, fact, tssol]; tssol = Solve[{equ2[\([1]\)] \[Equal] 0, equ2[\([2]\)] \[Equal] 0}, {t, s}] // Simplify;\), "\[IndentingNewLine]", \(s /. tssol\), "\[IndentingNewLine]", \(fact\ = \((\ s*Cos[k*\[Pi]]*\[ExponentialE]\^\(c\ k\ \[Pi]\))\) /. tssol\ [\([1]\)]\), "\[IndentingNewLine]", \(vec[c, k]\), "\[IndentingNewLine]", \(xycoords[c_, k_] = \ fact*{1, Tan[k*\[Pi]]}\)}], "Input"], Cell[BoxData[ \({\(2\ \[ExponentialE]\^\(c\ k\ \[Pi]\)\ Cos[k\ \[Pi]]\)\/\(1 + \ \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\)}\)], "Output"], Cell[BoxData[ \(\(2\ \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\ Cos[k\ \[Pi]]\^2\)\/\(1 + \ \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\)\)], "Output"], Cell[BoxData[ \({\[ExponentialE]\^\(c\ k\ \[Pi]\)\ Cos[ k\ \[Pi]], \[ExponentialE]\^\(c\ k\ \[Pi]\)\ Sin[ k\ \[Pi]]}\)], "Output"], Cell[BoxData[ \({\(2\ \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\ Cos[k\ \[Pi]]\^2\)\/\(1 + \ \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\), \(2\ \[ExponentialE]\^\(2\ c\ k\ \[Pi]\ \)\ Cos[k\ \[Pi]]\ Sin[k\ \[Pi]]\)\/\(1 + \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\ \)}\)], "Output"] }, Open ]], Cell[TextData[{ "The bending line with one endpoint at 1 and the other endpoint at ", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "]", StyleBox[" ", FontSlant->"Italic"], "is a vertical semicircle in upper half 3-space. We can determine a point \ on this semicircle by specifying the angle from the centre of the semicircle, \ with 0 angle representing 1 and angle \[Pi] representing ", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "]. Let ", StyleBox["cosAngInter ", FontSlant->"Italic"], "be the cosine of the angle at the intersection point \[Alpha]", StyleBox["\[Intersection]", "Title", FontSize->9], StyleBox["\[Gamma]", "Title", FontSize->12], ". This cosine can be read off in the plane from ", StyleBox["tssol ", FontSlant->"Italic"], "above. We will use the sine and cosine to determine \[Alpha]", StyleBox["\[Intersection]", "Title", FontSize->9], StyleBox["\[Gamma]", "Title", FontSize->12], " as a point of upper half 3-space. (1-2", StyleBox["t", FontSlant->"Italic"], ") is the quantity that tells us how far to go round the semicircle." }], "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[BoxData[ \(\(Simplify[Together[1 - 2 t /. tssol]]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \(\(\(-1\) + \[ExponentialE]\^\(2\ c\ k\ \[Pi]\)\)\/\(1 + \[ExponentialE]\ \^\(2\ c\ k\ \[Pi]\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "It's hard to get ", StyleBox["Mathematica", FontSlant->"Italic"], " to simplify this. So we give up and type in the values by hand." }], "Text"], Cell[BoxData[{ \(\(Clear[cosAngInt, sinAngInt];\)\), "\[IndentingNewLine]", \(\(cosAngInt\ = \ Tanh[c*k*\[Pi]];\)\), "\[IndentingNewLine]", \(\(sinAngInt = \ Sech[c*k*\[Pi]];\)\)}], "Input"], Cell[TextData[{ "The next thing to find is \[Alpha]", StyleBox["\[Intersection]", "Title", FontSize->9], StyleBox["\[Gamma]", "Title", FontSize->12], " in terms of ", StyleBox["c", FontSlant->"Italic"], " and ", StyleBox["k", FontSlant->"Italic"], ". We also check our previous calculations." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[c, k, centreSemiCircle, norm, horizVect, nHV, zcoord, alphaInterGamma]; centreSemiCircle = \ Flatten[{\(({1, 0}\ + \ vec[c, 2 k])\)/2, 0}];\), "\n", \(\(norm[{u_, v_, w_}]\ := \ Sqrt[u^2\ + \ v^2\ + \ w^2];\)\), "\n", \(\(horizVect\ = \ Flatten[{\(({1, 0}\ - \ vec[c, 2 k])\)/2, 0}];\)\), "\n", \(\(nHV\ = \ norm[horizVect];\)\), "\n", \(\(alphaInterGamma[c_, k_]\ = \ centreSemiCircle\ + \ cosAngInt*horizVect\ + \ sinAngInt*{0, 0, nHV};\)\), "\n", \(\(xycoords1[c_, k_] = \ Drop[alphaInterGamma[c, k], \(-1\)];\)\), "\n", \(\(zcoord[c_, k_]\ := \ \(alphaInterGamma[c, k]\)[\([3]\)];\)\), "\n", \(xycoords[c, k]\ - \ xycoords1[c, k] // Simplify\)}], "Input"], Cell[BoxData[ \({0, 0}\)], "Output"] }, Open ]], Cell["\<\ The formula for a point at angle \[Theta] along the bending line is\ \ \>", "Text"], Cell[BoxData[ \(centreSemiCircle + Cos[\[Theta]] . horizVect + sin[\[Theta]] . {0, 0, nHV}\)], "DisplayFormula"], Cell["\<\ The tangent vector to the bending line at \[Theta] is \ \>", "Text"], Cell[BoxData[ \(\(-sin[\[Theta]] . horizVect\) + cos[\[Theta]] . {0, 0, nHV}\)], "DisplayFormula"], Cell[TextData[{ "Now let's work out a tangent vector (not of unit hyperbolic length) to the \ bending line at the intersection point \[Alpha]", StyleBox["\[Intersection]", "Title", FontSize->9], StyleBox["\[Gamma]", "Title", FontSize->12], " , pointing from 1 to ", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "]." }], "Text"], Cell[BoxData[ \(\(\(\(Clear[ tangVectToBendingLine]\) \)\(;\)\(\(tangVectToBendingLine[c_, k_]\ = \ \(-Sech[c*k*\[Pi]]\)*horizVect\ + \ Tanh[c*k*\[Pi]] {0, 0, nHV}\) \)\(;\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[TextData[{ ".Now we go back to drawing some pictures, just to make sure that we \ haven't made some terrible blunder. Below we will show a plot of the circle \ containing the four endpoints. This is ", StyleBox["not", FontWeight->"Bold"], " the maximal circle. We show the vertical projection of the bending line \ and the vertical projection of the fixed points of the elliptic involution. \ The second of these is the interval through the origin.\n We first find the \ circle that contains the four endpoints 1, ", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "]", StyleBox[", vec", FontSlant->"Italic"], "[", StyleBox["c,k", FontSlant->"Italic"], "]", StyleBox[", -vec", FontSlant->"Italic"], "[", StyleBox["c,k", FontSlant->"Italic"], "]", "." }], "Text"], Cell[BoxData[{ \(\(Clear[circ, circ1, c, k, x, y, x0, y0];\)\), "\n", \(\(circ[{x_, y_}] := \((x - x0)\)^2\ + \ \((y - y0)\)^2\ - \ rSquared;\) (*the\ equation\ of\ the\ circle*) \), "\n", \(\(circ1[c_, k_] = {{x0, y0}, Sqrt[rSquared]} /. \(Solve[{circ[{1, 0}] \[Equal] 0, circ[vec[c, k]] \[Equal] 0, circ[\(-vec[c, k]\)] \[Equal] 0}, \[IndentingNewLine]{x0, y0, rSquared}]\)[\([1]\)];\)\ (*the\ solution\ for\ the\ centre\ \ and\ the\ radius\ squared\ of\ the\ circle\ containing\ the\ four\ \(\(points\ \)\(.\)\)*) \)}], "Input"], Cell[TextData[{ "Below we show a plot of the circle containing the four endpoints. This is \ ", StyleBox["not", FontWeight->"Bold"], " the maximal circle. We show the vertical projection of the bending line \ and the vertical projection of the fixed points of the elliptic involution. \ The second of these is the interval through the origin." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[circ2, k, c, l1, l2, pts, txt];\)\), "\n", \(c = .5; k = time[c];\), "\n", \(\(l1 = Line[{{1, 0}, vec[c, 2 k]}];\)\), "\n", \(\(l2 = Line[{vec[c, k], \(-vec[c, k]\)}];\)\), "\n", \(\(pts = {PointSize[ .015], Point[xycoords[c, k]], Point[\(circ1[c, k]\)[\([1]\)]], Point[vec[c, k]]};\)\), "\n", \(\ \(circ2 = \ Circle\ @@ circ1[c, k];\)\), "\n", \(\(str = StringJoin["\", ToString[c]];\)\), "\n", \(\(txt\ = \ Text[str, Scaled[{ .2, .5}]];\)\), "\n", \(\(spiral = ParametricPlot[{Re[sp[t]], Im[sp[t]]}, {t, \(-2\), 1.01}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(\(Show[{Graphics[{l1, l2, pts, circ2, txt}]}, \[IndentingNewLine]{\[IndentingNewLine]Axes \[Rule] True, \[IndentingNewLine]AspectRatio \[Rule] 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FontSlant->"Italic"], ", wewill find the angle at which the maximal circle meets its image after \ time ", StyleBox["t", FontSlant->"Italic"], " under the spiral flow. We firstw", "ork out the angle between two arbitrary circles" }], "Text"], Cell[BoxData[{ \(\(Clear[x, y, x1, y1, x2, y2, r1, r2];\)\), "\[IndentingNewLine]", \(\(Clear[intersectionPtSoln, intersectionPt, bending];\)\), "\[IndentingNewLine]", \(\(Simplify[\(Solve[{\[IndentingNewLine]\((x - x1)\)^2\ + \ \((y - y1)\)^2\ \[Equal] \ r1^2, \[IndentingNewLine]\((x - x2)\)^2\ + \ \((y - y2)\)^2\ \[Equal] \ r2^2\[IndentingNewLine]}, {x, y}]\)[\([1]\)]];\)\), "\[IndentingNewLine]", \(\({x, y} /. %;\)\), "\[IndentingNewLine]", \(\(Simplify[%, x1 > x2];\)\), "\[IndentingNewLine]", \(\(intersectionPt[{{x1_, y1_}, r1_}, {{x2_, y2_}, r2_}] = \ %;\)\)}], "Input"], Cell[TextData[{ "Note that the simplification two lines above does not work with x1 \ \[NotEqual] x2. This seems slightly unreasonable of ", StyleBox["Mathematica, ", FontSlant->"Italic"], "but it may be because it needs the square root of ", Cell[BoxData[ \(TraditionalForm\`\(\(\((x1 - x2)\)\^2\)\(.\)\)\)]], " We get an answer only up to sign, but this doesn't matter\[LongDash]" }], "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(bending[{{x1_, y1_}, r1_}, {{x2_, y2_}, r2_}] := \ Module[{v1, v2}, \[IndentingNewLine]v1\ = \ {x1, y1} - intersectionPt[{{x1, y1}, r1}, {{x2, y2}, r2}]; \[IndentingNewLine]v2\ = \ \ {x2, y2} - intersectionPt[{{x1, y1}, r1}, {{x2, y2}, r2}]; \[IndentingNewLine]ArcTan @@ v1\ - \ ArcTan @@ v2\ (*ArcTan\ can\ take\ two\ coordinates\ as\ argument\ in\ \ Mathematica, \ and\ then\ it\ gives\ a\ value\ in\ the\ quadrant\ of\ the\ \ corresponding\ vector*) \[IndentingNewLine]];\)\)\)], "Input"], Cell[TextData[{ "Applying the 1-parameter group, the circle moves according to the matrix \ derived from complex multiplication by ", Cell[BoxData[ FormBox[ StyleBox[\(\[ExponentialE]\^\(2 \[Pi]\ t\ \((\ \[ImaginaryI] + c)\)\)\), FontSize->14], TraditionalForm]]], ". The radius changes according to the absolute value only. The bending is \ then an analytic function of ", StyleBox["t.", FontSlant->"Italic"], " We are interested only in the linear coefficient. Actually the bending is \ positive. But putting in absolute values would destroy the analyticity, so we \ don't force the sign to be positive." }], "Text"], Cell[BoxData[{ \(\(Clear[c, k];\)\), "\n", \(\(mat[t_, c_]\ = \ Exp[2 Pi*t* c] {\[IndentingNewLine]{Cos[ 2 Pi*t], \(-Sin[2 Pi*t]\)}, \[IndentingNewLine]{Sin[ 2 Pi*t], Cos[2 Pi*t]}};\)\)}], "Input"], Cell[TextData[{ "We express the bending as a function of ", StyleBox["t", FontSlant->"Italic"], " and then take a power series. The derivative is the coefficient of the \ term ", StyleBox["t. ", FontSlant->"Italic"], "We use the function ", StyleBox["LeafCount ", FontSlant->"Italic"], " to see how complicated the expressions get" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ben\ = \ Series[bending[{centre[c, k], r[c, k]}, \[IndentingNewLine]\t\t\t{mat[t, c] . centre[c, k], Exp[2 Pi*t*c]*r[c, k]}], {t, 0, 1}];\)\), "\n", \(\(bendingDerivative[c_, k_] = SeriesCoefficient[ben, 1];\)\), "\n", \(LeafCount[ben]\), "\[IndentingNewLine]", \(LeafCount[bendingDerivative[c, k]]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \(59534\)], "Output"], Cell[BoxData[ \(59527\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Computing the hyperbolic speed.\ \>", "Section"], Cell[TextData[{ "Now we want to calculate the hyperbolic speed at ", StyleBox["t", FontSlant->"Italic"], "=0 of the special point \[Alpha]", StyleBox["\[Intersection]", "Title", FontSize->9], StyleBox["\[Gamma]", "Title", FontSize->12], " for which we have worked out coordinates. We work out the first two terms \ of the power series in ", StyleBox["t ", FontSlant->"Italic"], "and then concentrate on the derivative as a function of ", StyleBox["c ", FontSlant->"Italic"], "and ", StyleBox["time[c]", FontSlant->"Italic"], " " }], "Text"], Cell[BoxData[{ \(\(Clear[u, v, w, deriv, hypspeed];\)\), "\[IndentingNewLine]", \(\(\(Series[#, {t, 0, 1}]\ &\)\ /@ \[IndentingNewLine]Flatten[{mat[t, c] . xycoords[c, k], Exp[2 Pi*t*c]*zcoord[c, k]}];\)\), "\[IndentingNewLine]", \(\(deriv[c_, k_] = \(SeriesCoefficient[#, 1] &\)\ /@ %;\)\), "\[IndentingNewLine]", \(\(hypspeed[c_, k_]\ = norm[\ deriv[c, k]/ zcoord[c, k]];\) (*the\ hyperbolic\ length\ of\ the\ tangent\ vector\ to\ \ \[Gamma], \ when\ parametrized\ by\ t*) \)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Bending angle per unit time divided by hyperbolic speed.", "Section"], Cell["\<\ We take the bending derivative divided by the hyperbolic speed. \ Let's see how complicated the expression is\ \>", "Text"], 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we can read off \ the maximum more accurately. (This part commented out to reduce the time the \ program takes to run.)" }], "Text"], Cell[BoxData[ \( (*\[IndentingNewLine]Off[General::spell1]; \n Off[General::spell]; \[IndentingNewLine]xmin = .48558; \n\ xmax = .48622; \ ymin = .979894; \ ymax = .979898; \n xvalue = .48589; \ yvalue = .9798961; \n p = Plot[2*quot[c, time[c]]/Pi, {c, xmin, xmax}, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \n ls = Graphics[{Line[{{xmin\ , yvalue}, {xmax, yvalue}}], Line[{{xvalue, ymin}, {xvalue, ymax}}]}]; \n Show[{ls, p}, {\[IndentingNewLine]Axes \[Rule] True, \[IndentingNewLine]PlotRange \[Rule] {{xmin\ , xmax}, {ymin, ymax}}, \[IndentingNewLine]AxesOrigin \[Rule] {xmin\ , ymin}, \[IndentingNewLine]DisplayFunction \[Rule] \ $DisplayFunction\[IndentingNewLine]}]; \n On[General::spell1]; \[IndentingNewLine]On[ General::spell];\[IndentingNewLine]*) \)], "Input"], Cell[TextData[{ "We see that the maximum bending is .9798961\[Times] \[Pi] / 2. This occurs \ with the logarithmic spiral having ", StyleBox["c", FontSlant->"Italic"], "=.48589. Let's redo this with an accurate computation. The last line shows \ that the first 66 digits of the answer are correct. In particular, the \ maximum long range bending per unit hyperbolic length three lines up has all \ its digits correct." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(c = .48589`70; \ k = carefulTime[c]\), "\n", \(quot[c, k]*2/Pi\), "\n", \(Precision[%]\)}], "Input"], Cell[BoxData[ \(0.8727232338919547554419108816013907817454360705472660213764818760438214\ 84375985248`70\)], "Output"], Cell[BoxData[ \(0.9798960895786688304234634152109445821797144524937906790793319281507035\ 53953630396`66.2078\)], "Output"], Cell[BoxData[ \(66\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The Riemann mapping and the extremal qc map", "Section"], Cell[TextData[{ "We will work out the Riemann mapping ", StyleBox[" f", FontSlant->"Italic"], " : \[CapitalOmega] \[LongRightArrow]", Cell[BoxData[ \(TraditionalForm\`U\^2\)]], "with domain the complement of the logarithmic spiral and with values in \ the upper half plane . 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In hyperbolic coordinates, the speed of the curve is ", Cell[BoxData[ RowBox[{ StyleBox["\[Pi]", FontSlant->"Plain"], StyleBox[" ", FontSlant->"Plain"], RowBox[{ RowBox[{ StyleBox["(", FontSlant->"Plain"], RowBox[{ StyleBox["c", FontSlant->"Italic"], StyleBox["+", FontSlant->"Plain"], StyleBox[" ", FontSlant->"Plain"], StyleBox[ FractionBox["1", StyleBox["c", FontSlant->"Italic"]], "Title", FontSize->10]}], StyleBox[")", FontSlant->"Plain"]}], StyleBox[".", FontSlant->"Plain"]}]}]], TextAlignment->Center], "Now we work out the speed in hyperbolic 3-space divided by the speed in \ the floor, with the hyperbolic metric. The first plot confirms that the ratio \ of speeds, as ", StyleBox["c", FontSlant->"Italic"], " tends to \[Infinity], the ratio tends to 1/2. The second plot is rescaled \ vertically by \[Pi] / 2. This confirms that, as ", StyleBox["c", FontSlant->"Italic"], " tends to 0, the ratio of speeds tends to \[Pi] / 2, which is another \ confirmation that our computations are accurate. 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Here \[Gamma] \ is the positive ", StyleBox["y", FontSlant->"Italic"], "-axis.\nThe bending line is a semicircle with endpoints 1+cos(\[CurlyPhi]) \ and -1+cos(\[CurlyPhi]), corresponding to", StyleBox["\nvec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "] and to 1 respectively. Here cos( \[CurlyPhi] ) is given by the \ expression" }], "Text"], Cell[BoxData[ \(cosAngAlphaInterGamma[c, time[c]]\)], "DisplayFormula"], Cell[TextData[{ "We map the upper half plane by log to a horizontal strip of height \[Pi]. \ The endpoints of the bending line are mapped to 0 and to ", StyleBox["u", FontSlant->"Italic"], "=-log", StyleBox["(", FontSize->18], Cell[BoxData[ \(TraditionalForm\`\(1 + cos(\[CurlyPhi])\)\/\(1 - \ cos(\[CurlyPhi])\)\)], FontSize->14], StyleBox[") ", FontSize->18], "+ \[Pi] ", StyleBox["i", FontSlant->"Italic"], ". The point on the upper edge is to the left of the point on the lower \ edge. 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This time we apply the mapping\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["f", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{"(", StyleBox["z", FontSlant->"Italic"], ")"}]}], "=", " ", RowBox[{\(-\((\(c - i\)\/\(2 c\))\)\), " ", "log", RowBox[{ RowBox[{"(", StyleBox["z", FontSlant->"Italic"], ")"}], "."}]}]}]], "DisplayFormula", TextAlignment->Center], Cell[TextData[{ "which maps the complement of the logarithmic spiral to the horizontal \ strip of height \[Pi].\nUnder this map, 1 goes to 0, and the point of the \ spiral at time ", StyleBox["t ", FontSlant->"Italic"], "goes to " }], "Text", TextAlignment->Left], Cell[BoxData[ RowBox[{ RowBox[{\(-\((\(c - i\)\/\(2 c\))\)\), "2", StyleBox[ RowBox[{"\[Pi]", StyleBox["t", FontSlant->"Italic"]}]], \((c + i)\)}], " ", "=", RowBox[{\(-\ \[Pi]\), " ", StyleBox["t", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["(", FontSlant->"Italic"], RowBox[{ StyleBox["c", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["+", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], \(1\/c\)}], ")"}]}]}]], "DisplayFormula", TextAlignment->Center], Cell[TextData[{ "Because the segment from 1 to ", StyleBox["vec", FontSlant->"Italic"], "[", StyleBox["c,2k", FontSlant->"Italic"], "] together with the spiral curve from ", StyleBox["t=", FontSlant->"Italic"], "0 to ", StyleBox["t=k=time", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ") goes once around the origin, applying ", StyleBox["f ", FontSlant->"Italic"], "to the vertical projection of the bending line into \[CapitalOmega] gives \ a curve in the horizontal strip from 0 to ", Cell[BoxData[ RowBox[{\(-\((\(c - i\)\/\(2 c\))\)\), RowBox[{"(", RowBox[{"2", StyleBox[ RowBox[{"\[Pi]", StyleBox["t", FontSlant->"Italic"]}]], \((c + i)\)}]}]}]]], "-2\[Pi] ", StyleBox["i). ", FontSlant->"Italic"], "The image of the bending line in the horizontal strip has one endpoint at \ 0 and the other endpoint at ", StyleBox["v + i\[Pi].\n\n v", FontSlant->"Italic"], " is plotted below. 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We regard the dome as the domain and the floor \ \[CapitalOmega] as the range. The identity map on the logarithmic spiral \ between the boundary of the convex hull boundary and the boundary of the \ region \[CapitalOmega] translates to the restriction of an affine map from \ the boundary of the strip to the boundary of the strip. This is an inevitable \ consequence of the equivariance of the group action (which is a group of \ translations on the infinite strip). Now such boundary values correspond to a \ uniquely extremal affine map on the whole strip (Strebel) and so we can \ compute precisely the best possible qc map between convex hull boundary and \ region \[CapitalOmega]. We work in the infinite horizontal strip of width \ \[Pi]. The horizontal stretch factor is then \ \>", "Text"], Cell[BoxData[{ \(\(Clear[a, c, k];\)\), "\[IndentingNewLine]", \(\(a[c_, k_] := \((\[Pi]\ \((c\ + \ 1/c)\))\)/ hypspeed[c, k];\)\), "\[IndentingNewLine]", \(\(a[c_]\ := \ a[c, time[c]];\)\)}], "Input"], Cell[TextData[{ "We have to find ", StyleBox["b", FontSlant->"Italic"], " so that ", StyleBox["x-> ax+by, y->y ", FontSlant->"Italic"], "sends 0 to 0 on the lower edge,\nand {", StyleBox["u", FontSlant->"Italic"], ",\[Pi]} is sent to {", StyleBox["v", FontSlant->"Italic"], ",\[Pi]}" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[b, c, k, x, y];\)\), "\n", \(\(b[c_, k_] := \ \((\(-a[c, k]\)*\ u[c, k]\ + \ v[c, k])\)/\[Pi];\)\), "\[IndentingNewLine]", \(\(b[c_] := \ b[c, time[c]];\)\), "\[IndentingNewLine]", \(Plot[b[c], {c, 0.01, 3}]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 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expression \ for the bending. It has only 7745 terms. We will also do a careful \ computation of the dilatation at ", StyleBox["c=", FontSlant->"Italic"], "1. We see below that the expression 2.1 etc below for the dilatation is \ accurate to 67 digits. This shows that the Thurston ", StyleBox["K", FontSlant->"Italic"], "=2 conjecture is false." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(LeafCount[dil[c, k]]\), "\n", \(dil[1, carefulTime[1]]\), "\n", \(Precision[%]\)}], "Input"], Cell[BoxData[ \(6161\)], "Output"], Cell[BoxData[ \(2.1093361904888744432256469478020868539812750709833859303769028751790668\ 81488438387`67.4721\)], "Output"], Cell[BoxData[ \(67\)], "Output"] }, Open ]], Cell["\<\ Next we compute Lipschitz constants associated with the uniquely \ extremal qc map. 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.554 .244 .396 r F .6611 .41675 m .66184 .41142 L .64853 .41387 L F .67651 .41951 m .66138 .41715 L .65986 .42323 L .496 .147 .305 r F .66138 .41715 m .66105 .4171 L .66093 .4177 L F .66024 .42147 m .66093 .4177 L .65815 .42109 L .527 .186 .334 r F .65716 .42208 m .648 .41925 L .64738 .424 L F .66024 .42147 m .65815 .42109 L .65716 .42208 L F .65815 .42109 m .648 .41925 L .65716 .42208 L F .66138 .41715 m .66093 .4177 L .66079 .41843 L .496 .147 .305 r F .67462 .42442 m .66081 .42331 L .65846 .42818 L .432 .032 .178 r F .66081 .42331 m .65986 .42323 L .65964 .42399 L F .65927 .42554 m .65964 .42399 L .6574 .4253 L .489 .113 .253 r F .65549 .42631 m .64738 .424 L .64668 .4281 L F .65927 .42554 m .6574 .4253 L .65549 .42631 L F .6574 .4253 m .64738 .424 L .65549 .42631 L F .66081 .42331 m .65964 .42399 L .6596 .42414 L .432 .032 .178 r F .67249 .4281 m .65903 .42818 L .65688 .43186 L .326 0 0 r F .65903 .42818 m .65846 .42818 L .65835 .42845 L F .65577 .42944 m .64668 .4281 L .64589 .43152 L .434 .018 .147 r F .65819 .42894 m .65729 .42887 L .65577 .42944 L F .65729 .42887 m .64668 .4281 L .65577 .42944 L F .65903 .42818 m .65835 .42845 L .65833 .42848 L .326 0 0 r F .65701 .43163 m .64589 .43152 L .64502 .43422 L .355 0 .006 r F .65701 .43163 m .65819 .42894 L .64589 .43152 L F .67013 .43047 m .65741 .43181 L .65512 .43422 L .152 0 0 r F .65741 .43181 m .65688 .43186 L .65666 .43216 L F .65572 .43359 m .65666 .43216 L .6533 .43373 L .237 0 0 r F .65116 .43461 m .64502 .43422 L .64407 .43618 L F .65572 .43359 m .6533 .43373 L .65116 .43461 L F .6533 .43373 m .64502 .43422 L .65116 .43461 L F .65741 .43181 m .65666 .43216 L .65644 .43245 L .152 0 0 r F .64881 .43607 m .64407 .43618 L .64306 .43739 L .072 0 0 r F .65435 .4348 m .65243 .43506 L .64881 .43607 L F .65243 .43506 m .64407 .43618 L .64881 .43607 L F .66757 .4315 m .65647 .43393 L .65321 .43522 L 0 0 0 r F .65647 .43393 m .65611 .43401 L .65594 .43408 L F .65647 .43393 m .65594 .43408 L .65321 .43522 L F .65611 .43401 m .65512 .43422 L .65484 .43437 L F .65207 .43542 m .65352 .43506 L .64306 .43739 L F .65352 .43506 m .64881 .43607 L .64306 .43739 L F .65243 .43506 m .65534 .43392 L .64407 .43618 L .072 0 0 r F .64306 .43739 m .63215 .43861 L .6315 .43981 L .109 0 0 r F .64306 .43739 m .64407 .43618 L .63215 .43861 L F .6411 .43799 m .64306 .43739 L .6315 .43981 L 0 0 0 r F .64198 .43782 m .64209 .43778 L .6411 .43799 L F .64209 .43778 m .64306 .43739 L .6411 .43799 L F .6315 .43981 m .61997 .44085 L .61969 .44206 L .143 0 0 r F .6315 .43981 m .63215 .43861 L .61997 .44085 L F .62987 .44038 m .6315 .43981 L .61969 .44206 L 0 0 0 r F .6308 .44023 m .63087 .44019 L .62987 .44038 L F .63087 .44019 m .6315 .43981 L .62987 .44038 L F .61969 .44206 m .61997 .44085 L .60756 .4429 L .175 0 0 r F .6308 .44023 m .61969 .44206 L .61939 .44247 L 0 0 0 r F .63006 .43986 m .62024 .44227 L .61906 .44209 L .266 .774 .82 r F .62024 .44227 m .61939 .44247 L .61936 .44244 L F .61969 .44206 m .60756 .4429 L .60766 .44412 L .175 0 0 r F .61841 .44261 m .61969 .44206 L .60766 .44412 L 0 0 0 r F .61939 .44247 m .61942 .44243 L .61841 .44261 L F .61942 .44243 m .61969 .44206 L .61841 .44261 L F .61939 .44247 m .60766 .44412 L .60776 .44455 L F .61906 .44209 m .60865 .44435 L .60784 .44415 L .237 .755 .806 r F .60865 .44435 m .60776 .44455 L .60777 .44451 L F .60865 .44435 m .60777 .44451 L .60784 .44415 L F .62024 .44227 m .61936 .44244 L .61906 .44209 L .266 .774 .82 r F .64198 .43782 m .6315 .43981 L .6308 .44023 L 0 0 0 r F .64084 .43746 m .6316 .44001 L .63006 .43986 L .296 .793 .834 r F .6316 .44001 m .6308 .44023 L .63073 .4402 L F .6316 .44001 m .63073 .4402 L .63006 .43986 L F .6529 .43524 m .64306 .43739 L .64198 .43782 L 0 0 0 r F .65137 .43492 m .64272 .43759 L .64084 .43746 L .326 .811 .848 r F .64272 .43759 m .64198 .43782 L .64187 .43779 L F .64272 .43759 m .64187 .43779 L .64084 .43746 L F .6529 .43524 m .65305 .4352 L .65207 .43542 L 0 0 0 r F .65305 .4352 m .65352 .43506 L .65207 .43542 L F .65352 .43506 m .65435 .4348 L .64881 .43607 L F .65611 .43401 m .65484 .43437 L .65352 .43506 L F .65367 .43502 m .65352 .43506 L .65321 .43522 L F .65435 .4348 m .65484 .43437 L .65243 .43506 L .072 0 0 r F .65611 .43401 m .65352 .43506 L .65367 .43502 L 0 0 0 r F .65484 .43437 m .65534 .43392 L .65243 .43506 L .072 0 0 r F .65534 .43392 m .65572 .43359 L .65116 .43461 L F .65534 .43392 m .65116 .43461 L .64407 .43618 L F .65702 .43221 m .65644 .43245 L .65534 .43392 L .152 0 0 r F .65542 .43391 m .65534 .43392 L .65512 .43422 L F .65702 .43221 m .65534 .43392 L .65542 .43391 L F .65741 .43181 m .65644 .43245 L .65702 .43221 L F .6533 .43373 m .65693 .43174 L .64502 .43422 L .237 0 0 r F .65666 .43216 m .65693 .43174 L .6533 .43373 L F .65693 .43174 m .65701 .43163 L .65568 .43191 L F .65693 .43174 m .65568 .43191 L .64502 .43422 L F .659 .42823 m .65833 .42848 L .65693 .43174 L .326 0 0 r F .65695 .43174 m .65693 .43174 L .65688 .43186 L F .659 .42823 m .65693 .43174 L .65695 .43174 L F .65903 .42818 m .65833 .42848 L .659 .42823 L F .65729 .42887 m .65869 .42736 L .64668 .4281 L .434 .018 .147 r F .65819 .42894 m .65835 .42845 L .65729 .42887 L F .65835 .42845 m .65869 .42736 L .65729 .42887 L F .65869 .42736 m .65927 .42554 L .65549 .42631 L F .65869 .42736 m .65549 .42631 L .64668 .4281 L F .66069 .42356 m .6596 .42414 L .65869 .42736 L .432 .032 .178 r F .65883 .42741 m .65869 .42736 L .65846 .42818 L F .66069 .42356 m .65869 .42736 L .65883 .42741 L F .66081 .42331 m .6596 .42414 L .66069 .42356 L F .6574 .4253 m .66 .4225 L .64738 .424 L .489 .113 .253 r F .65964 .42399 m .66 .4225 L .6574 .4253 L F .66 .4225 m .66024 .42147 L .65716 .42208 L F .66 .4225 m .65716 .42208 L .64738 .424 L F .66115 .41806 m .66079 .41843 L .66 .4225 L .496 .147 .305 r F .66004 .42251 m .66 .4225 L .65986 .42323 L F .66115 .41806 m .66 .4225 L .66004 .42251 L F .66138 .41715 m .66079 .41843 L .66115 .41806 L F .65815 .42109 m .66107 .41694 L .648 .41925 L .527 .186 .334 r F .66093 .4177 m .66107 .41694 L .65815 .42109 L F .66107 .41694 m .6611 .41675 L .66044 .41688 L F .66107 .41694 m .66044 .41688 L .648 .41925 L F .66264 .41007 m .66194 .41049 L .66107 .41694 L .537 .23 .394 r F .66108 .41694 m .66107 .41694 L .66105 .4171 L F .66264 .41007 m .66107 .41694 L .66108 .41694 L F .66266 .40999 m .66194 .41049 L .66264 .41007 L F .66084 .41115 m .66215 .40848 L .64896 .40792 L .574 .29 .446 r F .66194 .41045 m .66215 .40848 L .66084 .41115 L F .66215 .40848 m .66246 .40552 L .65938 .40607 L F .66215 .40848 m .65938 .40607 L .64896 .40792 L F .66365 .40214 m .66264 .40307 L .66215 .40848 L .564 .291 .46 r F .66229 .4086 m .66215 .40848 L .66202 .40985 L F .66365 .40214 m .66215 .40848 L .66229 .4086 L F .66371 .40185 m .66264 .40307 L .66365 .40214 L F .6606 .40494 m .66284 .40057 L .6493 .40142 L .59 .327 .487 r F .66266 .40289 m .66284 .40057 L .6606 .40494 L F .66284 .40057 m .66295 .39909 L .66052 .3995 L F .66284 .40057 m .66052 .3995 L .6493 .40142 L F .6634 .39343 m .66322 .3938 L .66284 .40057 L .583 .338 .511 r F .66285 .40058 m .66284 .40057 L .66278 .40159 L F .6634 .39343 m .66284 .40057 L .66285 .40058 L F .66347 .39246 m .66322 .3938 L .6634 .39343 L F .66103 .39842 m .66331 .3923 L .64954 .39443 L .602 .359 .521 r F .66327 .39295 m .66331 .3923 L .66103 .39842 L F .66331 .3923 m .66331 .39216 L .66302 .39221 L F .66331 .3923 m .66302 .39221 L .64954 .39443 L F .66431 .38273 m .66357 .38336 L .66331 .3923 L .597 .376 .552 r F .66332 .3923 m .66331 .3923 L .6633 .3924 L F .66431 .38273 m .66331 .3923 L .66332 .3923 L F .66432 .38268 m .66357 .38336 L .66431 .38273 L F .66244 .38433 m .66361 .38064 L .64973 .37913 L .618 .409 .576 r F .66357 .38333 m .66361 .38064 L .66244 .38433 L F .66361 .38064 m .66366 .37701 L .66084 .37744 L F .66361 .38064 m .66084 .37744 L .64973 .37913 L F .66459 .37223 m .66365 .37353 L .66361 .38064 L .608 .408 .586 r F .66376 .38081 m .66361 .38064 L .6636 .38241 L F .66459 .37223 m .66361 .38064 L .66376 .38081 L F .6646 .3721 m .66365 .37353 L .66459 .37223 L F .66175 .37617 m .66365 .37057 L .64969 .37091 L .625 .43 .599 r F .66365 .37345 m .66365 .37057 L .66175 .37617 L F .66365 .37057 m .66364 .36888 L .6615 .36919 L F .66365 .37057 m .6615 .36919 L .64969 .37091 L F .68189 .36858 m .66354 .36046 L .66365 .37057 L .615 .435 .615 r F .66506 .37148 m .66365 .37057 L .66367 .37172 L F .68189 .36858 m .66365 .37057 L .66506 .37148 L F .66354 .36046 m .6635 .36044 L .6635 .36062 L F .66354 .36046 m .6635 .36062 L .66367 .37172 L F .6615 .36919 m .64954 .36239 L .64969 .37091 L .63 .449 .619 r F .66178 .36802 m .64954 .36239 L .6615 .36919 L F .48033 .39441 m .47964 .38541 L .4632 .38426 L .741 .509 .578 r F .49661 .39513 m .47964 .38541 L .48033 .39441 L .732 .502 .579 r F .59143 .40118 m .57591 .39438 L .57625 .40225 L .668 .434 .557 r F .59143 .40118 m .59119 .3934 L .57591 .39438 L F .6355 .38895 m .62093 .38266 L .62101 .39068 L .641 .428 .579 r F .6355 .38895 m .63548 .38101 L .62093 .38266 L F .51354 .40386 m .51278 .39555 L .49661 .39513 L .719 .475 .556 r F .52947 .40387 m .51278 .39555 L .51354 .40386 L .709 .467 .557 r F .60637 .39986 m .59119 .3934 L .59143 .40118 L .657 .425 .556 r F .60637 .39986 m .60623 .39217 L .59119 .3934 L F .44625 .37313 m .42962 .36106 L .4297 .37119 L .763 .557 .619 r F .44671 .38278 m .44625 .37313 L .4297 .37119 L .762 .541 .598 r F .4632 .38426 m .44625 .37313 L .44671 .38278 L .753 .534 .599 r F .4297 .37119 m .42962 .36106 L .41307 .35864 L .772 .564 .618 r F .49747 .40356 m .49661 .39513 L .48033 .39441 L .729 .483 .555 r F .51354 .40386 m .49661 .39513 L .49747 .40356 L .719 .475 .556 r F .46396 .39339 m .4632 .38426 L .44671 .38278 L .751 .517 .577 r F .48033 .39441 m .4632 .38426 L .46396 .39339 L .741 .509 .578 r F .62105 .39829 m .60623 .39217 L .60637 .39986 L .646 .416 .555 r F .62105 .39829 m .62101 .39068 L .60623 .39217 L F .64969 .38698 m .63548 .38101 L .6355 .38895 L .63 .419 .578 r F .64969 .38698 m .64973 .37913 L .63548 .38101 L F .68189 .36858 m .68173 .35746 L .6635 .36044 L .615 .435 .615 r F .54594 .4112 m .54525 .4036 L .52947 .40387 L .693 .436 .53 r F .5614 .41056 m .54525 .4036 L .54594 .4112 L .683 .428 .53 r F .5614 .41056 m .56085 .40306 L .54525 .4036 L F .57667 .40965 m .56085 .40306 L .5614 .41056 L .672 .419 .53 r F .57667 .40965 m .57625 .40225 L .56085 .40306 L F .4813 .40297 m .48033 .39441 L .46396 .39339 L .738 .49 .554 r F .49747 .40356 m .48033 .39441 L .4813 .40297 L .729 .483 .555 r F .5303 .41157 m .52947 .40387 L .51354 .40386 L .704 .445 .53 r F .54594 .4112 m .52947 .40387 L .5303 .41157 L .693 .436 .53 r F .59171 .40849 m .57625 .40225 L .57667 .40965 L .662 .41 .529 r F .59171 .40849 m .59143 .40118 L .57625 .40225 L F .63545 .39648 m .62101 .39068 L .62105 .39829 L .635 .406 .554 r F .63545 .39648 m .6355 .38895 L .62101 .39068 L F .51451 .41167 m .51354 .40386 L .49747 .40356 L .714 .453 .529 r F .5303 .41157 m .51354 .40386 L .51451 .41167 L .704 .445 .53 r F .4297 .37119 m .41307 .35864 L .41314 .36891 L .772 .564 .618 r F .43019 .38099 m .4297 .37119 L .41314 .36891 L .771 .549 .597 r F .44671 .38278 m .4297 .37119 L .43019 .38099 L .762 .541 .598 r F .60652 .40708 m .59143 .40118 L .59171 .40849 L .65 .401 .529 r F .60652 .40708 m .60637 .39986 L .59143 .40118 L F .44754 .39205 m .44671 .38278 L .43019 .38099 L .76 .524 .576 r F .46396 .39339 m .44671 .38278 L .44754 .39205 L .751 .517 .577 r F .41314 .36891 m .41307 .35864 L .39654 .35587 L .781 .572 .617 r F .46505 .40208 m .46396 .39339 L .44754 .39205 L .748 .498 .552 r F .4813 .40297 m .46396 .39339 L .46505 .40208 L .738 .49 .554 r F .49859 .41149 m .49747 .40356 L .4813 .40297 L .724 .46 .528 r F .51451 .41167 m .49747 .40356 L .49859 .41149 L .714 .453 .529 r F .64954 .39443 m .6355 .38895 L .63545 .39648 L .623 .396 .552 r F .64954 .39443 m .64969 .38698 L .6355 .38895 L F .62106 .40543 m .60637 .39986 L .60652 .40708 L .639 .391 .527 r F .62106 .40543 m .62105 .39829 L .60637 .39986 L F .68174 .37913 m .68189 .36858 L .66367 .37172 L .608 .408 .586 r F .69944 .36505 m .68173 .35746 L .68189 .36858 L .599 .422 .613 r F .69944 .36505 m .69929 .35407 L .68173 .35746 L F .63533 .40354 m .62105 .39829 L .62106 .40543 L .627 .381 .526 r F .63533 .40354 m .63545 .39648 L .62105 .39829 L F .56205 .41752 m .5614 .41056 L .54594 .4112 L .675 .401 .499 r F .57714 .41652 m .5614 .41056 L .56205 .41752 L .664 .392 .499 r F .57714 .41652 m .57667 .40965 L .5614 .41056 L F .54676 .41826 m .54594 .4112 L .5303 .41157 L .686 .41 .499 r F .56205 .41752 m .54594 .4112 L .54676 .41826 L .675 .401 .499 r F .48257 .41103 m .4813 .40297 L .46505 .40208 L .734 .468 .526 r F .49859 .41149 m .4813 .40297 L .48257 .41103 L .724 .46 .528 r F .43019 .38099 m .41314 .36891 L .41366 .37886 L .771 .549 .597 r F .43109 .3904 m .43019 .38099 L .41366 .37886 L .77 .532 .574 r F .44754 .39205 m .43019 .38099 L .43109 .3904 L .76 .524 .576 r F .59202 .41528 m .57667 .40965 L .57714 .41652 L .653 .383 .498 r F .59202 .41528 m .59171 .40849 L .57667 .40965 L F .5313 .41874 m .5303 .41157 L .51451 .41167 L .697 .418 .498 r F .54676 .41826 m .5303 .41157 L .5313 .41874 L .686 .41 .499 r F .41314 .36891 m .39654 .35587 L .3966 .3663 L .781 .572 .617 r F .41366 .37886 m .41314 .36891 L .3966 .3663 L .781 .557 .596 r F .44874 .40088 m .44754 .39205 L .43109 .3904 L .758 .505 .551 r F .46505 .40208 m .44754 .39205 L .44874 .40088 L .748 .498 .552 r F .60666 .41379 m .59171 .40849 L .59202 .41528 L .641 .373 .497 r F .60666 .41379 m .60652 .40708 L .59171 .40849 L F .51569 .41895 m .51451 .41167 L .49859 .41149 L .707 .427 .497 r F .5313 .41874 m .51451 .41167 L .51569 .41895 L .697 .418 .498 r F .3966 .3663 m .39654 .35587 L .38007 .35276 L .79 .58 .616 r F .6493 .40142 m .63545 .39648 L .63533 .40354 L .614 .37 .524 r F .6493 .40142 m .64954 .39443 L .63545 .39648 L F .46646 .41027 m .46505 .40208 L .44874 .40088 L .744 .476 .524 r F .48257 .41103 m .46505 .40208 L .46646 .41027 L .734 .468 .526 r F .62104 .41206 m .60652 .40708 L .60666 .41379 L .629 .362 .495 r F .62104 .41206 m .62106 .40543 L .60652 .40708 L F .68129 .389 m .68174 .37913 L .6636 .38241 L .597 .376 .552 r F .49995 .41888 m .49859 .41149 L .48257 .41103 L .718 .435 .496 r F .51569 .41895 m .49859 .41149 L .49995 .41888 L .707 .427 .497 r F .69922 .37547 m .68189 .36858 L .68174 .37913 L .59 .394 .583 r F .69922 .37547 m .69944 .36505 L .68189 .36858 L F .41366 .37886 m .3966 .3663 L .39716 .3764 L .781 .557 .596 r F .41463 .38843 m .41366 .37886 L .39716 .3764 L .78 .54 .572 r F .43109 .3904 m .41366 .37886 L .41463 .38843 L .77 .532 .574 r F .63515 .4101 m .62106 .40543 L .62104 .41206 L .617 .351 .493 r F .63515 .4101 m .63533 .40354 L .62106 .40543 L F .4324 .39938 m .43109 .3904 L .41463 .38843 L .768 .513 .549 r F .44874 .40088 m .43109 .3904 L .4324 .39938 L .758 .505 .551 r F .48411 .41854 m .48257 .41103 L .46646 .41027 L .728 .442 .494 r F .49995 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.45101 L .088 0 0 r F .2849 .34933 m .26895 .33241 L .26975 .34399 L .864 .623 .573 r F .28645 .36036 m .2849 .34933 L .26975 .34399 L .865 .604 .541 r F .78165 .37212 m .77207 .36996 L .76945 .37781 L .346 .081 .362 r F .78165 .37212 m .7845 .36429 L .77207 .36996 L F .46101 .45181 m .45651 .44904 L .44241 .44837 L .518 0 0 r F .4747 .45211 m .45651 .44904 L .46101 .45181 L .51 0 0 r F .30405 .37552 m .28645 .36036 L .28878 .37098 L .852 .573 .514 r F .30699 .38545 m .30405 .37552 L .28878 .37098 L .85 .546 .475 r F .40895 .44225 m .40393 .43738 L .38926 .43563 L .686 .219 .139 r F .42334 .44357 m .40393 .43738 L .40895 .44225 L .678 .218 .153 r F .54695 .45215 m .54473 .45159 L .53172 .45248 L .132 0 0 r F .5594 .45101 m .54473 .45159 L .54695 .45215 L .111 0 0 r F .3451 .41038 m .32582 .3986 L .32996 .40717 L .813 .468 .402 r F .3496 .41808 m .3451 .41038 L .32996 .40717 L .8 .424 .345 r F .49224 .45387 m .48838 .45218 L .4747 .45211 L .373 0 0 r F .50545 .45363 m .48838 .45218 L 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L .28 .133 .496 r F .82122 .32232 m .82601 .31176 L .82113 .32229 L .341 .199 .539 r F .82791 .32493 m .82811 .3206 L p .82122 .32232 L F P 0 g s .82811 .3206 m .82863 .3093 L p .82122 .32232 L .341 .199 .539 r F P 0 g s .83831 .30634 m .83651 .30559 L .83283 .31076 L .264 .157 .54 r F .83808 .30296 m .8448 .29167 L .83795 .30285 L .221 .103 .498 r F .8401 .30466 m .8383 .30314 L .83831 .30634 L .197 .105 .517 r F .83831 .30634 m .8383 .30314 L .83651 .30559 L .264 .157 .54 r F .84465 .29455 m .84466 .29442 L p .83651 .30559 L .221 .103 .498 r F P 0 g s .84466 .29442 m .8448 .29167 L p .83651 .30559 L .221 .103 .498 r F P 0 g s .8448 .29167 m .82883 .30485 L p .83506 .30757 L .221 .103 .498 r F P 0 g s .83651 .30559 m .83506 .30757 L .83283 .31076 L .264 .157 .54 r F .8361 .31442 m .83381 .3106 L .82791 .32493 L .221 .103 .498 r F .83982 .30965 m .83795 .30995 L .8361 .31442 L F .83795 .30995 m .83381 .3106 L .8361 .31442 L F .82791 .32493 m .84377 .31138 L p .83982 .30965 L F P 0 g s .83834 .30901 m .83831 .30634 L .83795 .30995 L .137 .016 .438 r F .83982 .30965 m .84315 .29722 L .83834 .30901 L .221 .103 .498 r F .83923 .30677 m .8401 .30466 L .83831 .30634 L .137 .016 .438 r F .84315 .29722 m .84342 .29622 L .83923 .30677 L .221 .103 .498 r F .84315 .29722 m .83923 .30677 L .83834 .30901 L F .83982 .30965 m .84465 .29455 L .84342 .29622 L F .8439 .3089 m .84465 .29455 L p .83982 .30965 L F P 0 g s .85114 .29321 m .84982 .29256 L .84768 .29709 L .116 .041 .487 r F .85107 .28979 m .85634 .2781 L .85104 .28976 L .044 0 .425 r F .85263 .29116 m .85113 .28985 L .85114 .29321 L .016 0 .446 r F .85114 .29321 m .85113 .28985 L .84982 .29256 L .116 .041 .487 r F .85626 .27938 m .85627 .27923 L p .84982 .29256 L .044 0 .425 r F P 0 g s .85627 .27923 m .85634 .2781 L p .84982 .29256 L .044 0 .425 r F P 0 g s .85634 .2781 m .8448 .29167 L p .84909 .29405 L .044 0 .425 r F P 0 g s .84982 .29256 m .84909 .29405 L .84768 .29709 L .116 .041 .487 r F .84785 .30572 m .84816 .29708 L .84377 .31138 L .044 0 .425 r F .85423 .29691 m .85073 .297 L .84785 .30572 L F .85073 .297 m .84816 .29708 L .84785 .30572 L F .8513 .29528 m .85114 .29321 L .85073 .297 L 0 0 .336 r F .85423 .29691 m .85613 .2805 L .8513 .29528 L .044 0 .425 r F .85181 .29365 m .85263 .29116 L .85114 .29321 L 0 0 .336 r F .85613 .2805 m .85621 .27985 L .85181 .29365 L .044 0 .425 r F .85613 .2805 m .85181 .29365 L .8513 .29528 L F .85524 .29685 m .85626 .27938 L p .85423 .29691 L F P 0 g s .85981 .27986 m .85898 .27928 L .85788 .28325 L 0 0 .39 r F .8587 .28024 m .85898 .27928 L .85634 .2781 L 0 0 .289 r F .85981 .27986 m .85981 .27641 L .85898 .27928 L 0 0 .39 r F .8632 .26473 m .86312 .26489 L p .85981 .27641 L 0 0 .289 r F P 0 g s .85981 .27642 m .8632 .26473 L .85981 .27641 L 0 0 .289 r F .86312 .26489 m .85634 .2781 L p .85898 .27928 L F P 0 g s .8608 .27745 m .85981 .27641 L .85981 .27986 L 0 0 .314 r F .8552 .29745 m .85706 .2937 L p .85808 .28332 L 0 0 .289 r F P 0 g s .85706 .2937 m .86163 .28449 L p .85936 .28374 L 0 0 .289 r F P 0 g s .85936 .28374 m .85808 .28332 L .85706 .2937 L 0 0 .289 r F .85991 .28134 m .85981 .27986 L .85936 .28374 L 0 0 .157 r F .86014 .28031 m .85981 .27986 L .85991 .28134 L F .85936 .28374 m .85981 .27986 L .85788 .28325 L 0 0 .258 r F .86014 .28031 m .8608 .27745 L .85981 .27986 L 0 0 .157 r F .86218 .28338 m .86306 .26734 L .85991 .28134 L 0 0 .289 r F .86151 .28217 m .86258 .27317 L .8608 .27745 L 0 0 .157 r F .86151 .28217 m .8608 .27745 L .86014 .28031 L F .85991 .28133 m .8632 .26473 L .8587 .28024 L 0 0 .289 r F .85991 .28134 m .85995 .28114 L .85991 .28133 L F .85995 .28114 m .8632 .26473 L .85991 .28133 L F .86306 .26734 m .8632 .26473 L .85991 .28134 L F .86015 .28032 m .86014 .28031 L .85991 .28134 L 0 0 .157 r F .86218 .28338 m .8587 .28024 L .85808 .28332 L 0 0 .289 r F .86163 .28449 m .86218 .28338 L p .85808 .28332 L F P 0 g s .86151 .28217 m .86015 .28032 L .85876 .28893 L 0 0 .157 r F .86015 .28032 m .85936 .28374 L .85876 .28893 L F .85634 .2781 m .85554 .29172 L p .8587 .28024 L 0 0 .289 r F P 0 g s .85898 .27928 m .8587 .28024 L .85788 .28325 L 0 0 .39 r F .85554 .29172 m .8552 .29745 L p .8587 .28024 L 0 0 .289 r F P 0 g s .85876 .28893 m .85936 .28374 L .85599 .28655 L 0 0 .258 r F .85936 .28374 m .85788 .28325 L .85599 .28655 L F .86258 .27317 m .85981 .2706 L .85981 .27641 L 0 0 .314 r F .86258 .27317 m .85981 .27641 L .8608 .27745 L F .85898 .27928 m .85981 .27642 L .85981 .27641 L 0 0 .289 r F .85898 .27928 m .85981 .2706 L .85599 .2772 L 0 0 .39 r F .85981 .27641 m .85981 .2706 L .85898 .27928 L F .85898 .27928 m .85897 .27928 L .8587 .28024 L F .86218 .28338 m .86338 .26439 L .8632 .26473 L p 0 0 .289 r F P 0 g s .85181 .29365 m .85114 .29321 L .8513 .29528 L 0 0 .336 r F .85073 .297 m .85114 .29321 L .84768 .29709 L .042 0 .395 r F .8513 .29528 m .85626 .27938 L .84909 .29405 L .044 0 .425 r F .85621 .27985 m .85626 .27938 L .8513 .29528 L F .85788 .28325 m .85599 .2772 L .85599 .28655 L 0 0 .39 r F .85897 .27928 m .85599 .2772 L .85788 .28325 L F .85497 .29569 m .85599 .28655 L .85263 .29116 L 0 0 .336 r F .85497 .29569 m .85263 .29116 L .85181 .29365 L F .85183 .29366 m .85181 .29365 L .8513 .29528 L F .85423 .29691 m .84909 .29405 L .84816 .29708 L .044 0 .425 r F .85497 .29569 m .85183 .29366 L .85014 .30242 L 0 0 .336 r F .85183 .29366 m .85073 .297 L .85014 .30242 L F .8448 .29167 m .84411 .30476 L p .84909 .29405 L .044 0 .425 r F P 0 g s .84411 .30476 m .84377 .31138 L p .84909 .29405 L .044 0 .425 r F P 0 g s .85072 .2971 m .85073 .297 L .84751 .29727 L .042 0 .395 r F .85073 .297 m .84768 .29709 L .84751 .29727 L F .84377 .31138 m .8552 .29745 L p .85423 .29691 L .044 0 .425 r F P 0 g s .8552 .29745 m .85524 .29685 L p .85423 .29691 L .044 0 .425 r F P 0 g s .85014 .30242 m .85072 .2971 L .84524 .29982 L .042 0 .395 r F .85072 .2971 m .84751 .29727 L .84524 .29982 L F .85599 .28655 m .85112 .28376 L .85113 .28985 L .016 0 .446 r F .85599 .28655 m .85113 .28985 L .85263 .29116 L F .84982 .29256 m .85107 .28979 L .84909 .29405 L .044 0 .425 r F .85107 .28979 m .85104 .28976 L .84909 .29405 L F .84982 .29256 m .85112 .28376 L .84521 .29027 L .116 .041 .487 r F .85113 .28985 m .85112 .28376 L .84982 .29256 L F .84982 .29256 m .84979 .29254 L .84909 .29405 L F .84768 .29709 m .84521 .29027 L .84524 .29982 L F .84979 .29254 m .84521 .29027 L .84768 .29709 L F .84428 .3091 m .84524 .29982 L .8401 .30466 L .137 .016 .438 r F .84377 .31138 m .8439 .3089 L p .83982 .30965 L .221 .103 .498 r F P 0 g s .84428 .3091 m .8401 .30466 L .83923 .30677 L .137 .016 .438 r F .83926 .30678 m .83923 .30677 L .83834 .30901 L F .83923 .30677 m .83831 .30634 L .83834 .30901 L F .83795 .30995 m .83831 .30634 L .83283 .31076 L .215 .08 .471 r F .83834 .30901 m .84342 .29622 L .83506 .30757 L .221 .103 .498 r F .83982 .30965 m .83506 .30757 L .83381 .3106 L F .84428 .3091 m .83926 .30678 L .83739 .31569 L .137 .016 .438 r F .83926 .30678 m .83795 .30995 L .83739 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.82601 .31176 L .341 .199 .539 r F .82113 .32229 m .8214 .31902 L .82111 .32235 L .28 .133 .496 r F .82246 .31944 m .8214 .31902 L .82113 .32229 L F .82111 .32235 m .8214 .31902 L .81307 .32413 L .333 .178 .516 r F .82113 .32229 m .82601 .31176 L .81667 .32055 L .341 .199 .539 r F .8225 .31945 m .82246 .31944 L .82113 .32229 L .28 .133 .496 r F .82948 .32218 m .8225 .31945 L .82057 .32853 L F .82122 .32232 m .81667 .32055 L .81505 .32369 L .341 .199 .539 r F .8225 .31945 m .82111 .32235 L .82057 .32853 L .28 .133 .496 r F .80855 .31738 m .8081 .32908 L p .81667 .32055 L .341 .199 .539 r F P 0 g s .8081 .32908 m .80777 .33787 L p .81667 .32055 L .341 .199 .539 r F P 0 g s .82111 .32235 m .82111 .32235 L .81245 .32451 L .333 .178 .516 r F .82111 .32235 m .81307 .32413 L .81245 .32451 L F .82057 .32853 m .82111 .32235 L .81145 .32513 L F .82111 .32235 m .81245 .32451 L .81145 .32513 L F .83035 .31275 m .82134 .30916 L .82138 .31609 L .32 .201 .556 r F .83035 .31275 m .82138 .31609 L .82329 .3177 L F .81919 .31817 m .82096 .31573 L .81667 .32055 L .341 .199 .539 r F .82096 .31573 m .82062 .31545 L .81667 .32055 L F .81919 .31817 m .82134 .30916 L .81138 .31516 L .367 .237 .569 r F .82138 .31609 m .82134 .30916 L .81919 .31817 L F .81919 .31817 m .81909 .31813 L .81667 .32055 L F .81307 .32413 m .81138 .31516 L .81145 .32513 L F .81909 .31813 m .81138 .31516 L .81307 .32413 L F .81145 .32513 m .80046 .32096 L .80053 .33104 L .407 .269 .579 r F .81145 .32513 m .81138 .31516 L .80046 .32096 L F .26895 .33241 m .26899 .32058 L .25415 .31444 L .873 .65 .597 r F .53439 .45309 m .53172 .45248 L .51862 .45316 L .151 0 0 r F .54695 .45215 m .53172 .45248 L .53439 .45309 L .132 0 0 r F .42836 .44746 m .42334 .44357 L .40895 .44225 L .619 .115 .03 r F .44241 .44837 m .42334 .44357 L .42836 .44746 L .612 .114 .044 r F .69847 .42212 m .68753 .42403 L .68459 .42643 L .037 0 0 r F .69847 .42212 m .70195 .41967 L .68753 .42403 L F .32582 .3986 m .30699 .38545 L .31066 .39484 L .833 .509 .441 r F .32996 .40717 m .32582 .3986 L .31066 .39484 L .824 .472 .391 r F .66652 .43173 m .67013 .43047 L .65512 .43422 L 0 0 0 r F .64042 .43309 m .64813 .43191 L .63841 .43438 L .627 .87 .988 r F .64644 .42924 m .64709 .43025 L .64042 .43309 L F .64709 .43025 m .64813 .43191 L .64042 .43309 L F .65899 .42629 m .64998 .43232 L .64727 .43017 L .619 .877 .99 r F .65127 .42847 m .64727 .43017 L .64674 .42975 L F .65899 .42629 m .64727 .43017 L .65127 .42847 L F .64998 .43232 m .64899 .43299 L .64867 .43253 L F .64998 .43232 m .64867 .43253 L .64674 .42975 L F .64618 .43292 m .64928 .43323 L .63965 .43632 L .57 .888 .989 r F .64813 .43191 m .64867 .43253 L .64618 .43292 L F .64867 .43253 m .64928 .43323 L .64618 .43292 L F .64928 .43323 m .64978 .4338 L .64028 .43616 L F .64928 .43323 m .64028 .43616 L .63965 .43632 L F .66197 .42942 m .65191 .43443 L .64941 .43319 L .53 .892 .975 r F .65586 .4311 m .64941 .43319 L .64899 .43299 L F .66197 .42942 m .64941 .43319 L .65586 .4311 L F .65191 .43443 m .65116 .43481 L .65085 .43455 L F .65191 .43443 m .65085 .43455 L .64899 .43299 L F .64686 .435 m .65124 .43483 L .64084 .43746 L .475 .878 .953 r F .64978 .4338 m .65085 .43455 L .64686 .435 L F .65085 .43455 m .65124 .43483 L .64686 .435 L F .65124 .43483 m .65137 .43492 L .64825 .43567 L F .65124 .43483 m .64825 .43567 L .64084 .43746 L F .66484 .43115 m .65411 .4349 L .65129 .43481 L .357 .83 .862 r F .654 .43405 m .65129 .43481 L .65116 .43481 L F .66484 .43115 m .65129 .43481 L .654 .43405 L F .65411 .4349 m .65321 .43522 L .65302 .43518 L F .65411 .4349 m .65302 .43518 L .65116 .43481 L F .66757 .4315 m .66784 .43139 L .66652 .43173 L 0 0 0 r F .66784 .43139 m .67013 .43047 L .66652 .43173 L F .52174 .45382 m .51862 .45316 L .50545 .45363 L .168 0 0 r F .53439 .45309 m .51862 .45316 L .52174 .45382 L .151 0 0 r F .79602 .3584 m .78662 .35572 L .7845 .36429 L .348 .121 .421 r F .79602 .3584 m .79831 .34987 L .78662 .35572 L F .7458 .40122 m .73664 .40147 L 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.34399 m .26895 .33241 L .2541 .32647 L .876 .633 .567 r F .50903 .45436 m .50545 .45363 L .49224 .45387 L .182 0 0 r F .52174 .45382 m .50545 .45363 L .50903 .45436 L .168 0 0 r F .77813 .3791 m .76945 .37781 L .76619 .38479 L .282 0 .254 r F .77813 .3791 m .78165 .37212 L .76945 .37781 L F .36942 .42757 m .3496 .41808 L .35468 .42507 L .771 .368 .29 r F .37471 .4336 m .36942 .42757 L .35468 .42507 L .742 .302 .207 r F .38926 .43563 m .36942 .42757 L .37471 .4336 L .734 .301 .223 r F .39466 .44067 m .38926 .43563 L .37471 .4336 L .693 .219 .122 r F .40895 .44225 m .38926 .43563 L .39466 .44067 L .686 .219 .139 r F .58486 .44777 m .58391 .44815 L .57173 .44967 L .157 .698 .767 r F .59644 .44605 m .59593 .44644 L .58391 .44815 L .182 .716 .779 r F .68142 .42751 m .67013 .43047 L .66757 .4315 L 0 0 0 r F .68142 .42751 m .68459 .42643 L .67013 .43047 L F .57312 .44931 m .57173 .44967 L .5594 .45101 L .132 .68 .756 r F .58486 .44777 m .57173 .44967 L .57312 .44931 L .157 .698 .767 r F .3496 .41808 m .32996 .40717 L .33473 .41508 L .8 .424 .345 r F .35468 .42507 m .3496 .41808 L .33473 .41508 L .78 .37 .275 r F .60784 .44415 m .60776 .44455 L .59593 .44644 L .209 .735 .792 r F .71172 .41754 m .70195 .41967 L .69847 .42212 L 0 0 0 r F .71172 .41754 m .71574 .41505 L .70195 .41967 L F .56124 .45067 m .5594 .45101 L .54695 .45215 L .11 .663 .747 r F .57312 .44931 m .5594 .45101 L .56124 .45067 L .132 .68 .756 r F .41438 .44629 m .40895 .44225 L .39466 .44067 L .625 .115 .014 r F .42836 .44746 m .40895 .44225 L .41438 .44629 L .619 .115 .03 r F .61906 .44209 m .61939 .44247 L .60776 .44455 L .237 .755 .806 r F .46576 .4537 m .46101 .45181 L .44734 .45127 L .391 0 0 r F .479 .4539 m .46101 .45181 L .46576 .4537 L .383 0 0 r F .54925 .45184 m .54695 .45215 L .53439 .45309 L .088 .647 .739 r F .56124 .45067 m .54695 .45215 L .54925 .45184 L .11 .663 .747 r F .32996 .40717 m .31066 .39484 L .31503 .40363 L .824 .472 .391 r F .33473 .41508 m .32996 .40717 L .31503 .40363 L .81 .427 .331 r F .81887 .33756 m .81746 .33707 L .81125 .34241 L .291 .106 .449 r F .81893 .33537 m .82791 .32493 L .81846 .33508 L .247 .043 .394 r F .82054 .33635 m .81924 .33556 L .81887 .33756 L .229 .051 .421 r F .81887 .33756 m .81924 .33556 L .81746 .33707 L .291 .106 .449 r F .82668 .32955 m .82679 .32915 L p .81746 .33707 L .247 .043 .394 r F P 0 g s .82679 .32915 m .82791 .32493 L p .81746 .33707 L .247 .043 .394 r F P 0 g s .82791 .32493 m .80777 .33787 L p .81393 .33995 L .247 .043 .394 r F P 0 g s .81746 .33707 m .81393 .33995 L .81125 .34241 L .291 .106 .449 r F .81793 .34066 m .81887 .33756 L .81125 .34241 L .235 .017 .364 r F .81697 .34098 m .81255 .34209 L .80345 .35616 L .247 .043 .394 r F .80777 .33787 m .80641 .34363 L p .81393 .33995 L F P 0 g s .80641 .34363 m .80544 .34775 L p .81125 .34241 L .247 .043 .394 r F P 0 g s .81393 .33995 m .80641 .34363 L .81125 .34241 L .247 .043 .394 r F .80345 .35616 m .81131 .35091 L p .81697 .34098 L F P 0 g s .81131 .35091 m .82309 .34305 L p .81761 .3412 L .247 .043 .394 r F P 0 g s .81761 .3412 m .81697 .34098 L .81131 .35091 L .247 .043 .394 r F .82309 .34305 m .82429 .33855 L p .81793 .34066 L F P 0 g s .81761 .3412 m .81793 .34066 L .81697 .34098 L .247 .043 .394 r F .82309 .34305 m .81793 .34066 L .81761 .3412 L F .81962 .33781 m .81887 .33756 L .81793 .34066 L .162 0 .324 r F .81793 .34066 m .81125 .34241 L .81697 .34098 L .235 .017 .364 r F .81697 .34098 m .8195 .33801 L .81393 .33995 L .247 .043 .394 r F .8195 .33801 m .82668 .32955 L .82285 .33268 L F .81962 .33781 m .82054 .33635 L .81887 .33756 L .162 0 .324 r F .8195 .33801 m .82285 .33268 L .81393 .33995 L .247 .043 .394 r F .82429 .33855 m .82668 .32955 L p .81697 .34098 L F P 0 g s .83552 .32463 m .83443 .32419 L .83043 .32888 L .153 0 .385 r F .83582 .32228 m .84377 .31138 L .83562 .32216 L .082 0 .306 r F .83713 .32307 m .83599 .32239 L .83552 .32463 L .062 0 .338 r F .83552 .32463 m .83599 .32239 L .83443 .32419 L .153 0 .385 r F .84275 .31488 m .84288 .31442 L p .83443 .32419 L .082 0 .306 r F P 0 g s .84288 .31442 m .84377 .31138 L p .83443 .32419 L .082 0 .306 r F P 0 g s .84377 .31138 m .82791 .32493 L p .83218 .32671 L .082 0 .306 r F P 0 g s .83443 .32419 m .83218 .32671 L .83043 .32888 L .153 0 .385 r F .83278 .33086 m .83104 .32875 L .82309 .34305 L .082 0 .306 r F .8351 .32794 m .83437 .32808 L .83278 .33086 L F .83437 .32808 m .83104 .32875 L .83278 .33086 L F .82309 .34305 m .83853 .32937 L p .8351 .32794 L F P 0 g s .83458 .32772 m .83552 .32463 L .83437 .32808 L 0 0 .208 r F .8351 .32794 m .8387 .32027 L .83458 .32772 L .082 0 .306 r F .8361 .32489 m .83713 .32307 L .83552 .32463 L 0 0 .208 r F .8387 .32027 m .83955 .31846 L .8361 .32489 L .082 0 .306 r F .8387 .32027 m .8361 .32489 L .83458 .32772 L F .8351 .32794 m .84275 .31488 L .83955 .31846 L F .8393 .32674 m .84275 .31488 L p .8351 .32794 L F P 0 g s .84815 .31127 m .84744 .31087 L .84491 .31502 L 0 0 .276 r F .84863 .30882 m .8552 .29745 L .84857 .30878 L 0 0 .158 r F .8495 .30936 m .84869 .30886 L .84815 .31127 L 0 0 .196 r F .84815 .31127 m .84869 .30886 L .84744 .31087 L 0 0 .276 r F .85448 .29979 m .85464 .29927 L p .84744 .31087 L 0 0 .158 r F P 0 g s .85464 .29927 m .8552 .29745 L p .84744 .31087 L 0 0 .158 r F P 0 g s .8552 .29745 m .84377 .31138 L p .84612 .31296 L 0 0 .158 r F P 0 g s .84744 .31087 m .84612 .31296 L .84491 .31502 L 0 0 .276 r F .84265 .32358 m .84517 .315 L .83853 .32937 L 0 0 .158 r F .84888 .31481 m .84686 .31491 L .84265 .32358 L F .84686 .31491 m .84517 .315 L .84265 .32358 L F .84739 .31381 m .84815 .31127 L .84686 .31491 L 0 0 .022 r F .84888 .31481 m .85213 .3039 L .84739 .31381 L 0 0 .158 r F .84844 .31155 m .8495 .30936 L .84815 .31127 L 0 0 .022 r F .85213 .3039 m .85236 .30314 L .84844 .31155 L 0 0 .158 r F .85213 .3039 m .84844 .31155 L .84739 .31381 L F .84888 .31481 m .85448 .29979 L .85236 .30314 L F .84988 .31466 m .85448 .29979 L p .84888 .31481 L F P 0 g s .84988 .31288 m .85293 .3045 L .8495 .30936 L 0 0 .022 r F .84988 .31288 m .8495 .30936 L .84844 .31155 L F .84848 .31158 m .84844 .31155 L .84739 .31381 L F .84844 .3115