(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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We find that many graphs are too complicated to be graphed and understood this way. Mathematicians have learned to graph on a different coordinate system based on the unit circle. :[font = input; preserveAspect; groupLikeNormal] Clear[x,y,t] {x[t_],y[t_]}={Cos[t],Sin[t]}; Clear[crcl]; crcl= ParametricPlot[{x[t],y[t]},{t,0,2Pi}, PlotStyle->{RGBColor[1,0,0]}, AspectRatio->Automatic, PlotRange->{{-2,2},{-1.5,1.5}}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 263] %! %%Creator: Mathematica %%AspectRatio: .75 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.25 0.375 0.25 [ [(-2)] 0 .375 0 2 Msboxa [(-1.5)] .125 .375 0 2 Msboxa [(-1)] .25 .375 0 2 Msboxa [(-0.5)] .375 .375 0 2 Msboxa [(0.5)] .625 .375 0 2 Msboxa [(1)] .75 .375 0 2 Msboxa [(1.5)] .875 .375 0 2 Msboxa [(2)] 1 .375 0 2 Msboxa [(-1.5)] .4875 0 1 0 Msboxa [(-1)] .4875 .125 1 0 Msboxa [(-0.5)] .4875 .25 1 0 Msboxa [(0.5)] .4875 .5 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa 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Pi) ||(t==Pi/2)||(t==3 Pi/2)||(t==0)) ,0, If[((t >0) && (t < Pi/2)) || ((t > 3 Pi/2) && (t < 2Pi)),0.6,-0.6]] mapprsy[t_] := If[((t == 0) || (t == Pi)|| (t ==2 Pi)|| (t==Pi/2)),0.15,If[(t==3Pi/2),-0.15,0]] Clear[ordprgrph]; ordprgrph= Show[crcl, Table[ordpr[t],{t,0,2Pi,Pi/4}], Table[ordpr[t],{t,0,2Pi,Pi/6}], PlotRange->{{-2,2},{-1.5,1.5}}, AspectRatio->Automatic, Ticks->None, AxesLabel->None]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 263; endGroup] %! %%Creator: Mathematica %%AspectRatio: .75 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.25 0.375 0.25 [ [ -0.001 -0.001 0 0 ] [ 1.001 .751 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .375 m 1 .375 L s P p .002 w .5 0 m .5 .75 L s P P 0 0 m 1 0 L 1 .75 L 0 .75 L closepath clip newpath p p 1 0 0 r p .004 w .75 .375 m .74999 .37705 L .74997 .37909 L .74992 .38114 L .74987 .38318 L .7497 .38727 L .74946 .39135 L .74916 .39543 L .7488 .3995 L .74786 .40763 L .74666 .41572 L .7452 .42377 L .74148 .4397 L .73673 .45536 L .73097 .47067 L .71651 .5 L .69834 .52719 L .67678 .55178 L .65219 .57334 L .625 .59151 L .61057 .59922 L .59567 .60597 L .58036 .61173 L .5647 .61648 L .54877 .6202 L .54072 .62166 L .53263 .62286 L .5245 .6238 L .52043 .62416 L .51635 .62446 L .51227 .6247 L .51022 .62479 L .50818 .62487 L .50614 .62492 L .50409 .62497 L .50205 .62499 L .5 .625 L .49795 .62499 L .49591 .62497 L .49386 .62492 L .49182 .62487 L .48978 .62479 L .48773 .6247 L .48365 .62446 L .47957 .62416 L .4755 .6238 L .46737 .62286 L .45928 .62166 L .45123 .6202 L .4353 .61648 L Mistroke .41964 .61173 L .40433 .60597 L .375 .59151 L .34781 .57334 L .32322 .55178 L .30166 .52719 L .28349 .5 L .27578 .48557 L .26903 .47067 L .26327 .45536 L .25852 .4397 L .2548 .42377 L .25334 .41572 L .25214 .40763 L .2512 .3995 L .25084 .39543 L .25054 .39135 L .2503 .38727 L .25021 .38522 L .25013 .38318 L .25008 .38114 L .25003 .37909 L .25001 .37705 L .25 .375 L .25001 .37295 L .25003 .37091 L .25008 .36886 L .25013 .36682 L .2503 .36273 L .25054 .35865 L .25084 .35457 L .2512 .3505 L .25214 .34237 L .25334 .33428 L .2548 .32623 L .25852 .3103 L .26327 .29464 L .26903 .27933 L .28349 .25 L .30166 .22281 L .32322 .19822 L .34781 .17666 L .375 .15849 L .38943 .15078 L .40433 .14403 L .41964 .13827 L .4353 .13352 L .45123 .1298 L .45928 .12834 L .46737 .12714 L Mistroke .4755 .1262 L .47957 .12584 L .48365 .12554 L .48773 .1253 L .48978 .12521 L .49182 .12513 L .49386 .12508 L .49591 .12503 L .49795 .12501 L .5 .125 L .50205 .12501 L .50409 .12503 L .50614 .12508 L .50818 .12513 L .51227 .1253 L .51635 .12554 L .52043 .12584 L .5245 .1262 L .53263 .12714 L .54072 .12834 L .54877 .1298 L .5647 .13352 L .58036 .13827 L .59567 .14403 L .625 .15849 L .65219 .17666 L .67678 .19822 L .69834 .22281 L .71651 .25 L .72422 .26443 L .73097 .27933 L .73673 .29464 L .74148 .3103 L .7452 .32623 L .74666 .33428 L .74786 .34237 L .7488 .3505 L .74916 .35457 L .74946 .35865 L .7497 .36273 L .74979 .36478 L .74987 .36682 L .74992 .36886 L .74997 .37091 L .74999 .37295 L .75 .375 L Mfstroke P P p .015 w .75 .375 Mdot [({1., 0})] .75 .4125 0 0 Mshowa P p .015 w .67678 .55178 Mdot [({0.7071, 0.7071})] .82678 .55178 0 0 Mshowa P p .015 w .5 .625 Mdot [({0, 1.})] .5 .6625 0 0 Mshowa P p .015 w .32322 .55178 Mdot [({-0.7071, 0.7071})] .17322 .55178 0 0 Mshowa P p .015 w .25 .375 Mdot [({-1., 0})] .25 .4125 0 0 Mshowa P p .015 w .32322 .19822 Mdot [({-0.7071, -0.7071})] .17322 .19822 0 0 Mshowa P p .015 w .5 .125 Mdot [({0, -1.})] .5 .0875 0 0 Mshowa P p .015 w .67678 .19822 Mdot [({0.7071, -0.7071})] .82678 .19822 0 0 Mshowa P p .015 w .75 .375 Mdot [({1., 0})] .75 .4125 0 0 Mshowa P p .015 w .75 .375 Mdot [({1., 0})] .75 .4125 0 0 Mshowa P p .015 w .71651 .5 Mdot [({0.866, 0.5})] .86651 .5 0 0 Mshowa P p .015 w .625 .59151 Mdot [({0.5, 0.866})] .775 .59151 0 0 Mshowa P p .015 w .5 .625 Mdot [({0, 1.})] .5 .6625 0 0 Mshowa P p .015 w .375 .59151 Mdot [({-0.5, 0.866})] .225 .59151 0 0 Mshowa P p .015 w .28349 .5 Mdot [({-0.866, 0.5})] .13349 .5 0 0 Mshowa P p .015 w .25 .375 Mdot [({-1., 0})] .25 .4125 0 0 Mshowa P p .015 w .28349 .25 Mdot [({-0.866, -0.5})] .13349 .25 0 0 Mshowa P p .015 w .375 .15849 Mdot [({-0.5, -0.866})] .225 .15849 0 0 Mshowa P p .015 w .5 .125 Mdot [({0, -1.})] .5 .0875 0 0 Mshowa P p .015 w .625 .15849 Mdot [({0.5, -0.866})] .775 .15849 0 0 Mshowa P p .015 w .71651 .25 Mdot [({0.866, -0.5})] .86651 .25 0 0 Mshowa P p .015 w .75 .375 Mdot [({1., 0})] .75 .4125 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] These values should look familiar. Let's put these values in a table with another set of ordered pairs. :[font = input; preserveAspect; startGroup] tbl4= N[ Table[{x[t],y[t]},{t,0,Pi,Pi/4}],4]; tbl6=N[Table[{x[t],y[t]},{t,0,Pi,Pi/6}],4]; Clear[tlist] tlist=Union[tbl4,tbl6]; tbl1= TableForm[tlist, TableHeadings->{None,{value x,value y}}]; tbl2= TableForm[{{tlist,tlist}}, TableHeadings->{None, {"value x value y","Cos[x] Sin[x]"}}] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] TableForm[{{{{-1., 0}, {-0.8660254037844386468, 0.5}, {-0.7071067811865475244, 0.7071067811865475244}, {-0.5, 0.8660254037844386468}, {0, 1.}, {0.5, 0.8660254037844386468}, {0.7071067811865475244, 0.7071067811865475244}, {0.8660254037844386468, 0.5}, {1., 0}}, {{-1., 0}, {-0.8660254037844386468, 0.5}, {-0.7071067811865475244, 0.7071067811865475244}, {-0.5, 0.8660254037844386468}, {0, 1.}, {0.5, 0.8660254037844386468}, {0.7071067811865475244, 0.7071067811865475244}, {0.8660254037844386468, 0.5}, {1., 0}}}}, TableHeadings -> {None, {"value x value y", "Cos[x] Sin[x]"}}] ;[o] value x value y Cos[x] Sin[x] -1. 0 -1. 0 -0.866 0.5 -0.866 0.5 -0.7071 0.7071 -0.7071 0.7071 -0.5 0.866 -0.5 0.866 0 1. 0 1. 0.5 0.866 0.5 0.866 0.7071 0.7071 0.7071 0.7071 0.866 0.5 0.866 0.5 1. 0 1. 0 :[font = text; inactive; preserveAspect] The values of the ordered pairs of the unit circle are the same as the values of the Cosine and Sine functions. The Cosine is the x-coordinate and the Sine is the y-coordinate. To understand this you have to remember your right triangle trigonometry. Sine is y/r and Cosine is x/r. Look at a right triangle inscribed in the unit circle. Remember the unit circle has a radius of one. :[font = input; preserveAspect; startGroup] Clear[trngl] trngl=Graphics[{Thickness[0.005],RGBColor[0,0,1], Line[{{0,0},{x[Pi/4],y[Pi/4]}}], Line[{{0,0},{x[Pi/4],0}}], Line[{{x[Pi/4],0},{x[Pi/4],y[Pi/4]}}], Text[{N[x[Pi/4],4],N[y[Pi/4],4]}, {0.9,0.707},{-1,0}], Text["x",{0.5,-0.05}], Text["y",{0.86,0.35}], Text["r",{0.35,0.5}], Text["a",{0.2,0.1}]}]; Show[crcl,trngl, AspectRatio->Automatic, PlotRange->{{-2,2},{-1.5,1.5}}, AxesLabel->None, Ticks->None]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 263; endGroup] %! %%Creator: Mathematica %%AspectRatio: .75 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.25 0.375 0.25 [ [ -0.001 -0.001 0 0 ] [ 1.001 .751 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .375 m 1 .375 L s P p .002 w .5 0 m .5 .75 L s P P 0 0 m 1 0 L 1 .75 L 0 .75 L closepath clip newpath p p 1 0 0 r p .004 w .75 .375 m .74999 .37705 L .74997 .37909 L .74992 .38114 L .74987 .38318 L .7497 .38727 L .74946 .39135 L .74916 .39543 L .7488 .3995 L .74786 .40763 L .74666 .41572 L .7452 .42377 L .74148 .4397 L .73673 .45536 L .73097 .47067 L .71651 .5 L .69834 .52719 L .67678 .55178 L .65219 .57334 L .625 .59151 L .61057 .59922 L .59567 .60597 L .58036 .61173 L .5647 .61648 L .54877 .6202 L .54072 .62166 L .53263 .62286 L .5245 .6238 L .52043 .62416 L .51635 .62446 L .51227 .6247 L .51022 .62479 L .50818 .62487 L .50614 .62492 L .50409 .62497 L .50205 .62499 L .5 .625 L .49795 .62499 L .49591 .62497 L .49386 .62492 L .49182 .62487 L .48978 .62479 L .48773 .6247 L .48365 .62446 L .47957 .62416 L .4755 .6238 L .46737 .62286 L .45928 .62166 L .45123 .6202 L .4353 .61648 L Mistroke .41964 .61173 L .40433 .60597 L .375 .59151 L .34781 .57334 L .32322 .55178 L .30166 .52719 L .28349 .5 L .27578 .48557 L .26903 .47067 L .26327 .45536 L .25852 .4397 L .2548 .42377 L .25334 .41572 L .25214 .40763 L .2512 .3995 L .25084 .39543 L .25054 .39135 L .2503 .38727 L .25021 .38522 L .25013 .38318 L .25008 .38114 L .25003 .37909 L .25001 .37705 L .25 .375 L .25001 .37295 L .25003 .37091 L .25008 .36886 L .25013 .36682 L .2503 .36273 L .25054 .35865 L .25084 .35457 L .2512 .3505 L .25214 .34237 L .25334 .33428 L .2548 .32623 L .25852 .3103 L .26327 .29464 L .26903 .27933 L .28349 .25 L .30166 .22281 L .32322 .19822 L .34781 .17666 L .375 .15849 L .38943 .15078 L .40433 .14403 L .41964 .13827 L .4353 .13352 L .45123 .1298 L .45928 .12834 L .46737 .12714 L Mistroke .4755 .1262 L .47957 .12584 L .48365 .12554 L .48773 .1253 L .48978 .12521 L .49182 .12513 L .49386 .12508 L .49591 .12503 L .49795 .12501 L .5 .125 L .50205 .12501 L .50409 .12503 L .50614 .12508 L .50818 .12513 L .51227 .1253 L .51635 .12554 L .52043 .12584 L .5245 .1262 L .53263 .12714 L .54072 .12834 L .54877 .1298 L .5647 .13352 L .58036 .13827 L .59567 .14403 L .625 .15849 L .65219 .17666 L .67678 .19822 L .69834 .22281 L .71651 .25 L .72422 .26443 L .73097 .27933 L .73673 .29464 L .74148 .3103 L .7452 .32623 L .74666 .33428 L .74786 .34237 L .7488 .3505 L .74916 .35457 L .74946 .35865 L .7497 .36273 L .74979 .36478 L .74987 .36682 L .74992 .36886 L .74997 .37091 L .74999 .37295 L .75 .375 L Mfstroke P P p 0 0 1 r .005 w .5 .375 m .67678 .55178 L s .5 .375 m .67678 .375 L s .67678 .375 m .67678 .55178 L s [({0.7071, 0.7071})] .725 .55175 -1 0 Mshowa [(x)] .625 .3625 0 0 Mshowa [(y)] .715 .4625 0 0 Mshowa [(r)] .5875 .5 0 0 Mshowa [(a)] .55 .4 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] If we assume this is the unit circle, then r=1. Using right triangle trigonometry, we find: Sin[a]=¨’ç=y and Cos[a]=¨’ô=x It follows that for any angle, the sine is y and the cosine is x. :[font = text; inactive; preserveAspect] The equation of a circle; xÛ+yÛ=rÛ helps verify this. If Cosine=x and Sine=y, then substituting gives the equation Cos^2Û+SinÛ=rÛ. Remember that r=1. Then the equation looks familiar: CosÛ+SinÛ=1. This we know is true because it is one of the basic trigonometric identities. :[font = text; inactive; preserveAspect] Now lets see the progress of this triangle as it is graphed around the unit circle. Animate the following... :[font = input; preserveAspect; startGroup] Clear[mvngtngl] mvngtngl[t_]:=Graphics[{{Thickness[0.001],RGBColor[0,0,1], {Line[{{0,0},{x[t],y[t]}}]}}, {Thickness[0.005], Line[{{0,0},{x[t],0}}], Line[{{x[t],0},{x[t],y[t]}}], Text[{N[x[t],3],N[y[t],3]}, {0.9,0.707},{-1,0}], Text[{"angle=",t},{-1.5,1}]}}]; JoesList={0,Pi/6,Pi/4,Pi/3,Pi/2, 2Pi/3,3Pi/4,5Pi/6,Pi, Pi+Pi/6,Pi+Pi/4,Pi+Pi/3,Pi+Pi/2, Pi+2Pi/3,Pi+3Pi/4,Pi+5Pi/6,2Pi}; Clear[pthold] pthold[t_]:=Graphics[{PointSize[0.025], Point[{x[t],y[t]}]}] Do[Show[crcl,mvngtngl[JoesList[[t]]], pthold[JoesList[[t]]], AspectRatio->Automatic, PlotRange->{{-2,2},{-1.5,1.5}}, AxesLabel->None, Ticks->None], {t,1,Length[JoesList]}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 263; startGroup; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .75 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.25 0.375 0.25 [ [ -0.001 -0.001 0 0 ] [ 1.001 .751 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .375 m 1 .375 L s P p .002 w .5 0 m .5 .75 L s P P 0 0 m 1 0 L 1 .75 L 0 .75 L closepath clip newpath p p 1 0 0 r p .004 w .75 .375 m .74999 .37705 L .74997 .37909 L .74992 .38114 L .74987 .38318 L .7497 .38727 L .74946 .39135 L .74916 .39543 L .7488 .3995 L .74786 .40763 L .74666 .41572 L .7452 .42377 L .74148 .4397 L .73673 .45536 L .73097 .47067 L .71651 .5 L .69834 .52719 L .67678 .55178 L .65219 .57334 L .625 .59151 L .61057 .59922 L .59567 .60597 L .58036 .61173 L .5647 .61648 L .54877 .6202 L .54072 .62166 L .53263 .62286 L .5245 .6238 L .52043 .62416 L .51635 .62446 L .51227 .6247 L .51022 .62479 L .50818 .62487 L .50614 .62492 L .50409 .62497 L .50205 .62499 L .5 .625 L .49795 .62499 L .49591 .62497 L .49386 .62492 L .49182 .62487 L .48978 .62479 L .48773 .6247 L .48365 .62446 L .47957 .62416 L .4755 .6238 L .46737 .62286 L .45928 .62166 L .45123 .6202 L .4353 .61648 L Mistroke .41964 .61173 L .40433 .60597 L .375 .59151 L .34781 .57334 L .32322 .55178 L .30166 .52719 L .28349 .5 L .27578 .48557 L .26903 .47067 L .26327 .45536 L .25852 .4397 L .2548 .42377 L .25334 .41572 L .25214 .40763 L .2512 .3995 L .25084 .39543 L .25054 .39135 L .2503 .38727 L .25021 .38522 L .25013 .38318 L .25008 .38114 L .25003 .37909 L .25001 .37705 L .25 .375 L .25001 .37295 L .25003 .37091 L .25008 .36886 L .25013 .36682 L .2503 .36273 L .25054 .35865 L .25084 .35457 L .2512 .3505 L .25214 .34237 L .25334 .33428 L .2548 .32623 L .25852 .3103 L .26327 .29464 L .26903 .27933 L .28349 .25 L .30166 .22281 L .32322 .19822 L .34781 .17666 L .375 .15849 L .38943 .15078 L .40433 .14403 L .41964 .13827 L .4353 .13352 L .45123 .1298 L .45928 .12834 L .46737 .12714 L Mistroke .4755 .1262 L .47957 .12584 L .48365 .12554 L .48773 .1253 L .48978 .12521 L .49182 .12513 L .49386 .12508 L .49591 .12503 L .49795 .12501 L .5 .125 L .50205 .12501 L .50409 .12503 L .50614 .12508 L .50818 .12513 L .51227 .1253 L .51635 .12554 L .52043 .12584 L .5245 .1262 L .53263 .12714 L .54072 .12834 L .54877 .1298 L .5647 .13352 L .58036 .13827 L .59567 .14403 L .625 .15849 L .65219 .17666 L .67678 .19822 L .69834 .22281 L .71651 .25 L .72422 .26443 L .73097 .27933 L .73673 .29464 L .74148 .3103 L .7452 .32623 L .74666 .33428 L .74786 .34237 L .7488 .3505 L .74916 .35457 L .74946 .35865 L .7497 .36273 L .74979 .36478 L .74987 .36682 L .74992 .36886 L .74997 .37091 L .74999 .37295 L .75 .375 L Mfstroke P P p p 0 0 1 r p .001 w .5 .375 m .75 .375 L s P P p .005 w .5 .375 m .75 .375 L s .75 .375 m .75 .375 L s [({1., 0})] .725 .55175 -1 0 Mshowa [({angle=, 0})] .125 .625 0 0 Mshowa P P p .025 w .75 .375 Mdot P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 263; animationSpeed = 12] %! 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:[font = text; inactive; preserveAspect] We can also progress through the Sine curve. Animate this... :[font = input; preserveAspect; startGroup] Clear[sinpoint]; sinpoint[t_] := Graphics[ {PointSize[.025], Point[{t,Sin[t]}]}] ticks = Table[i,{i,0,2Pi,Pi/2}]; pointlist = {}; Do[ AppendTo[pointlist,sinpoint[JoesList[[t]]]]; {Show[pointlist,Ticks->{ticks,Automatic}, PlotRange->{{0,2Pi},{-1,1}}, Axes->True],}, {t,1,Length[JoesList]}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 216; startGroup; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.159155 0.309017 0.309017 [ [(Pi)(--)(2)] .25 .30902 0 2 Msboxa [(Pi)] .5 .30902 0 2 Msboxa [(3 Pi)(----)( 2)] .75 .30902 0 2 Msboxa [(2 Pi)] 1 .30902 0 2 Msboxa [(-1)] -0.0125 0 1 0 Msboxa [(-0.75)] -0.0125 .07725 1 0 Msboxa [(-0.5)] -0.0125 .15451 1 0 Msboxa [(-0.25)] -0.0125 .23176 1 0 Msboxa [(0)] -0.0125 .30902 1 0 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P p .001 w 0 .37082 m .00375 .37082 L s P p .001 w 0 .40172 m .00375 .40172 L s P p .001 w 0 .41717 m .00375 .41717 L s P p .001 w 0 .43262 m .00375 .43262 L s P p .001 w 0 .44807 m .00375 .44807 L s P p .001 w 0 .47898 m .00375 .47898 L s P p .001 w 0 .49443 m .00375 .49443 L s P p .001 w 0 .50988 m .00375 .50988 L s P p .001 w 0 .52533 m .00375 .52533 L s P p .001 w 0 .55623 m .00375 .55623 L s P p .001 w 0 .57168 m .00375 .57168 L s P p .001 w 0 .58713 m .00375 .58713 L s P p .001 w 0 .60258 m .00375 .60258 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .025 w 0 .30902 Mdot P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 216; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.159155 0.309017 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