(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 59776, 1149] NotebookOptionsPosition[ 43329, 855] NotebookOutlinePosition[ 59862, 1151] CellTagsIndexPosition[ 59819, 1148] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`fcn$$ = $CellContext`f3, $CellContext`n$$ = 10, $CellContext`opacity$$ = 0.877, $CellContext`POINTS$$ = 2, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fcn$$], $CellContext`f2, Style[ Row[{ Spacer[50], "f(x) = "}], 20, Bold]}, {$CellContext`f1 -> Row[{ Style[ TraditionalForm[$CellContext`x^2], 20], Style[ ", a=0 \[LessEqual] x \[LessEqual] b=1", 20]}], $CellContext`f2 -> Row[{ Style[ TraditionalForm[ Sin[$CellContext`x]], 20], Style[ ", a=0 \[LessEqual] x \[LessEqual] b=2\[Pi]", 20]}], $CellContext`f3 -> Row[{ Style[ TraditionalForm[ Cos[$CellContext`x]], 20], Style[ ", a=-\[Pi] \[LessEqual] x \[LessEqual] b=\[Pi]/2", 20]}], $CellContext`f4 -> Row[{ Style[ TraditionalForm[2 - E^$CellContext`x], 20], Style[", a=0 \[LessEqual] x \[LessEqual] b=1", 20]}], Null}}, {{ Hold[$CellContext`POINTS$$], 2, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"SAMPLE\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"POINTS\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, { 1 -> Style["Left\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 15, Bold}], 2 -> Style["Midpoints\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 15, Bold}], 3 -> Style["Right ", { GrayLevel[0], 15, Bold}], 4 -> Row[{ Style["Area ", { GrayLevel[0], 15, Bold}], Invisible["12345678910121314151617181920"], "\!\(\*\nStyleBox[\"\[Copyright]\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\".\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Bykov\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"SJ\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Delta\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"College\",\n\ FontColor->RGBColor[0, 0, 1]]\)"}]}}, { Hold[ "\!\(\*\nStyleBox[\"Color\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 0, 1]]\)"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`opacity$$], 0.5, ""}, 0, 1}, {{ Hold[$CellContext`n$$], 10, Row[{ Spacer[350], "\!\(\*\nStyleBox[\"n\",\nFontSize->24,\nFontColor->RGBColor[0, 0, \ 1]]\)"}]}, 1, 100, 1}}, Typeset`size$$ = {1010., {277.5, 282.5}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`fcn$1744$$ = False, $CellContext`POINTS$1745$$ = False, $CellContext`opacity$1746$$ = 0, $CellContext`n$1747$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`fcn$$ = $CellContext`f2, $CellContext`n$$ = 10, $CellContext`opacity$$ = 0.5, $CellContext`POINTS$$ = 2}, "ControllerVariables" :> { Hold[$CellContext`fcn$$, $CellContext`fcn$1744$$, False], Hold[$CellContext`POINTS$$, $CellContext`POINTS$1745$$, False], Hold[$CellContext`opacity$$, $CellContext`opacity$1746$$, 0], Hold[$CellContext`n$$, $CellContext`n$1747$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Framed[ Pane[Switch[$CellContext`fcn$$, $CellContext`f1, $CellContext`f[ Pattern[$CellContext`x, Blank[]]] := $CellContext`x^2; $CellContext`a = 0; $CellContext`b = 1, $CellContext`f2, $CellContext`f[ Pattern[$CellContext`x, Blank[]]] := Sin[$CellContext`x]; $CellContext`a = 0; $CellContext`b = 2 Pi, $CellContext`f3, $CellContext`f[ Pattern[$CellContext`x, Blank[]]] := Cos[$CellContext`x]; $CellContext`a = -Pi; $CellContext`b = Pi/2, $CellContext`f4, $CellContext`f[ Pattern[$CellContext`x, Blank[]]] := 2 - Exp[$CellContext`x]; $CellContext`a = 0; $CellContext`b = 1]; $CellContext`d = ($CellContext`b - \ $CellContext`a)/$CellContext`n$$; $CellContext`Int = Integrate[ $CellContext`f[$CellContext`x], {$CellContext`x, $CellContext`a, \ $CellContext`b}]; $CellContext`Curve = Plot[ $CellContext`f[$CellContext`x], {$CellContext`x, $CellContext`a, \ $CellContext`b}, PlotStyle -> {Blue, Thick}, Filling -> Axis]; If[$CellContext`POINTS$$ == 1, $CellContext`SamplePoints = Table[$CellContext`a + ($CellContext`i - 1) $CellContext`d, {$CellContext`i, 1, $CellContext`n$$}]]; If[$CellContext`POINTS$$ == 2, $CellContext`SamplePoints = Table[$CellContext`a + ($CellContext`i - 1/ 2) $CellContext`d, {$CellContext`i, 1, $CellContext`n$$}]]; If[$CellContext`POINTS$$ == 3, $CellContext`SamplePoints = Table[$CellContext`a + $CellContext`i $CellContext`d, \ {$CellContext`i, 1, $CellContext`n$$}]]; If[ Or[$CellContext`POINTS$$ == 1, $CellContext`POINTS$$ == 2, $CellContext`POINTS$$ == 3], {$CellContext`S = $CellContext`d Sum[ $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]], \ {$CellContext`i, 1, $CellContext`n$$}]; If[$CellContext`n$$ < 20, $CellContext`R = Graphics[{ EdgeForm[Black], Opacity[$CellContext`opacity$$], Table[{ Inset[ If[$CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]] < 0, "\!\(\*\nStyleBox[\"-\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->36,\nFontColor->RGBColor[1, 0, 0]]\)", "\!\(\*\nStyleBox[\"+\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->36,\nFontColor->RGBColor[0, 0, 1]]\)"], {$CellContext`a + \ ($CellContext`i - 1/4) $CellContext`d, 0.5 $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}], Inset[ Row[{"\!\(\*\nStyleBox[\"R\",\nFontFamily->\"Grenoble \ Heavy SF\",\nFontSize->16]\)", $CellContext`i}], {$CellContext`a + \ ($CellContext`i - 3/4) $CellContext`d, 0.5 $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}], Hue[ RandomReal[]], Rectangle[{$CellContext`a + ($CellContext`i - 1) $CellContext`d, 0}, {$CellContext`a + $CellContext`i $CellContext`d, $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}]}, \ {$CellContext`i, 1, $CellContext`n$$}]}], $CellContext`R = Graphics[{ EdgeForm[Black], Opacity[$CellContext`opacity$$], Table[{ Hue[ RandomReal[]], Rectangle[{$CellContext`a + ($CellContext`i - 1) $CellContext`d, 0}, {$CellContext`a + $CellContext`i $CellContext`d, $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}]}, \ {$CellContext`i, 1, $CellContext`n$$}]}]]; $CellContext`P1 = Graphics[ Table[{ RGBColor[0, 0, 1], PointSize[0.01], Point[{ Part[$CellContext`SamplePoints, $CellContext`i], $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}]}, \ {$CellContext`i, 1, $CellContext`n$$}]]; $CellContext`P2 = Graphics[ Table[{ RGBColor[1, 0, 0], PointSize[0.01], Point[{ Part[$CellContext`SamplePoints, $CellContext`i], 0}]}, {$CellContext`i, 1, $CellContext`n$$}]]; $CellContext`P3 = Graphics[ Table[{ RGBColor[1, 0, 0], PointSize[0.01], Point[{0, $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}]}, \ {$CellContext`i, 1, $CellContext`n$$}]]; $CellContext`L1 = Graphics[ Table[{Dashed, Line[{{ Part[$CellContext`SamplePoints, $CellContext`i], 0}, { Part[$CellContext`SamplePoints, $CellContext`i], $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}}]}, \ {$CellContext`i, 1, $CellContext`n$$}]]; $CellContext`L2 = Graphics[ Table[{Dashed, Line[{{0, $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}, { Part[$CellContext`SamplePoints, $CellContext`i], $CellContext`f[ Part[$CellContext`SamplePoints, $CellContext`i]]}}]}, \ {$CellContext`i, 1, $CellContext`n$$}]]; Null}, {$CellContext`R = Graphics[]; $CellContext`P1 = Graphics[]; $CellContext`P2 = Graphics[]; $CellContext`P3 = Graphics[]; $CellContext`L1 = Graphics[]; $CellContext`L2 = Graphics[]; Null}]; Row[{ Show[$CellContext`Curve, $CellContext`R, $CellContext`P1, \ $CellContext`P2, $CellContext`P3, $CellContext`L1, $CellContext`L2, Axes -> True, AxesLabel -> {"X", "Y"}, ImageSize -> {550, 600}], Style[ Column[{ Row[{"\!\(\*\nStyleBox[\"NET\",\nFontFamily->\"Grenoble Heavy \ SF\",\nFontColor->RGBColor[0.5019607843137255, 0.5019607843137255, 1.]]\)\!\(\ \*\nStyleBox[\" \",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontColor->RGBColor[0.5019607843137255, 0.5019607843137255, 1.]]\)\!\(\*\n\ StyleBox[\"AREA\",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontColor->RGBColor[0.5019607843137255, 0.5019607843137255, 1.]]\) \!\(\*\n\ StyleBox[\"=\",\nFontFamily->\"Grenoble Light SF\"]\)\!\(\*\nStyleBox[\" \",\n\ FontFamily->\"Grenoble Light SF\"]\)\!\(\*\nStyleBox[SubsuperscriptBox[\"\ \[Integral]\", \"a\", \"b\"],\nFontFamily->\"Grenoble Light SF\",\n\ FontSize->18]\)\!\(\*\nStyleBox[\"f\",\nFontFamily->\"Grenoble Light SF\",\n\ FontSize->18]\)\!\(\*\nStyleBox[\"(\",\nFontFamily->\"Grenoble Light SF\",\n\ FontSize->18]\)\!\(\*\nStyleBox[\"x\",\nFontFamily->\"Grenoble Light SF\",\n\ FontSize->18]\)\!\(\*\nStyleBox[\")\",\nFontFamily->\"Grenoble Light SF\",\n\ FontSize->18]\)\!\(\*\nStyleBox[\"\[DifferentialD]\",\nFontFamily->\"Grenoble \ Light SF\",\nFontSize->18]\)\!\(\*\nStyleBox[\"x\",\nFontFamily->\"Grenoble \ Light SF\",\nFontSize->18]\)\!\(\*\nStyleBox[\" \",\nFontFamily->\"Grenoble \ Light SF\",\nFontSize->16]\)\!\(\*\nStyleBox[\"=\",\nFontFamily->\"Grenoble \ Light SF\",\nFontSize->16]\) ", N[$CellContext`Int]}], Null, Null, If[ Or[$CellContext`POINTS$$ == 1, $CellContext`POINTS$$ == 2, $CellContext`POINTS$$ == 3], Row[{"\!\(\*\nStyleBox[\"RIEMANN\",\nFontFamily->\"Grenoble \ Heavy SF\"]\)\!\(\*\nStyleBox[\" \",\nFontFamily->\"Grenoble Heavy \ SF\"]\)\!\(\*\nStyleBox[\"SUM\",\nFontFamily->\"Grenoble Heavy SF\"]\) \!\(\*\ \nStyleBox[\"=\",\nFontFamily->\"Grenoble Light SF\"]\)\!\(\*\n\ StyleBox[UnderoverscriptBox[\n RowBox[{\" \", \"\[Sum]\"}], \n \ RowBox[{\"i\", \"=\", \"1\"}], \"n\"],\nFontFamily->\"Grenoble Light \ SF\"]\)\!\(\*\nStyleBox[\"\[PlusMinus]\",\nFontFamily->\"Grenoble Light SF\"]\ \)\!\(\*\nStyleBox[\"Area\",\nFontFamily->\"Grenoble Light SF\"]\)\!\(\*\n\ StyleBox[\"(\",\nFontFamily->\"Grenoble Light SF\"]\)\!\(\*\n\ StyleBox[SubscriptBox[\"R\", \"i\"],\nFontFamily->\"Grenoble Light \ SF\"]\)\!\(\*\nStyleBox[\")\",\nFontFamily->\"Grenoble Light SF\"]\)\!\(\*\n\ StyleBox[\" \",\nFontFamily->\"Grenoble Light SF\"]\)\!\(\*\nStyleBox[\"=\",\n\ FontFamily->\"Grenoble Light SF\"]\) ", N[$CellContext`S]}]], Null, Null, If[ Or[$CellContext`POINTS$$ == 1, $CellContext`POINTS$$ == 2, $CellContext`POINTS$$ == 3], Row[{ If[$CellContext`S < $CellContext`Int, Row[{"\!\(\*\nStyleBox[\"Underestimate\",\n\ FontFamily->\"Grenoble Light SF\",\nFontColor->RGBColor[0, 0, 1]]\) by ", N[$CellContext`Int - $CellContext`S]}], Row[{"\!\(\*\nStyleBox[\"Overestimate\",\n\ FontFamily->\"Grenoble Light SF\",\nFontColor->RGBColor[1, 0, 0]]\) by ", N[$CellContext`S - $CellContext`Int]}]]}]]}, Frame -> True], 20]}], {1000, 550}, Alignment -> {Left}, ImageSizeAction -> Automatic]], "Specifications" :> {{{$CellContext`fcn$$, $CellContext`f2, Style[ Row[{ Spacer[50], "f(x) = "}], 20, Bold]}, {$CellContext`f1 -> Row[{ Style[ TraditionalForm[$CellContext`x^2], 20], Style[ ", a=0 \[LessEqual] x \[LessEqual] b=1", 20]}], $CellContext`f2 -> Row[{ Style[ TraditionalForm[ Sin[$CellContext`x]], 20], Style[ ", a=0 \[LessEqual] x \[LessEqual] b=2\[Pi]", 20]}], $CellContext`f3 -> Row[{ Style[ TraditionalForm[ Cos[$CellContext`x]], 20], Style[ ", a=-\[Pi] \[LessEqual] x \[LessEqual] b=\[Pi]/2", 20]}], $CellContext`f4 -> Row[{ Style[ TraditionalForm[2 - E^$CellContext`x], 20], Style[", a=0 \[LessEqual] x \[LessEqual] b=1", 20]}], Null}, ControlType -> PopupMenu, FieldSize -> {25, 1.5}}, {{$CellContext`POINTS$$, 2, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"SAMPLE\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"POINTS\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, { 1 -> Style["Left\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 15, Bold}], 2 -> Style["Midpoints\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 15, Bold}], 3 -> Style["Right ", { GrayLevel[0], 15, Bold}], 4 -> Row[{ Style["Area ", { GrayLevel[0], 15, Bold}], Invisible["12345678910121314151617181920"], "\!\(\*\nStyleBox[\"\[Copyright]\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\".\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Bykov\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"SJ\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Delta\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"College\",\n\ FontColor->RGBColor[0, 0, 1]]\)"}]}, ControlType -> RadioButtonBar}, "\!\(\*\nStyleBox[\"Color\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 0, 1]]\)", {{$CellContext`opacity$$, 0.5, ""}, 0, 1, ImageSize -> Large, AutoAction -> True, ControlType -> VerticalSlider}, {{$CellContext`n$$, 10, Row[{ Spacer[350], "\!\(\*\nStyleBox[\"n\",\nFontSize->24,\nFontColor->RGBColor[0, \ 0, 1]]\)"}]}, 1, 100, 1, Appearance -> "Open", ImageSize -> Large, AutoAction -> True}}, "Options" :> { ControlPlacement -> { Top, Top, Right, Right, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, 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{Pi/4, 0.4045084971874737}], Hue[0.2108582315195846], Rectangle[{Pi/5, 0}, {(2 Pi)/5, (1 + Sqrt[5])/4}]}, { Inset[ "\!\(\*\nStyleBox[\"+\",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontSize->36,\nFontColor->RGBColor[0, 0, 1]]\)", {(11 Pi)/20, 0.5}], Inset[ Row[{"\!\(\*\nStyleBox[\"R\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->16]\)", 3}], {(9 Pi)/20, 0.5}], Hue[0.6477210425571438], Rectangle[{(2 Pi)/5, 0}, {(3 Pi)/5, 1}]}, { Inset[ "\!\(\*\nStyleBox[\"+\",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontSize->36,\nFontColor->RGBColor[0, 0, 1]]\)", {(3 Pi)/4, 0.4045084971874737}], Inset[ Row[{"\!\(\*\nStyleBox[\"R\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->16]\)", 4}], {(13 Pi)/20, 0.4045084971874737}], Hue[0.28148038653927476`], Rectangle[{(3 Pi)/5, 0}, {(4 Pi)/5, (1 + Sqrt[5])/4}]}, { Inset[ "\!\(\*\nStyleBox[\"+\",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontSize->36,\nFontColor->RGBColor[0, 0, 1]]\)", {(19 Pi)/20, 0.15450849718747373`}], Inset[ Row[{"\!\(\*\nStyleBox[\"R\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->16]\)", 5}], {(17 Pi)/20, 0.15450849718747373`}], Hue[0.8410146219589938], Rectangle[{(4 Pi)/5, 0}, {Pi, (-1 + Sqrt[5])/4}]}, { Inset[ "\!\(\*\nStyleBox[\"-\",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontSize->36,\nFontColor->RGBColor[1, 0, 0]]\)", {(23 Pi)/ 20, -0.15450849718747373`}], Inset[ Row[{"\!\(\*\nStyleBox[\"R\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->16]\)", 6}], {(21 Pi)/20, -0.15450849718747373`}], Hue[0.9135453887335949], Rectangle[{Pi, 0}, {(6 Pi)/5, (1 - Sqrt[5])/4}]}, { Inset[ "\!\(\*\nStyleBox[\"-\",\nFontFamily->\"Grenoble Heavy SF\",\n\ FontSize->36,\nFontColor->RGBColor[1, 0, 0]]\)", {(27 Pi)/ 20, -0.4045084971874737}], Inset[ Row[{"\!\(\*\nStyleBox[\"R\",\nFontFamily->\"Grenoble Heavy SF\ \",\nFontSize->16]\)", 7}], {(5 Pi)/4, -0.4045084971874737}], Hue[0.14910270539067616`], Rectangle[{(6 Pi)/5, 0}, {(7 Pi)/5, (-1 - Sqrt[5])/4}]}, { Inset[ 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Hue[0.1851415221632935], Rectangle[{(9 Pi)/5, 0}, { 2 Pi, (1 - Sqrt[5])/4}]}}}], $CellContext`P1 = Graphics[{{ RGBColor[0, 0, 1], PointSize[0.01], Point[{Pi/10, (-1 + Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(3 Pi)/10, (1 + Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{Pi/2, 1}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(7 Pi)/10, (1 + Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(9 Pi)/10, (-1 + Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(11 Pi)/10, (1 - Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(13 Pi)/10, (-1 - Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(3 Pi)/2, -1}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(17 Pi)/10, (-1 - Sqrt[5])/4}]}, { RGBColor[0, 0, 1], PointSize[0.01], Point[{(19 Pi)/10, (1 - Sqrt[5])/4}]}}], $CellContext`P2 = Graphics[{{ RGBColor[1, 0, 0], PointSize[0.01], Point[{Pi/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(3 Pi)/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{Pi/2, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(7 Pi)/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(9 Pi)/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(11 Pi)/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(13 Pi)/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(3 Pi)/2, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(17 Pi)/10, 0}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{(19 Pi)/10, 0}]}}], $CellContext`P3 = Graphics[{{ RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (-1 + Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (1 + Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, 1}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (1 + Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (-1 + Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (1 - Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (-1 - Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, -1}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (-1 - Sqrt[5])/4}]}, { RGBColor[1, 0, 0], PointSize[0.01], Point[{0, (1 - Sqrt[5])/4}]}}], $CellContext`L1 = Graphics[{{ Dashing[{Small, Small}], Line[{{Pi/10, 0}, {Pi/10, (-1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{(3 Pi)/10, 0}, {(3 Pi)/10, (1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{Pi/2, 0}, {Pi/2, 1}}]}, { Dashing[{Small, Small}], Line[{{(7 Pi)/10, 0}, {(7 Pi)/10, (1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{(9 Pi)/10, 0}, {(9 Pi)/10, (-1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{(11 Pi)/10, 0}, {(11 Pi)/10, (1 - Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{(13 Pi)/10, 0}, {(13 Pi)/10, (-1 - Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{(3 Pi)/2, 0}, {(3 Pi)/2, -1}}]}, { Dashing[{Small, Small}], Line[{{(17 Pi)/10, 0}, {(17 Pi)/10, (-1 - Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{(19 Pi)/10, 0}, {(19 Pi)/10, (1 - Sqrt[5])/ 4}}]}}], $CellContext`L2 = Graphics[{{ Dashing[{Small, Small}], Line[{{0, (-1 + Sqrt[5])/4}, {Pi/10, (-1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, (1 + Sqrt[5])/4}, {(3 Pi)/10, (1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, 1}, {Pi/2, 1}}]}, { Dashing[{Small, Small}], Line[{{0, (1 + Sqrt[5])/4}, {(7 Pi)/10, (1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, (-1 + Sqrt[5])/4}, {(9 Pi)/10, (-1 + Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, (1 - Sqrt[5])/4}, {(11 Pi)/10, (1 - Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, (-1 - Sqrt[5])/4}, {(13 Pi)/10, (-1 - Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, -1}, {(3 Pi)/2, -1}}]}, { Dashing[{Small, Small}], Line[{{0, (-1 - Sqrt[5])/4}, {(17 Pi)/10, (-1 - Sqrt[5])/4}}]}, { Dashing[{Small, Small}], Line[{{0, (1 - Sqrt[5])/4}, {(19 Pi)/10, (1 - Sqrt[5])/ 4}}]}}]}; {$CellContext`st1 = { Black, 15, Bold}; $CellContext`st2 = 20; Null}}; Typeset`initDone$$ = True), SynchronousInitialization->True, 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