(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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This is an indication that the user is invited and encouraged to \ replace these values with others to investigate the resulting effects.", Evaluatable->False, FontSize->14] }], "Title", Evaluatable->False], Cell[TextData[{ StyleBox[ "Please feel free to contact the author, Paul R. McCreary, with any \ comments or suggestions. He can be reached at the Department of Mathematics, \ UIUC, Urbana, IL 61801 or by email address: paulmcc@", Evaluatable->False], StyleBox["@", Evaluatable->False, FontFamily->"Chicago"], StyleBox["symcom.math.uiuc.edu", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Comments for the 6/6 Presentation"], "Subsubtitle", Evaluatable->False], Cell[TextData[ "Much of the mathematics in this presentation was unknown 150 years ago, \ which is a very short time in the history of all mathematics. There are even \ unanswered questions about the material that mathematicians still work on \ today.\n\nSince most of the material will be familiar to the professors \ present, we will ask them to take on the special role of observor, so as not \ to spoil the fun of discovery. The particular question they should concern \ themselves with is, \"How do the descriptions and ideas evolve, with whom do \ they originate and who reshapes them?\" An especially good role will be to \ encourage the discussion of others while not discussing content yourself.\n\n\ "], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Initializations: ", PageWidth->Infinity, Evaluatable->False, InitializationCell->True], StyleBox["All of the included cells should be executed.", PageWidth->Infinity, Evaluatable->False, InitializationCell->True, FontColor->RGBColor[0.732433, 0.263157, 0.284169]] }], "Subsubsection", PageWidth->Infinity, Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData["2-D Graphics"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["Unit Disc Created"], "SmallText", Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData[ "Eliminating the number 1, which is sent to ComplexInfinity by g[z] when \ fixedPoint1 = 1."], "Text", Evaluatable->False, InitializationCell->True], Cell[BoxData[ \(\(rotation = \(1 + .001\ I\)\/Abs[1 + .001\ I]; 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.92709 .41468 L s .92709 .41468 m 1 .95087 L s 1 .95087 m .70609 .58851 L s .70609 .58851 m .67218 0 L s .08841 .24526 m 0 .80647 L s 0 .80647 m .70609 .58851 L s .70609 .58851 m .67218 0 L s .67218 0 m .08841 .24526 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{264.562, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False, ImageRangeCache->{{{0, 263.562}, {287, 0}} -> {0.0495908, -0.000233974, 0.00324785, 0.00324785}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Misc graphics items"], "SmallText", Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData["Axes"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(axes = Graphics3D[{RGBColor[1, 0, 0], Line[{{1.5, 0, 0}, {\(-1.5\), 0, 0}}], RGBColor[0, 0, 1], Line[{{0, 0, 1.5}, {0, 0, \(-1.5\)}}], RGBColor[0, 1, 0], Line[{{0, 1.5, 0}, {0, \(-1.5\), 0}}]}]; \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(axes\)\" is similar to \ existing symbol \"\!\(Axes\)\"."\)], "Message"] }, Open ]] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Functions"], "Text", Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData["Basic Functions defined"], "Text", Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData[ "mTrans, the Mobius transformation defined along with its inverse."], "SmallText", Evaluatable->False, InitializationCell->True], Cell[BoxData[{ \(\(mTrans[z_] := N[\(a\ z + b\)\/\(c\ z + d\)]; \)\), \(\(mTransInv[z_] := N[\(d\ z - b\)\/\(\(-c\)\ z + a\)]; \)\)}], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Define g & gInverse "], "SmallText", Evaluatable->False, InitializationCell->True], Cell[TextData["FixedPoint1 -> \\ ; FixedPoint2 -> 0."], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(g[z_] := N[\(z - fixedPoint2\)\/\(z - fixedPoint1\)]; \)\), \(\(gInv[z_] := N[\(fixedPoint1\ z - fixedPoint2\)\/\(z - 1\)]; \)\), \(\(parabolicG[z_] := N[1\/\(z - fixedPoint\)]; \)\), \(\(parabolicGInv[z_] := N[\(1 + z\ fixedPoint\)\/z]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(fixedPoint\)\" is \ similar to existing symbol \"\!\(FixedPoint\)\"."\)], "Message"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Stereographic Projection Code"], "Text", Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData["Computation for the projection"], "SmallText", Evaluatable->False, InitializationCell->True], Cell[TextData[ "xs=2 x/(Abs[z]^2 + 1);\nys=2 y/(Abs[z]^2 + 1);\nzs=(Abs[z]^2 - 1)/(Abs[z]^2 \ + 1);"], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "stereoProjToSphere[ ]:\n\tList of Complex Numbers\t\t-> List of 3-D \ points."], "SmallText", Evaluatable->False, InitializationCell->True], Cell[BoxData[ \(\(stereoProjToSphere[numbersList_List] := Table[{\(2\ Re[numbersList\[LeftDoubleBracket]m\[RightDoubleBracket]] \)\/\(Abs[ numbersList\[LeftDoubleBracket]m\[RightDoubleBracket]]\^2 + 1\), \(2\ Im[numbersList\[LeftDoubleBracket]m\[RightDoubleBracket]] \)\/\(Abs[ numbersList\[LeftDoubleBracket]m\[RightDoubleBracket]]\^2 + 1\), \(Abs[ numbersList\[LeftDoubleBracket]m\[RightDoubleBracket]]\^2 - 1\)\/\(Abs[ numbersList\[LeftDoubleBracket]m\[RightDoubleBracket]]\^2 + 1\)}, {m, Length[numbersList]}]; \)\)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "pointsOnSphere[numbersList,RGBColor[r,g,b],pointSizeFactor] \t\tList of \ Complex Numbers -> Graphics of points after \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tstreographic projection"], "SmallText", Evaluatable->False, InitializationCell->True], Cell[BoxData[ \(pointsOnSphere[numbersList_, RGBColor[r_, g_, b_], pointSizeFactor_] := Block[{pts1, pSizes, pSizes1, pSizes2}, pts1 = stereoProjToSphere[numbersList]; pSizes1 = Table[1\/2 + n\/\(2\ Floor[Length[pts1]\/2]\), {n, Floor[Length[pts1]\/2]}]; pSizes2 = Table[\(Length[pts1] - n\/2\)\/Length[pts1], {n, Ceiling[Length[pts1]\/2], Length[pts1]}]; pSizes = Join[pSizes1, pSizes2]; Return[Graphics3D[{RGBColor[r, g, b], Table[{PointSize[ pointSizeFactor\ pSizes\[LeftDoubleBracket]n\[RightDoubleBracket]], Point[pts1\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[pts1]}]}]]; ]\)], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Basic Functions for Modular Group Picture"], "SmallText", Evaluatable->False], Cell[BoxData[ \(\(planeToDisc[z_] := \(I\ \((z - I)\)\)\/\(z + I\); \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(discToPlane[z_] := \(z + I\)\/\(z\ I + 1\); \)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(parabolicRS[z_] := N[1 + z]; \)\), \(\(parabolicLS[z_] := N[\(-1\) + z]; \)\)}], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicLS\)\" is \ similar to existing symbol \"\!\(parabolicRS\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(elliptic[z_] := N[\(-\(1\/z\)\)]; \)\), \(\(parabolicR[z_, n_] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1\[RightDoubleBracket] + z\[LeftDoubleBracket]Round[Length[z]\/2] \[RightDoubleBracket])\)] < 0, N[n + 1 + z], N[n + z]]; \)\), \(\(parabolicL[z_, n_] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1\[RightDoubleBracket] + z\[LeftDoubleBracket]Round[Length[z]\/2] \[RightDoubleBracket])\)] > 0, N[\(-n\) - 1 + z], N[\(-n\) + z]]; \)\), \(\(ellParaRR[z_, 0] := z; \)\), \(\(ellParaRR[z_, 1] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1, 1\[RightDoubleBracket] + \(z\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]Round[ 1\/2\ Length[ z\[LeftDoubleBracket]1\[RightDoubleBracket]]] \[RightDoubleBracket])\)] < 0, N[\(-\(1\/\(2 + z\)\)\)], N[\(-\(1\/\(1 + z\)\)\)]]; \)\), \(\(ellParaRR[z_, n_] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1, 1\[RightDoubleBracket] + \(z\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]Round[ 1\/2\ Length[ z\[LeftDoubleBracket]1\[RightDoubleBracket]]] \[RightDoubleBracket])\)] < 0, ellParaR[ellParaR[parabolic[z], n - 1], 1], ellParaR[ellParaR[z, n - 1], 1]]; \)\), \(\(ellParaLL[z_, 0] := z; \)\), \(\(ellParaLL[z_, 1] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1, 1\[RightDoubleBracket] + \(z\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]Round[ 1\/2\ Length[ z\[LeftDoubleBracket]1\[RightDoubleBracket]]] \[RightDoubleBracket])\)] > 0, N[\(-\(1\/\(\(-2\) + z\)\)\)], N[\(-\(1\/\(\(-1\) + z\)\)\)]]; \)\), \(\(ellParaLL[z_, n_] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1, 1\[RightDoubleBracket] + \(z\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]Round[ 1\/2\ Length[ z\[LeftDoubleBracket]1\[RightDoubleBracket]]] \[RightDoubleBracket])\)] > 0, ellParaL[ellParaL[parabolicL[z], n - 1], 1], ellParaL[ellParaL[z, n - 1], 1]]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicR\)\" is \ similar to existing symbols \!\({parabolicG, parabolicRS}\)."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicL\)\" is \ similar to existing symbols \!\({parabolicG, parabolicLS, parabolicR}\)."\)], "Message"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(ellParaR\)\" is similar \ to existing symbol \"\!\(ellParaRR\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolic\)\" is \ similar to existing symbols \!\({parabolicG, parabolicL, parabolicR}\)."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(ellParaL\)\" is similar \ to existing symbols \!\({ellParaLL, ellParaR}\)."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(parabolicRListFast[z_, n_] := N[n + z]; \)\), \(\(parabolicLListFast[z_, n_] := N[\(-n\) + z]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicLListFast\)\" \ is similar to existing symbol \"\!\(parabolicRListFast\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(parabolicRList[z_, n_] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1, 1\[RightDoubleBracket] + \(z\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]Round[ 1\/2\ Length[ z\[LeftDoubleBracket]1\[RightDoubleBracket]]] \[RightDoubleBracket])\)] < 0, N[n + 1 + z], N[n + z]]; \)\), \(\(parabolicLList[z_, n_] := If[Re[1\/2\ \((z\[LeftDoubleBracket]1, 1\[RightDoubleBracket] + \(z\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]Round[ 1\/2\ Length[ z\[LeftDoubleBracket]1\[RightDoubleBracket]]] \[RightDoubleBracket])\)] > 0, N[\(-n\) - 1 + z], N[\(-n\) + z]]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicLList\)\" is \ similar to existing symbol \"\!\(parabolicRList\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sssssssssssssssssssssssssssssssssssssssssssssss\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(sssssssssssssssssssssssssssssssssssssssssssssss\)], "Output"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(elliptic[z_] := N[\(-\(1\/z\)\)]; \)\), \(\(parabolicRN[z_, n_] := N[n + z]; \)\), \(\(parabolicLN[z_, n_] := N[\(-n\) + z]; \)\), \(\(ellParaR[z_, 0] := z; \)\), \(\(ellParaR[z_, 1] := N[\(-\(1\/\(1 + z\)\)\)]; \)\), \(\(ellParaL[z_, 0] := z; \)\), \(\(ellParaL[z_, 1] := N[\(-\(1\/\(\(-1\) + z\)\)\)]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicRN\)\" is \ similar to existing symbols \!\({parabolicR, parabolicRS}\)."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(parabolicLN\)\" is \ similar to existing symbols \!\({parabolicL, parabolicLS, parabolicRN}\)."\)], "Message"] }, Open ]], Cell[BoxData[{ \(\(parabolicRList[z_, n_] := N[n + z]; \)\), \(\(parabolicLList[z_, n_] := N[\(-n\) + z]; \)\)}], "Input", InitializationCell->True] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Initialize Tables"], "Text", PageWidth->Infinity, Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData["Misc"], "SmallText", Evaluatable->False], Cell[BoxData[ \(\(flowerPict = Table[{}, {n, 100}]; \)\)], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[BoxData[ \(\(flowerPict2 = Table[{}, {n, 100}]; \)\)], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(flowerPict2B = Table[{}, {n, 100}]; \)\)], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(flowerPict2B\)\" is \ similar to existing symbol \"\!\(flowerPict2\)\"."\)], "Message"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Color tables"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["Loren's Colors"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(Color = {{}, {}, {}, {}, {}, {}}; \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(Color\)\" is similar to \ existing symbol \"\!\(color\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(background = RGBColor[0.990, 1.000, 0.602]\ \((really\ pale\ yellow)\)\), \(Color\[LeftDoubleBracket]1\[RightDoubleBracket] = RGBColor[0.665, 0.081, 1.000]\), \(Color\[LeftDoubleBracket]2\[RightDoubleBracket] = RGBColor[1.000, 0.040, 0.392]\), \(Color\[LeftDoubleBracket]3\[RightDoubleBracket] = RGBColor[0.093, 0.302, 0.700]\), \(Color\[LeftDoubleBracket]4\[RightDoubleBracket] = RGBColor[0.065, 0.758, 0.409]\), \(Color\[LeftDoubleBracket]5\[RightDoubleBracket] = RGBColor[0.467, 0.064, 0.385]\), \(\(Color\[LeftDoubleBracket]6\[RightDoubleBracket] = RGBColor[0.510, 1.000, 1.000]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(background\)\" is \ similar to existing symbol \"\!\(Background\)\"."\)], "Message"], Cell[BoxData[ RowBox[{"pale", " ", "really", " ", "yellow", " ", RowBox[{"RGBColor", "[", RowBox[{ StyleBox["0.990000000000000035`", StyleBoxAutoDelete->True, PrintPrecision->3], ",", StyleBox["1.`", StyleBoxAutoDelete->True, PrintPrecision->4], ",", StyleBox["0.601999999999999957`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "]"}]}]], "Output"], Cell[BoxData[ RowBox[{"RGBColor", "[", RowBox[{ StyleBox["0.665000000000000035`", StyleBoxAutoDelete->True, PrintPrecision->3], ",", StyleBox["0.0810000000000000142`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["1.`", StyleBoxAutoDelete->True, PrintPrecision->4]}], "]"}]], "Output"], Cell[BoxData[ RowBox[{"RGBColor", "[", RowBox[{ StyleBox["1.`", StyleBoxAutoDelete->True, PrintPrecision->4], ",", StyleBox["0.04`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.391999999999999992`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "]"}]], "Output"], Cell[BoxData[ RowBox[{"RGBColor", "[", RowBox[{ StyleBox["0.0929999999999999893`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.302000000000000001`", StyleBoxAutoDelete->True, PrintPrecision->3], ",", StyleBox["0.7`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "]"}]], "Output"], Cell[BoxData[ RowBox[{"RGBColor", "[", RowBox[{ StyleBox["0.065`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.758000000000000007`", StyleBoxAutoDelete->True, PrintPrecision->3], ",", StyleBox["0.408999999999999985`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "]"}]], "Output"], Cell[BoxData[ RowBox[{"RGBColor", "[", RowBox[{ StyleBox["0.466999999999999992`", StyleBoxAutoDelete->True, PrintPrecision->3], ",", StyleBox["0.0640000000000000035`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.385`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "]"}]], "Output"] }, Open ]], Cell[BoxData[ \(\(RGBColor[0.800, 0.329, 0.406]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["old picks"], "SmallText", Evaluatable->False], Cell[BoxData[ \(\(color = Table[{}, {n, 0, 100}]; \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(con = .75; \)\)], "Input", InitializationCell->True], Cell[BoxData[{ \(\(color\[LeftDoubleBracket]0\[RightDoubleBracket] = RGBColor[0.510, 1.000, 1.000]; \)\), \(\(color\[LeftDoubleBracket]1\[RightDoubleBracket] = RGBColor[0.139, 0.114, 1.000]; \)\), \(\(color\[LeftDoubleBracket]2\[RightDoubleBracket] = RGBColor[0.103, 0.667, 1.000]; \)\), \(\(color\[LeftDoubleBracket]3\[RightDoubleBracket] = RGBColor[0.105, 1.000, 0.728]; \)\), \(\(color\[LeftDoubleBracket]4\[RightDoubleBracket] = RGBColor[0.098, 1.000, 0.119]; \)\), \(\(color\[LeftDoubleBracket]5\[RightDoubleBracket] = RGBColor[0.684, 1.000, 0.099]; \)\), \(\(color\[LeftDoubleBracket]6\[RightDoubleBracket] = RGBColor[1.000, 0.992, 0.094]; \)\), \(\(color\[LeftDoubleBracket]7\[RightDoubleBracket] = RGBColor[1.000, 0.695, 0.105]; \)\), \(\(color\[LeftDoubleBracket]8\[RightDoubleBracket] = RGBColor[1.000, 0.109, 0.090]; \)\), \(\(color\[LeftDoubleBracket]9\[RightDoubleBracket] = RGBColor[1.000, 0.090, 0.596]; \)\), \(\(color\[LeftDoubleBracket]10\[RightDoubleBracket] = RGBColor[0.814, 0.064, 1.000]; \)\), \(\(color\[LeftDoubleBracket]11\[RightDoubleBracket] = RGBColor[0.804, 0.590, 1.000]; \)\), \(\(color\[LeftDoubleBracket]12\[RightDoubleBracket] = RGBColor[0.510, 1.000, 1.000]; \)\), \(\(color\[LeftDoubleBracket]13\[RightDoubleBracket] = RGBColor[0.139, 0.114, 1.000]; \)\), \(\(color\[LeftDoubleBracket]14\[RightDoubleBracket] = RGBColor[0.103, 0.667, 1.000]; \)\), \(\(color\[LeftDoubleBracket]15\[RightDoubleBracket] = RGBColor[0.105, 1.000, 0.728]; \)\), \(\(color\[LeftDoubleBracket]16\[RightDoubleBracket] = RGBColor[0.098, 1.000, 0.119]; \)\), \(\(color\[LeftDoubleBracket]17\[RightDoubleBracket] = RGBColor[0.684, 1.000, 0.099]; \)\), \(\(color\[LeftDoubleBracket]18\[RightDoubleBracket] = RGBColor[1.000, 0.992, 0.094]; \)\), \(\(color\[LeftDoubleBracket]19\[RightDoubleBracket] = RGBColor[1.000, 0.695, 0.105]; \)\), \(\(color\[LeftDoubleBracket]20\[RightDoubleBracket] = RGBColor[1.000, 0.109, 0.090]; \)\), \(\(color\[LeftDoubleBracket]21\[RightDoubleBracket] = RGBColor[1.000, 0.090, 0.596]; \)\), \(\(color\[LeftDoubleBracket]22\[RightDoubleBracket] = RGBColor[0.814, 0.064, 1.000]; \)\), \(\(color\[LeftDoubleBracket]23\[RightDoubleBracket] = RGBColor[0.804, 0.590, 1.000]; \)\), \(\(color\[LeftDoubleBracket]50\[RightDoubleBracket] = RGBColor[con\ 0.510, con\ 1.000, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]51\[RightDoubleBracket] = RGBColor[con\ 0.139, con\ 0.114, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]52\[RightDoubleBracket] = RGBColor[con\ 0.103, con\ 0.667, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]53\[RightDoubleBracket] = RGBColor[con\ 0.105, con\ 1.000, con\ 0.728]; \)\), \(\(color\[LeftDoubleBracket]54\[RightDoubleBracket] = RGBColor[con\ 0.098, con\ 1.000, con\ 0.119]; \)\), \(\(color\[LeftDoubleBracket]55\[RightDoubleBracket] = RGBColor[con\ 0.684, con\ 1.000, con\ 0.099]; \)\), \(\(color\[LeftDoubleBracket]56\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.992, con\ 0.094]; \)\), \(\(color\[LeftDoubleBracket]57\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.695, con\ 0.105]; \)\), \(\(color\[LeftDoubleBracket]58\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.109, con\ 0.090]; \)\), \(\(color\[LeftDoubleBracket]59\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.090, con\ 0.596]; \)\), \(\(color\[LeftDoubleBracket]60\[RightDoubleBracket] = RGBColor[con\ 0.814, con\ 0.064, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]61\[RightDoubleBracket] = RGBColor[con\ 0.804, con\ 0.590, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]62\[RightDoubleBracket] = RGBColor[con\ 0.510, con\ 1.000, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]63\[RightDoubleBracket] = RGBColor[con\ 0.139, con\ 0.114, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]64\[RightDoubleBracket] = RGBColor[con\ 0.103, con\ 0.667, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]65\[RightDoubleBracket] = RGBColor[con\ 0.105, con\ 1.000, con\ 0.728]; \)\), \(\(color\[LeftDoubleBracket]66\[RightDoubleBracket] = RGBColor[con\ 0.098, con\ 1.000, con\ 0.119]; \)\), \(\(color\[LeftDoubleBracket]67\[RightDoubleBracket] = RGBColor[con\ 0.684, con\ 1.000, con\ 0.099]; \)\), \(\(color\[LeftDoubleBracket]68\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.992, con\ 0.094]; \)\), \(\(color\[LeftDoubleBracket]69\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.695, con\ 0.105]; \)\), \(\(color\[LeftDoubleBracket]60\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.109, con\ 0.090]; \)\), \(\(color\[LeftDoubleBracket]61\[RightDoubleBracket] = RGBColor[con\ 1.000, con\ 0.090, con\ 0.596]; \)\), \(\(color\[LeftDoubleBracket]62\[RightDoubleBracket] = RGBColor[con\ 0.814, con\ 0.064, con\ 1.000]; \)\), \(\(color\[LeftDoubleBracket]63\[RightDoubleBracket] = RGBColor[con\ 0.804, con\ 0.590, con\ 1.000]; \)\)}], "Input", CellOpen->False, InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Julie's Color pick"], "SmallText", Evaluatable->False], Cell[BoxData[ \(\(color\[LeftDoubleBracket]1\[RightDoubleBracket] = \(color\[LeftDoubleBracket]0\[RightDoubleBracket] = RGBColor[1.000, 0.348, 0.796]\); \)\)], "Input", PageWidth->PaperWidth, InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Talk revisions"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(color\[LeftDoubleBracket]11\[RightDoubleBracket] = RGBColor[1, 0, 0]; \)\), \(\(color\[LeftDoubleBracket]12\[RightDoubleBracket] = RGBColor[0, 1, 0]; \)\), \(\(color\[LeftDoubleBracket]13\[RightDoubleBracket] = RGBColor[0, 0, 1]; \)\), \(red = 11; green = 12; blue = 13; \), \(\(color\[LeftDoubleBracket]14\[RightDoubleBracket] = GrayLevel[ .2]; \)\), \(\(color\[LeftDoubleBracket]15\[RightDoubleBracket] = GrayLevel[ .4]; \)\), \(\(color\[LeftDoubleBracket]16\[RightDoubleBracket] = GrayLevel[ .6]; \)\), \(xDot = 14; yDot = 16; grDot = 15; \)}], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(xDot\)\" is similar to \ existing symbol \"\!\(Dot\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(yDot\)\" is similar to \ existing symbols \!\({Dot, xDot}\)."\)], "Message"] }, Open ]] }, Closed]], Cell[BoxData[ \(\(colorChart = { Graphics[PointSize[ .1], Table[{color\[LeftDoubleBracket]k\[RightDoubleBracket], Point[{k, 0}]}, {k, 0, 11}]]}; \)\)], "Input", InitializationCell->True] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Canonical and Bounded Fundamental Regions created"], "Text", Evaluatable->False, InitializationCell->True], Cell[BoxData[{ \(\(vertices = 10; \)\), \(\(theta1[n_] := ArcCos[ .5] + \(\((\[Pi]\/2 - ArcCos[ .5])\)\ n\)\/vertices; \)\), \(\(theta2[n_] := \[Pi] + \(\((ArcCos[\(- .5\)] - \[Pi])\)\ n\)\/\(2\ vertices\); \)\), \(\(boundedFundamentalRegionNumbers = Join[Table[N[Cos[theta1[n]] + I\ Sin[theta1[n]]], {n, 0, vertices}], Table[N[I\ \((1 - n\/\(2\ vertices\))\)], {n, 2\ vertices - 1}], { .001\ I}, Table[N[Cos[theta2[n]] + I\ Sin[theta2[n]] + 1], {n, 2\ vertices}]]; \)\), \(\(boundedFundamentalRegionPoints = Transpose[{Re[boundedFundamentalRegionNumbers], Im[boundedFundamentalRegionNumbers]}]; \)\), \(\(boundedFundamentalRegionPict = Graphics[{color\[LeftDoubleBracket]0\[RightDoubleBracket], Polygon[boundedFundamentalRegionPoints]}]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[{ \(\(fundamentalRegionNumbers = elliptic[boundedFundamentalRegionNumbers]; \)\), \(\(fundamentalRegionPoints = Transpose[{Re[fundamentalRegionNumbers], Im[fundamentalRegionNumbers]}]; \)\), \(\(fundamentalRegionPict = Graphics[{color\[LeftDoubleBracket]0\[RightDoubleBracket], Polygon[fundamentalRegionPoints]}]; \)\)}], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[BoxData[ \(\(RGBColor[0.149, 1.000, 0.581]; \)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Misc"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["Invariant Circle Data"], "Text", Evaluatable->False, InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(pole = \(center = \(-\(d\/c\)\)\); \)\), \(\(zero = \(-\(b\/a\)\); \)\), \(\(radius = Abs[1\/c]; \)\)}], "Input", CellHorizontalScrolling->False, InitializationCell->True], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(pole\)\" is similar to \ existing symbol \"\!\(pale\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(center\)\" is similar \ to existing symbol \"\!\(Center\)\"."\)], "Message"] }, Open ]] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Examples From the Modular Group"], "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["The Modular Group Tesselation (Abreviated)"], "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["Notes"], "SmallText", Evaluatable->False], Cell[TextData[ "Code to generate a more complete picture is in the \"Modular Group 5/21\" \ Notebook."], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[TextData["First Flower"], "SmallText", Evaluatable->False], Cell[BoxData[ \(\(bNum = 7; \)\)], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[BoxData[{ \(\(tempNumsList = Join[Reverse[NestList[parabolicRS, fundamentalRegionNumbers, bNum]], Drop[NestList[parabolicLS, fundamentalRegionNumbers, bNum], 1]]; \)\), \(\(flowerNums = elliptic[tempNumsList]; \)\), \(\(pointsList = Table[Transpose[{ Re[flowerNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[flowerNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], {n, Length[flowerNums]}]; \)\), \(\(firstFlowerPict = Table[Graphics[{color\[LeftDoubleBracket]0\[RightDoubleBracket], Polygon[pointsList\[LeftDoubleBracket]m\[RightDoubleBracket]]}], { m, Length[pointsList]}]; \)\)}], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Second Layer"], "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(mm = 1; \)\), \(\(While[mm < Length[flowerNums]\/4, newNumsList = Drop[Drop[flowerNums, 2\ mm], \(-\((2\ mm)\)\)]; tempNums = Join[ellParaL[parabolicLList[newNumsList, mm], 1], ellParaR[parabolicRList[newNumsList, mm], 1]]; flowerPoints = Table[Transpose[{ Re[tempNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[tempNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], {n, Length[tempNums]}]; flowerPict\[LeftDoubleBracket]mm\[RightDoubleBracket] = Table[Graphics[{color\[LeftDoubleBracket]1\[RightDoubleBracket], Polygon[flowerPoints\[LeftDoubleBracket]k \[RightDoubleBracket]]}], {k, Length[flowerPoints]}]; mm = mm + 1; ]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Third Layer"], "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(mm = 1; \)\), \(\(While[mm < Length[flowerNums]\/6, newNumsList = Drop[Drop[flowerNums, 3\ mm], \(-\((3\ mm)\)\)]; tempNums = Join[elliptic[ parabolicLList[ellParaL[parabolicLList[newNumsList, mm], 1], 3]], elliptic[ parabolicRList[ellParaR[parabolicRList[newNumsList, mm], 1], 3]]]; flowerPoints = Table[Transpose[{ Re[tempNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[tempNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], {n, Length[tempNums]}]; flowerPict2\[LeftDoubleBracket]mm\[RightDoubleBracket] = Table[Graphics[{color\[LeftDoubleBracket]1\[RightDoubleBracket], Polygon[flowerPoints\[LeftDoubleBracket]k \[RightDoubleBracket]]}], {k, Length[flowerPoints]}]; mm = mm + 1; ]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Fourth Layer (approx. 82 sec)"], "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(mm = 1; \)\), \(\(While[mm < Length[flowerNums]\/6, newNumsList = Drop[Drop[flowerNums, 3\ mm], \(-\((3\ mm)\)\)]; tempNums = Join[elliptic[ parabolicLList[ellParaL[parabolicLList[newNumsList, mm], 1], 4]], elliptic[ parabolicRList[ellParaR[parabolicRList[newNumsList, mm], 1], 4]]]; flowerPoints = Table[Transpose[{ Re[tempNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[tempNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], {n, Length[tempNums]}]; flowerPict2B\[LeftDoubleBracket]mm\[RightDoubleBracket] = Table[Graphics[{color\[LeftDoubleBracket]5\[RightDoubleBracket], Polygon[flowerPoints\[LeftDoubleBracket]k \[RightDoubleBracket]]}], {k, Length[flowerPoints]}]; mm = mm + 1; ]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Picture of All Layers"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[fundamentalRegionPict, flowerPict, flowerPict2, flowerPict2B, firstFlowerPict, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(- .6\), .6}, {\(- .01\), 1.5}}]; \)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{ \(Show::"gcomb"\), \( : \ \), "\<\"An error was encountered in combining the graphics objects in \ \\!\\(Show[\\(\\*TagBox[\\(\\\\[SkeletonIndicator] Graphics \ \\\\[SkeletonIndicator]\\), False, Rule[Editable, False]], \ \\(\\\\[LeftSkeleton] 5 \\\\[RightSkeleton]\\), \\(PlotRange \\\\[Rule] \ \\({\\({\\(-\\*StyleBox[\\\"0.6`\\\", Rule[PrintPrecision, 1], \ Rule[StyleBoxAutoDelete, True]]\\), \\*StyleBox[\\\"0.6`\\\", \ Rule[PrintPrecision, 1], Rule[StyleBoxAutoDelete, True]]}\\), \ \\({\\(-\\*StyleBox[\\\"0.01`\\\", Rule[PrintPrecision, 1], \ Rule[StyleBoxAutoDelete, True]]\\), \\*StyleBox[\\\"1.5`\\\", \ Rule[PrintPrecision, 2], Rule[StyleBoxAutoDelete, True]]}\\)}\\)\\)\\)]\\).\"\ \>"}]], "Message"] }, Closed]] }, Open ]] }, Closed]], Cell[BoxData[ \(\(factor = 4; \)\)], "Input", PageWidth->PaperWidth, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData[ "Preliminaries for the Modular Group(On the real axis)"], "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["Functions Defined"], "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(el1[x_] := 1\/x; \)\), \(\(el2[x_] := \(x - 1\)\/x; \)\), \(\(par1[x_] := x\/\(x + 1\); \)\), \(\(hyp1[x_] := \(2\ x + 3\)\/\(x + 2\); \)\)}], "Input", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Graphs"], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["el1[x] = 1/x"], "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(windowWidth = 4; \)\), \(\(el1Graph = Plot[el1[x], {x, \(-windowWidth\), windowWidth}, PlotRange \[Rule] {{\(-windowWidth\), windowWidth}, {\(-windowWidth\), windowWidth}}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] {color\[LeftDoubleBracket]xDot\[RightDoubleBracket]}]; \)\)}], "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(\(factor = 1\n\)\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[BoxData[{ \(beginningValue = \(-4.01\); \n\(delta = 4\/\(5\ factor\); \)\), \(\(movieFrames = factor\ 10; \)\)}], "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[TextData["\"Normal\" (With 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\)\), \(\(vertex2Pict = Graphics[{color\[LeftDoubleBracket]blue\[RightDoubleBracket], PointSize[ .02], Point[{0, 1}]}]; \)\), \(\(regionsPict = {fundamentalRegionPict}; \)\), \(\(vertices1Pict = {vertex1Pict}; \)\), \(\(vertices2Pict = {vertex2Pict}; \)\), \(\(vertex1Num = \(- .5\) + \(I\ \@3\)\/2; \)\), \(\(vertex2Num = I; \)\), \(\(newRegionNums = fundamentalRegionNumbers; \)\), \(\(nn = 1; \)\), \(\(While[nn < iterationNum, newRegionNums = hypF[newRegionNums]; newRegionPoints = Transpose[{Re[newRegionNums], Im[newRegionNums]}]; regionsPict = Join[regionsPict, { Graphics[{color\[LeftDoubleBracket]0\[RightDoubleBracket], Polygon[newRegionPoints]}]}]; vertex1Num = hypF[vertex1Num]; vertex1 = {Re[vertex1Num], Im[vertex1Num]}; vertices1Pict = Join[vertices1Pict, { Graphics[{color\[LeftDoubleBracket]red\[RightDoubleBracket], PointSize[ .02], Point[vertex1]}]}]; vertex2Num = hypF[vertex2Num]; vertex2 = {Re[vertex2Num], Im[vertex2Num]}; vertices2Pict = Join[vertices2Pict, { Graphics[{color\[LeftDoubleBracket]blue\[RightDoubleBracket], PointSize[ .02], Point[vertex2]}]}]; nn = nn + 1; ]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Pictures"], "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(center = \@3; \)\), \(\(width = center + 1; \)\), \(\(nn = 1; \)\), \(\(While[nn \[LessEqual] iterationNum, Show[regionsPict, vertices1Pict, vertices2Pict, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{center - width, center + width}, {0, 2\ width}}, Axes \[Rule] {center, 0}]; width = width\/10; nn = nn + 1; ]; \)\)}], "Input", PageWidth->PaperWidth] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Terms"], "Section", Evaluatable->False], Cell[TextData["Fixed points\n\nasymptotes"], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[TextData["More Q's"], "Section", Evaluatable->False], Cell[TextData[ "Sketch the graphs:\t\t\t\ty = -1/x ,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ y = (x-1)/x ,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ty = x/(x+1) ,\n\t\t\t\t\t\ \t\t\t\t\t\t\t\t\t\t\t\t\t\ty = (2x+3)/(x+2)."], "SmallText", Evaluatable->False], Cell[TextData[ "Construct a transformation, (ax + b)/(cx + d)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ \twith ad - bc = 1."], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["General Examples", "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["Hyperbolic transformations. ", "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell["Text:", "Text", Evaluatable->False], Cell["\<\ There are two distinct fixed points for hyperbolic transformations. \ The line through the fixed points is a member of the pencil of invariant \ circles. \ \>", "SmallText", Evaluatable->False], Cell[TextData[{ StyleBox[ "If a,b,c,d are all real, then the real axis is mapped onto itself and the \ fixed points will be on the real axis. \nWe will assume that the \ transformation is normalized, so that ad-bc=1. Thus, d=(1+bc)/a.\nNote: If \ the real axis is to be invariant, the number a must have absolute value \ greater than 1, for if a<1 and both fixed points are real, b & c will be \ non-real!!\nNote also that the larger a's value, the faster the iterated \ values of the transformation converge to the attractor fixed point.\nWe \ construct the Mobius transformation from the following data: 1) a real value \ greater than one for the coefficient ", Evaluatable->False], StyleBox["a", Evaluatable->False, FontWeight->"Bold"], StyleBox[ " and \n2) complex values for the two fixed points, purely real values if \ we want the real axis to be invariant.", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Code", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(Clear[a, b, c, d, fixedPoint1, fixedPoint2, bc, kay]; \)\), \(\(fixedPoint1 = .25; \)\), \(\(fixedPoint2 = \(- .25\); \)\), \(\(a = \(-1.007\); \)\), \(bc = Solve[{fixedPoint1 == \(a - \(1 + b\ c\)\/a + \@\(\((\(1 + b\ c\)\/a + a)\)\^2 - 4\)\)\/\(2\ c\), fixedPoint2 == \(a - \(1 + b\ c\)\/a - \@\(\((\(1 + b\ c\)\/a + a)\)\^2 - 4\)\)\/\(2\ c\)}, {b, c}] \), \(\(b = \(\(bc\[LeftDoubleBracket]1\[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]2\[RightDoubleBracket]; \)\), \(\(c = \(\(bc\[LeftDoubleBracket]1\[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]\)\[LeftDoubleBracket]2\[RightDoubleBracket]; \)\), \(\(d = \(1 + b\ c\)\/a; \)\), \(\(kay = \(a - c\ fixedPoint2\)\/\(a - c\ fixedPoint1\); \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes: ", "Text", Evaluatable->False], Cell["\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tExecute these cells", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ One characterization of a hyperbolic transformations is that its \ trace, i.e., a+d is real with absolute value greater than 2.\ \>", "SmallText",\ Evaluatable->False], Cell[BoxData[ \(a + d\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ If kay=K > 1, then fixedPoint2 will be the attractor for the \ iterated points. \ \>", "SmallText", Evaluatable->False], Cell[BoxData[ \(kay\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["We also note that any transformation that has all ", Evaluatable->False], StyleBox["real", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[ " coefficients when in its normal form, i.e., with \nad - bc = 1, leaves \ the real axis invariant. ", Evaluatable->False] }], "SmallText", Evaluatable->False], Cell[BoxData[ \({a, b, c, d}\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["The transformation operating on points and circles", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ Family of circles through the fixed points (hyperbolic pencil). \ \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Technical notes about the graphic illustration", "SmallText", Evaluatable->False], Cell["\<\ r10=red for the RGBColor[] of beginning circle of the hyperbolic \ family, r20=red for final circle, etc. \ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Hyperbolic pencil of circles generating code (approx. 15 sec)\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(numberOfHyperbolicCircles = 10\), \(r10 = 1; g10 = 0.090; b10 = .118; r20 = 1; g20 = 0.630; b20 = 0.090; \), \(\(centers = Table[\(-\((fixedPoint1 - fixedPoint2)\)\)\ I\ y\ .2, {y, \(-Floor[numberOfHyperbolicCircles\/2]\), Ceiling[numberOfHyperbolicCircles\/2]}]; \)\), \(\(radii = Table[Abs[ fixedPoint1 - centers\[LeftDoubleBracket]n\[RightDoubleBracket]], { n, numberOfHyperbolicCircles}]; \)\), \(\(hypCircleNums = Table[N[radii\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUDShort + centers\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, numberOfHyperbolicCircles}]; \)\), \(\(hypCirclePointsList = Table[Transpose[{ Re[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[hypCircleNums]}]; \)\), \(\(hyperbolicFamilyOfCirclesPict = Graphics[Table[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hypCirclePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[hypCirclePointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Choose an initial point and its iterates under the transformation \ will be displayed along with the hyperbolic pencil of circles.\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPoint = {0.320227, 0.10933}; \)\), \(\(numberOfIterations = 12; \)\), \(\(numbersList = NestList[mTrans, beginningPoint\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ beginningPoint\[LeftDoubleBracket]2\[RightDoubleBracket], numberOfIterations]; \)\), \(\(pointsList = Table[{Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[numbersList]}]; \)\), \(\(iteratedPointsPict = Table[Graphics[{ RGBColor[0, 0, \(Length[pointsList] - n\/2\)\/Length[pointsList]], PointSize[ \( .02\ \((Length[pointsList] - n\/2)\)\)\/Length[pointsList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[pointsList]}]; \)\)}], "Input"], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[hyperbolicFamilyOfCirclesPict, iteratedPointsPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, Ticks \[Rule] None]; \)\)], "Input"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Family of circles about the fixed points, the elliptic pencil.\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Notes:", "Text", Evaluatable->False], Cell[TextData[{ StyleBox[ "The code constructs a circle on which the transformation will operate. You \ may select a point on that circle. The default point will be the initial \ point from above, so that the iterated points will lie on the circles. The ", Evaluatable->False], StyleBox["elliptic pencil of circles", Evaluatable->False, FontWeight->"Bold"], StyleBox[ " will be generated and shown along with the hyperbolic pencil and images \ of the initial point.", Evaluatable->False] }], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Technical notes about the graphics illustration", "SmallText", Evaluatable->False], Cell[TextData[{ StyleBox[ "r1=red for the RGBColor[] of beginning circle of the elliptic family,\n\ r2=red for final circle, etc.\nThe pre-image of the unit circle with respect \ to the normalizing function ", Evaluatable->False], StyleBox["g(z)", Evaluatable->False, FontWeight->"Bold"], StyleBox[" is the imaginary axis.", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Code (approx. 15 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(r1 = 0.043; g1 = 0.113; b1 = 1; r2 = 0.489; g2 = 0.740; b2 = 1; \), \(\(pointOnBeginningCircle = pointsList\[LeftDoubleBracket]1\[RightDoubleBracket]; \)\), \(\(begPtNum = pointOnBeginningCircle\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ pointOnBeginningCircle\[LeftDoubleBracket]2\[RightDoubleBracket]; \)\), \(\(ellipticRadius = Abs[g[begPtNum]]; \)\), \(\(beginningCircleNums = gInv[ellipticRadius\ numbersUDShort]; \)\), \(\(numberOfEllipticCircles = Length[pointsList]; \)\), \(\(ellipticCircleFamilyNums = NestList[mTrans, beginningCircleNums, numberOfEllipticCircles - 1]; \)\), \(ellipticCircleFamilyPoints = Table[Transpose[{ Re[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCircleFamilyNums]}]; Null; \), \(\(ellipticFamilyOfCirclesPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCircleFamilyPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCircleFamilyPoints]}]; \)\), \(\(iteratedPointsPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((Length[pointsList] - n)\)\ r1\)\/Length[ pointsList], \(n\ g2 + \((Length[pointsList] - n)\)\ g1\)\/Length[ pointsList], \(n\ b2 + \((Length[pointsList] - n)\)\ b1\)\/Length[ pointsList]], PointSize[ \( .02\ \((Length[pointsList] - n\/2)\)\)\/Length[pointsList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[pointsList]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[ellipticFamilyOfCirclesPict, hyperbolicFamilyOfCirclesPict, iteratedPointsPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(- .5\), 1.5}}]; \)\)], "Input"] }, Closed]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Changing the setting (Integral Forms & Stereographic Projections)\ \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Picture of the ", Evaluatable->False], StyleBox["integral", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" transformation which is similar to this one. ", Evaluatable->False] }], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsNormalized = Transpose[{Re[g[numbersList]], Im[g[numbersList]]}]; \)\), \(\(iteratedPointsNormalizedPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfIterations - n)\)\ r1\)\/numberOfIterations, \(n\ g2 + \((numberOfIterations - n)\)\ g1\)\/numberOfIterations, \(n\ b2 + \((numberOfIterations - n)\)\ b1\)\/numberOfIterations], PointSize[ .03], Point[iteratedPointsNormalized\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsNormalized]}]; \)\), \(\(ellipticCirclesNormalizedPoints = Table[Transpose[{ Re[g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], Im[g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]]}], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(ellipticCirclesNormalizedPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesNormalizedPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesNormalizedPoints]}]; \)\), \(\(hypCirclesNormalizedNumsList = Table[g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[hypCircleNums]}]; \)\), \(\(hypCirclesNormalizedPointsList = Table[Transpose[{ Re[hypCirclesNormalizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[hypCirclesNormalizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hypCirclesNormalizedNumsList]}]; \)\), \(\(hyperbolicCirclesNormalizedPict = Graphics[Table[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hypCirclesNormalizedPointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[hypCirclesNormalizedPointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[hyperbolicCirclesNormalizedPict, iteratedPointsNormalizedPict, ellipticCirclesNormalizedPict, AspectRatio \[Rule] Automatic]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes on the itegral form.", "Text", Evaluatable->False], Cell["\<\ In the transfomation's normalized form, the hyperbolic pencil of \ circles appear as straight lines through the origin and the elliptic pencil \ of circles as concentric circles about the origin.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Clearing commands", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsNormalized = {}; \)\), \(\(iteratedPointsNormalizedPict = {}; \)\), \(\(ellipticCirclesNormalizedPoints = {}; \)\), \(\(ellipticCirclesNormalizedPict = {}; \)\), \(\(hypCirclesNormalizedNumsList = {}; \)\), \(\(hyperbolicCirclesNormalizedPict = {}; \)\)}], "Input"], Cell[BoxData[ \(\(Clear[iteratedPointsNormalized, iteratedPointsNormalizedPict, ellipticCirclesNormalizedPoints, ellipticCirclesNormalizedPict, hypCirclesNormalizedNumsList, hyperbolicCirclesNormalizedPict]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (original \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[numbersList]; \)\), \(\(iteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(ellipticCirclesOnSpherePoints = Table[stereoProjToSphere[ ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(ellipticCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesOnSpherePoints]}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[hypCircleNums]}]; \)\), \(\(hyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[iteratedPointsOnSpherePict, ellipticCirclesOnSpherePict, hyperbolicCirclesOnSpherePict, Boxed \[Rule] False, ViewPoint \[Rule] {1.079, 3.006, \(-8.414\)}]; \)\)], "Input"], Cell[BoxData[ \(Show::"gcomb" \( : \ \) "An error was encountered in combining the graphics objects in \ \!\(\\[LeftSkeleton] 1 \\[RightSkeleton]\)."\)], "Message"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Clear Command", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Clear[iteratedPointsOnSphere, iteratedPointsOnSpherePict, ellipticCirclesOnSpherePoints, ellipticCirclesOnSpherePict, hyperbolicCirclesOnSpherePoints, hyperbolicCirclesOnSpherePict]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (integral \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 100 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[g[numbersList]]; \)\), \(\(integralIteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(ellipticCirclesOnSpherePoints = Table[stereoProjToSphere[ g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(integralEllipticCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesOnSpherePoints]}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]], {n, Length[hypCircleNums]}]; \)\), \(\(integralHyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 40 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[integralIteratedPointsOnSpherePict, integralEllipticCirclesOnSpherePict, integralHyperbolicCirclesOnSpherePict, Boxed \[Rule] False]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes on the integral transformation.", "Text", Evaluatable->False], Cell["\<\ Since the two fixed points in the integral form are antipodal on \ the Riemann Sphere, the hyperbolic pencil of circles are transformed into \ great circles through the fixed points while the elliptic pencil of circles \ are transformed into circles with their centers on the axis which passes \ through the two fixed points.\ \>", "SmallText", Evaluatable->False] }, Closed]] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Elliptic transformations. ", "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell["Text", "SmallText", Evaluatable->False], Cell[TextData[ "Elliptic Transformations have two fixed points. If the real axis is \ invariant, the two fixed points are conjugates of each other.\nAs always we \ assume the transformation is normalized so that ad-bc=1. Further, the trace \ of an elliptic transformation has the following restrictions: a + d = +/- 2 \ Cos(theta/2)\nWe will construct our transformation to have a finite cycle, \ thus ` must be a rational multiple of theta.\nFor the code below, the fixed \ point must be chosen to be pure imaginary.\nSince the sum of the fixed points \ = (a-d)/c, and since we are setting up the situation where the sum of the \ fixed points = 0, we must have that a = d. \nFurther we have that \n\t\t\t\t\ E\[CapitalEGrave]\[NonBreakingSpace]\[CapitalEHat]\[CapitalOAcute]\:2030\ \[CapitalEHat]\[CapitalARing] =(a - c fixedPoint1)/(a - c fixedPoint2).\n\ Thus, the fixed point and theta alone determine this elliptic transformation. \ "], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Code", "Text", Evaluatable->False], Cell[BoxData[{ \(\(Clear[a, b, c, d, theta, fixedPoint1, fixedPoint2, p]; \)\), \(\(p = 7; \)\), \(\(theta = \(2\ \[Pi]\)\/p; \)\), \(\(fixedPoint2 = \(-\(1\/4\)\); \)\), \(\(fixedPoint1 = 1\/4; \)\), \(\(a = \(d = Cos[theta\/2]\); \)\), \(\(c = \(a\ \((1 - E\^\(I\ theta\))\)\)\/\(fixedPoint2 - E\^\(I\ theta\)\ fixedPoint1\); \)\), \(\(b = \(a\ d - 1\)\/c; \)\), \(N[{a, b, c, d}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["The transformation operating on points and circles", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ Choose an initial point and its iterated images under the elliptic \ transformation will be displayed.\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPoint = {0.914696, 0.186053}; \)\), \(\(numbersList = NestList[mTrans, beginningPoint\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ beginningPoint\[LeftDoubleBracket]2\[RightDoubleBracket], p - 1]; \)\), \(\(pointsList = Table[{Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[numbersList]}]; \)\), \(\(pointsPict = Graphics[Table[{ RGBColor[\(Length[pointsList] - n\/2\)\/Length[pointsList], 0, 0], PointSize[ \( .02\ \((Length[pointsList] - n\/2)\)\)\/Length[pointsList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[pointsList]}]]; \)\), \(\(Show[pointsPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; \)\)}], "Input"], Cell["Not much to see without additional data.", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Hyperbolic Pencil of circles through the fixed points", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["General (approx. 10 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(numberOfHyperbolicCircles = 10\), \(r10 = 1; g10 = 0.090; b10 = .118; r20 = 1; g20 = 0.630; b20 = 0.090; \), \(\(centers = Table[N[1\/2\ I\ \((fixedPoint2 - fixedPoint1)\)\ n], {n, \(-Floor[numberOfHyperbolicCircles\/2]\), Ceiling[numberOfHyperbolicCircles\/2]}]; \)\), \(\(radii = Table[N[Abs[ fixedPoint1 - centers\[LeftDoubleBracket]n\[RightDoubleBracket]]], {n, numberOfHyperbolicCircles}]; \)\), \(\(hypCircleNums = Table[N[radii\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUD + centers\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, numberOfHyperbolicCircles}]; \)\), \(\(hypCirclePointsList = Table[Transpose[{ Re[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[hypCircleNums]}]; \)\), \(\(hyperbolicFamilyOfCirclesPict = Graphics[Table[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hypCirclePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[hypCirclePointsList]}]]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[pointsPict, hyperbolicFamilyOfCirclesPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, Ticks \[Rule] None]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Family of Geodesics through the Fixed Point and the iterates from \ above\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(centersList = Table[{\(\(\(Solve[ \((Re[fixedPoint2] - x)\)\^2 + Im[fixedPoint2]\^2 == \((x - Re[numbersList\[LeftDoubleBracket]n \[RightDoubleBracket]])\)\^2 + Im[numbersList\[LeftDoubleBracket]n \[RightDoubleBracket]]\^2, {x}] \)\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], 0}, {n, Length[numbersList]}]; \)\), \(\(radiiList = Table[\[Sqrt]\(( \((Re[fixedPoint2] - \(centersList\[LeftDoubleBracket]n \[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket])\)\^2 + Im[fixedPoint2]\^2)\), { n, Length[numbersList]}]; \)\), \(\(circleNumsList = Table[radiiList\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUDShort, {n, Length[numbersList]}]; \)\), \(\(circlePointsListTemp = Table[Transpose[{ Re[circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[numbersList]}]; \)\), \(\(circlePointsList = Table[Table[ \(circlePointsListTemp\[LeftDoubleBracket]n \[RightDoubleBracket]\)\[LeftDoubleBracket]k \[RightDoubleBracket] + centersList\[LeftDoubleBracket]n\[RightDoubleBracket], {k, Length[circlePointsListTemp\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[circlePointsListTemp]}]; \)\), \(\(circlesPict = Table[Graphics[{RGBColor[0, 1, 0], Line[circlePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[circlePointsList]}]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[hyperbolicFamilyOfCirclesPict, circlesPict, pointsPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic]; \)\)], "Input"], Cell[CellGroupData[{ Cell["\<\ Arcs of the geodesics corresponding to rays from the fixed point \ through the iterates\ \>", "Text", Evaluatable->False], Cell[BoxData[{ \(\(centerNums = Table[\(centersList\[LeftDoubleBracket]n \[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket] + I\ \(centersList\[LeftDoubleBracket]n \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], {n, Length[centersList]}]; \)\), \(\(secondRay = Table[If[Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]] < Re[fixedPoint2], {\(-1\), 0}, {1, 0}, help], {n, Length[centersList]}]; \)\), \(\(angles = Table[ArcCos[ \(({Re[fixedPoint2], Im[fixedPoint2]} - centersList\[LeftDoubleBracket]n\[RightDoubleBracket])\) . secondRay\[LeftDoubleBracket]n\[RightDoubleBracket]/ \((\[Sqrt]\(( \(({Re[fixedPoint2], Im[fixedPoint2]} - centersList\[LeftDoubleBracket]n\[RightDoubleBracket]) \) . \(({Re[fixedPoint2], Im[fixedPoint2]} - centersList\[LeftDoubleBracket]n\[RightDoubleBracket]) \))\))\)], {n, Length[centersList]}]; \)\), \(\(beginningAnglesPre = Table[ArcCos[ \(({Re[fixedPoint2], Im[fixedPoint2]} - centersList\[LeftDoubleBracket]n\[RightDoubleBracket])\) . { 1, 0}/\(( \[Sqrt]\(( \(({Re[fixedPoint2], Im[fixedPoint2]} - centersList\[LeftDoubleBracket]n\[RightDoubleBracket]) \) . \(({Re[fixedPoint2], Im[fixedPoint2]} - centersList\[LeftDoubleBracket]n\[RightDoubleBracket]) \))\))\)], {n, Length[centersList]}]; \)\), \(\(beginningAngles = Table[If[Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]] < 0, beginningAnglesPre\[LeftDoubleBracket]n\[RightDoubleBracket], 0], { n, Length[pointsList]}]; \)\), \(\(jump = 10; \)\), \(\(arcsNums = Table[Table[ N[radiiList\[LeftDoubleBracket]n\[RightDoubleBracket]\ E\^\(I\ \(( beginningAngles\[LeftDoubleBracket]n \[RightDoubleBracket] + \(m\ angles\[LeftDoubleBracket]n \[RightDoubleBracket]\)\/jump)\)\) + centerNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {m, 0, jump}], {n, Length[pointsList]}]; \)\), \(\(arcsPointsList = Table[ Transpose[{Re[arcsNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[arcsNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], {n, Length[arcsNums]}]; \)\), \(\(arcsPict = Table[Graphics[{RGBColor[N[1\/\( .5\ n + .5\)], 0, 0], Line[arcsPointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[arcsPointsList]}]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[pointsPict, arcsPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic]; \)\)], "Input"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Elliptic pencil of circles about the fixed points", "SmallText", Evaluatable->False], Cell[BoxData[{ \(r1 = 0.043; g1 = 0.113; b1 = 1; r2 = 0.489; g2 = 0.740; b2 = 1; \), \(\(numberOfEllipticCircles = 10; \)\), \(\(radiiList = Table[n\ .3, {n, numberOfEllipticCircles}]; \)\), \(\(ellipticCircleFamilyNums = Table[gInv[ radiiList\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUDShort], {n, numberOfEllipticCircles}]; \)\), \(ellipticCircleFamilyPoints = Table[Transpose[{ Re[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, numberOfEllipticCircles}]; Null; \), \(\(ellipticFamilyOfCirclesPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCircleFamilyPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCircleFamilyPoints]}]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[pointsPict, hyperbolicFamilyOfCirclesPict, ellipticFamilyOfCirclesPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-1\), 1}, {\(-1. \), 1. }}, Ticks \[Rule] None]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Changing the setting (Normalizing & Stereographic Projecting)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Picture of the normalized transformation", "SmallText", Evaluatable->False], Cell["\<\ The fixed point1 is moved to \\ and the fixed point2 is moved to \ the origin.\ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(numberOfIterations = Length[numbersList]; \)\), \(\(iteratedPointsNormalized = Transpose[{Re[g[numbersList]], Im[g[numbersList]]}]; \)\), \(\(iteratedPointsNormalizedPict = Table[Graphics[{ RGBColor[ \(n\ r20 + \((numberOfIterations - n)\)\ r10\)\/numberOfIterations, \(n\ g20 + \((numberOfIterations - n)\)\ g10\)\/numberOfIterations, \(n\ b20 + \((numberOfIterations - n)\)\ b10\)\/numberOfIterations], PointSize[ .03], Point[iteratedPointsNormalized\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsNormalized]}]; \)\), \(\(ellipticCirclesNormalizedPoints = Table[Transpose[{ Re[g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], Im[g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]]}], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(ellipticCirclesNormalizedPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesNormalizedPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesNormalizedPoints]}]; \)\), \(\(hypCirclesNormalizedNumsList = Table[g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[hypCircleNums]}]; \)\), \(\(hypCirclesNormalizedPointsList = Table[Transpose[{ Re[hypCirclesNormalizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[hypCirclesNormalizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hypCirclesNormalizedNumsList]}]; \)\), \(\(hyperbolicCirclesNormalizedPict = Table[Graphics[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hypCirclesNormalizedPointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hypCirclesNormalizedPointsList]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[hyperbolicCirclesNormalizedPict, ellipticCirclesNormalizedPict, iteratedPointsNormalizedPict, AspectRatio \[Rule] Automatic]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (original \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[numbersList]; \)\), \(\(iteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((Length[numbersList] - n)\)\ r10\)\/Length[ numbersList], \(n\ g20 + \((Length[numbersList] - n)\)\ g10\)\/Length[ numbersList], \(n\ b20 + \((Length[numbersList] - n)\)\ b10\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(ellipticCirclesOnSpherePoints = Table[stereoProjToSphere[ ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(ellipticCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesOnSpherePoints]}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[hypCircleNums]}]; \)\), \(\(hyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[iteratedPointsOnSpherePict, ellipticCirclesOnSpherePict, hyperbolicCirclesOnSpherePict, Boxed \[Rule] False, ViewPoint \[Rule] {1.660, \(-2.267\), \(-4.518\)}]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (normalized \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 100 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[g[numbersList]]; \)\), \(\(normalizedIteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((Length[numbersList] - n)\)\ r10\)\/Length[ numbersList], \(n\ g20 + \((Length[numbersList] - n)\)\ g10\)\/Length[ numbersList], \(n\ b20 + \((Length[numbersList] - n)\)\ b10\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(ellipticCirclesOnSpherePoints = Table[stereoProjToSphere[ g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(normalizedEllipticCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesOnSpherePoints]}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]], {n, Length[hypCircleNums]}]; \)\), \(\(normalizedHyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Code to add more points to hyperbolic circles on sphere (not very \ effective)\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(vertices = 100; \)\), \(\(zee = Cos[\(2\ \[Pi]\)\/vertices] + I\ Sin[\(2\ \[Pi]\)\/vertices]; \)\), \(\(numbersUDHyp = Table[N[rotation\ zee\^k], {k, 0, vertices}]; \)\), \(numberOfHyperbolicCircles = 10\), \(r10 = 1; g10 = 0.090; b10 = .118; r20 = 1; g20 = 0.630; b20 = 0.090; \), \(\(centers = Table[x, {x, \(-Floor[numberOfHyperbolicCircles\/2]\), Ceiling[numberOfHyperbolicCircles\/2]}]; \)\), \(\(radii = Table[Abs[ fixedPoint1 - centers\[LeftDoubleBracket]n\[RightDoubleBracket]], { n, numberOfHyperbolicCircles}]; \)\), \(\(hypCircleNums = Table[N[radii\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUDHyp + centers\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, numberOfHyperbolicCircles}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]], {n, Length[hypCircleNums]}]; \)\), \(\(normalizedHyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 40 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[normalizedIteratedPointsOnSpherePict, normalizedEllipticCirclesOnSpherePict, normalizedHyperbolicCirclesOnSpherePict, Boxed \[Rule] False]; \)\)], "Input"] }, Closed]] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Parabolic transformations. ", "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell["Notes:", "Text", Evaluatable->False], Cell["\<\ Parabolic transformations have a single fixed point. This fixed point will be chosen to be on the real axis and the coefficients \ will be constructed so that the real axis is invariant.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Code", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(Clear[fixedPoint, a, b, c, d]; \)\), \(\(fixedPoint = 1.0; \)\), \(\(a = .0005; \)\), \(\(c = \(a - 1\)\/fixedPoint; \)\), \(\(d = 2 - a; \)\), \(\(b = \(a\ d - 1\)\/c; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes:", "Text", Evaluatable->False], Cell["\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tExecute these cells.", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ The trace of a normalized parabolic transformation is equal 2 or \ -2.\ \>", "SmallText", Evaluatable->False], Cell[BoxData[ \(a + d\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ If the coefficients are all real, the real axis will be invariant\ \ \>", "SmallText", Evaluatable->False], Cell[BoxData[ \({a, b, c, d}\)], "Input"] }, Open ]], Cell["\<\ In the code below you may select a beginning point. The iterates \ of that point under the transformation will be displayed.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Code", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(Clear[beginningPoint, numbersList, pointsList]; \)\), \(\(beginningPoint = {1.18176, 0.0760107}; \)\), \(\(numberOfIterations = 10; \)\), \(\(numbersList = NestList[mTrans, beginningPoint\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ beginningPoint\[LeftDoubleBracket]2\[RightDoubleBracket], numberOfIterations]; \)\), \(\(pointsList = Table[{Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[numbersList]}]; \)\), \(\(iteratedPointsPict = Table[Graphics[{ RGBColor[0, 0, \(Length[pointsList] - n\/2\)\/Length[pointsList]], PointSize[ \( .02\ \((Length[pointsList] - n\/2)\)\)\/Length[pointsList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[pointsList]}]; \)\), \(\(Show[iteratedPointsPict, Axes \[Rule] {1, 0}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Invariant pencil & Interchanged pencil of circles generated & shown \ with iterated points \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Family of invariant circles", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(numberOfInvariantCircles = 11; \)\), \(r10 = 1; g10 = 0.090; b10 = .118; r20 = 1; g20 = 0.630; b20 = 0.090; \), \(\(centers = Table[fixedPoint - .075\ n\ I, {n, \(-Floor[numberOfInvariantCircles\/2]\), Ceiling[numberOfInvariantCircles\/2]}]; \)\), \(\(radii = Table[Abs[ fixedPoint - centers\[LeftDoubleBracket]n\[RightDoubleBracket]], { n, numberOfInvariantCircles}]; \)\), \(\(invariantCirclesNums = Table[N[radii\[LeftDoubleBracket]n\[RightDoubleBracket]\ E\^\(-I\)\ numbersUDShort + centers\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, numberOfInvariantCircles}]; \)\), \(\(invariantCirclesPoints = Table[Transpose[{ Re[invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[invariantCirclesNums]}]; \)\), \(\(invariantCirclesPict = Graphics[Table[{ RGBColor[ \(n\ r20 + \((numberOfInvariantCircles - n)\)\ r10 \)\/numberOfInvariantCircles, \(n\ g20 + \((numberOfInvariantCircles - n)\)\ g10 \)\/numberOfInvariantCircles, \(n\ b20 + \((numberOfInvariantCircles - n)\)\ b10 \)\/numberOfInvariantCircles], Line[invariantCirclesPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[invariantCirclesPoints]}]]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[iteratedPointsPict, invariantCirclesPict, Axes \[Rule] {1, 0}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{Re[fixedPoint] - .4, Re[fixedPoint] + .4}, { Im[fixedPoint] - .3, Im[fixedPoint] + .8}}]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Family of \"pushed\" or interchanged circles", "SmallText", Evaluatable->False], Cell[BoxData[{ \(r1 = 0.043; g1 = 0.113; b1 = 1; r2 = 0.489; g2 = 0.740; b2 = 1; \), \(\(centersList = Table[\(\(\(Solve[ \((fixedPoint - x)\)\^2 == \((x - Re[numbersList\[LeftDoubleBracket]n \[RightDoubleBracket]])\)\^2 + Im[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]] \^2, {x}]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], {n, Length[numbersList]}]; \)\), \(\(radiiList = Table[Abs[ fixedPoint - centersList\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[numbersList]}]; \)\), \(\(circleNumsList = Table[radiiList\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUDShort + centersList\[LeftDoubleBracket]n\[RightDoubleBracket], {n, Length[numbersList]}]; \)\), \(\(circlePointsList = Table[Transpose[{ Re[circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[numbersList]}]; \)\), \(\(interchangedCirclesPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], Line[circlePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[circlePointsList]}]; \)\), \(\(iteratedPointsPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ \( .02\ \((Length[numbersList] - n\/2)\)\)\/Length[ numbersList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[numbersList]}]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[iteratedPointsPict, invariantCirclesPict, interchangedCirclesPict, Axes \[Rule] {1, 0}, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, PlotRange \[Rule] {{Re[fixedPoint] - .4, Re[fixedPoint] + .4}, { Im[fixedPoint] - .3, Im[fixedPoint] + .8}}]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Changing the setting (Integral Forms & Stereographic Projections)\ \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Picture of the ", Evaluatable->False], StyleBox["integral", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" transformation which is similar to this one. ", Evaluatable->False] }], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 30 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsNormalized = Transpose[{Re[parabolicG[numbersList]], Im[parabolicG[numbersList]]}]; \)\), \(\(iteratedPointsIntegralizedPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfIterations - n)\)\ r1\)\/numberOfIterations, \(n\ g2 + \((numberOfIterations - n)\)\ g1\)\/numberOfIterations, \(n\ b2 + \((numberOfIterations - n)\)\ b1\)\/numberOfIterations], PointSize[ .03], Point[iteratedPointsNormalized\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsNormalized]}]; \)\), \(\(invariantCirclesIntegralizedPoints = Table[If[\(invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket] \[NotEqual] fixedPoint, Transpose[{ Re[parabolicG[ invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], Im[parabolicG[ invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]]]}], Table[{0, 0}, {m, Length[invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]]}], help], {n, Length[invariantCirclesNums]}]; \)\), \(\(num = Length[invariantCirclesIntegralizedPoints]; \)\), \(\(invariantCirclesIntegralizedPict = Table[Graphics[{ RGBColor[\(n\ r20 + \((num - n)\)\ r10\)\/num, \(n\ g20 + \((num - n)\)\ g10\)\/num, \(n\ b20 + \((num - n)\)\ b10\)\/num], Line[invariantCirclesIntegralizedPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, num}]; \)\), \(\(interchangedCirclesIntegralizedNumsList = Table[parabolicG[ circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[circleNumsList]}]; \)\), \(\(interchangedCirclesIntegralizedPointsList = Table[Transpose[{ Re[interchangedCirclesIntegralizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[interchangedCirclesIntegralizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[interchangedCirclesIntegralizedNumsList]}]; \)\), \(\(num = Length[interchangedCirclesIntegralizedPointsList]; \)\), \(\(interchangedCirclesIntegralizedPict = Graphics[Table[{ RGBColor[\(n\ r2 + \((num - n)\)\ r1\)\/num, \(n\ g2 + \((num - n)\)\ g1\)\/num, \(n\ b2 + \((num - n)\)\ b1\)\/num], Line[interchangedCirclesIntegralizedPointsList \[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[interchangedCirclesIntegralizedPointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[interchangedCirclesIntegralizedPict, invariantCirclesIntegralizedPict, iteratedPointsIntegralizedPict, Axes \[Rule] {0, 0}, AspectRatio \[Rule] Automatic]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes on the integral form.", "Text", Evaluatable->False], Cell["\<\ The fixed point was sent to \\ by G(z), so G will transform all \ circles that went through the fixed point into straight lines.\ \>", "SmallText", Evaluatable->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (original \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[numbersList]; \)\), \(\(iteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(interchangedCirclesOnSpherePoints = Table[stereoProjToSphere[ circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[circleNumsList]}]; \)\), \(\(num = Length[interchangedCirclesOnSpherePoints]; \)\), \(\(interchangedCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[\(n\ r2 + \((num - n)\)\ r1\)\/num, \(n\ g2 + \((num - n)\)\ g1\)\/num, \(n\ b2 + \((num - n)\)\ b1\)\/num], Line[interchangedCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[interchangedCirclesOnSpherePoints]}]; \)\), \(\(invariantCirclesOnSpherePoints = Table[stereoProjToSphere[ invariantCirclesNums\[LeftDoubleBracket]n\[RightDoubleBracket]], { n, Length[invariantCirclesNums]}]; \)\), \(\(num = Length[invariantCirclesOnSpherePoints]; \)\), \(\(invariantCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[\(n\ r20 + \((num - n)\)\ r10\)\/num, \(n\ g20 + \((num - n)\)\ g10\)\/num, \(n\ b20 + \((num - n)\)\ b10\)\/num], Line[invariantCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[invariantCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(spherePict\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Show[iteratedPointsOnSpherePict, invariantCirclesOnSpherePict, interchangedCirclesOnSpherePict, Boxed \[Rule] False, ViewPoint \[Rule] {4.043, \(-1.611\), 0.600}]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (integral \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 100 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[parabolicG[numbersList]]; \)\), \(\(integralIteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(interchangedCirclesOnSpherePoints = Table[stereoProjToSphere[ parabolicG[ circleNumsList\[LeftDoubleBracket]n\[RightDoubleBracket]]], {n, Length[circleNumsList]}]; \)\), \(\(num = Length[interchangedCirclesOnSpherePoints]; \)\), \(\(interchangedCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[\(n\ r2 + \((num - n)\)\ r1\)\/num, \(n\ g2 + \((num - n)\)\ g1\)\/num, \(n\ b2 + \((num - n)\)\ b1\)\/num], Line[interchangedCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[interchangedCirclesOnSpherePoints]}]; \)\), \(\(invariantCirclesOnSpherePoints = Table[stereoProjToSphere[ parabolicG[ invariantCirclesNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], {n, Length[invariantCirclesNums]}]; \)\), \(\(num = Length[invariantCirclesOnSpherePoints]; \)\), \(\(invariantCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[\(n\ r20 + \((num - n)\)\ r10\)\/num, \(n\ g20 + \((num - n)\)\ g10\)\/num, \(n\ b20 + \((num - n)\)\ b10\)\/num], Line[invariantCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[invariantCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 40 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[integralIteratedPointsOnSpherePict, interchangedCirclesOnSpherePict, invariantCirclesOnSpherePict, Boxed \[Rule] False]; \)\)], "Input"] }, Closed]], Cell["Notes on the integral transformation.", "Text", Evaluatable->False] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Loxodromic transformations. ", "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell["Text", "SmallText", Evaluatable->False], Cell["\<\ Loxodromic transormations cannot fix the upper half-plane. We \ include an example of these transformations here for completeness and because \ they are a graphic curiosity. The invariant curves are the charming \ Bernoulli curves.\ \>", "Text", Evaluatable->False], Cell[TextData[ "There are two distinct fixed points for loxodromic transformations.\nAs \ always we assume the transformation to be normalized so that ad-bc=1. Thus, \ d=(1+bc)/a.\nFor a loxodromic transformation we have that \n\t\t\tA*E\ \[CapitalEGrave]\[NonBreakingSpace]\[CapitalEHat]\[CapitalOAcute]\:2030\ \[CapitalEHat]\[CapitalARing] =(a - c fixedPoint1)/(a - c fixedPoint2).\nHere \ A is a real number whose absolute value is not equal 1."], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Code", "Text", Evaluatable->False], Cell[BoxData[{ \(\(Clear[A, a, b, c, d, ad, theta, fixedPoint1, fixedPoint2, b1]; \)\), \(\(A = 1.2; \)\), \(\(theta = \[Pi]\/3; \)\), \(\(fixedPoint1 = .25; \)\), \(\(fixedPoint2 = \(- .25\); \)\), \(b1 = N[\@\(2 + \((A + 1\/A)\)\ Cos[theta] + I\ \((A - 1\/A)\)\ Sin[theta]\)]; b2 = \(1 - A\ E\^\(I\ theta\)\)\/\(fixedPoint1 - fixedPoint2\ A\ E\^\(I\ theta\)\); \), \(\(ad = Solve[{aa + dd == b1, \(aa - dd\)\/cc == fixedPoint1 + fixedPoint2, cc == aa\ b2}, {aa, cc, dd}]; \)\), \(\(a = N[\(\(ad\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]]; \)\), \(\(c = N[\(\(ad\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]]; \)\), \(\(d = N[\(\(ad\[LeftDoubleBracket]1 \[RightDoubleBracket]\)\[LeftDoubleBracket]3 \[RightDoubleBracket]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]]; \)\), \(\(b = \(a\ d - 1\)\/c; \)\), \(\(kay = \(a - c\ fixedPoint2\)\/\(a - c\ fixedPoint1\); \)\), \({a, b, c, d}\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes ", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ One characteristic of a loxodromic transformations is that its \ trace, i.e., a+d is never real.\ \>", "Input"], Cell[BoxData[ \(a + d\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K is also not real, causing a spiraling of points iterated by the \ transformation\ \>", "SmallText", Evaluatable->False], Cell[BoxData[ \(kay\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["The transformation operating on points and circles", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ Choose an initial point and its iterates under the loxodromic \ transformation will be displayed\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPoint = {1.04375\/4, 0.00757682\/4}; \)\), \(\(numbersList = NestList[mTrans, beginningPoint\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ beginningPoint\[LeftDoubleBracket]2\[RightDoubleBracket], 40]; \)\), \(\(LoxPointsList = Table[{Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[numbersList]}]; \)\), \(\(LoxPointsPict = Graphics[Table[{ RGBColor[0, 0, \(Length[LoxPointsList] - n\/2\)\/Length[LoxPointsList]], PointSize[ \( .013\ \((Length[LoxPointsList] - n\/2)\)\)\/Length[ LoxPointsList]], Point[LoxPointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, { n, Length[LoxPointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ The Bernoulli curve connecting the iterated points is \ generated\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPointIndex = 6; \)\), \(\(num1 = Length[numbersList] - 2\ beginningPointIndex; \)\), \(\(steps = 5; \)\), \(\(preArcNums = Table[N[E\^\(\(n\ Log[kay]\)\/steps\)\ g[numbersList\[LeftDoubleBracket]beginningPointIndex \[RightDoubleBracket]]], {n, 0, num1\ steps}]; \)\), \(\(arcNums = gInv[preArcNums]; \)\), \(\(arcPoints = Transpose[{Re[arcNums], Im[arcNums]}]; \)\), \(\(arcPict = Graphics[{RGBColor[1.000, 0.403, 0.121], Line[arcPoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Family of interchanged circles: each passes through one of the \ iterated points and each circle is an iterate of the others.\ \>", "SmallText",\ Evaluatable->False], Cell[BoxData[{ \(\(numberOfLoxCircles = Length[numbersList]\/4; \)\), \(r10 = 1; g10 = 0.090; b10 = .118; r20 = 1; g20 = 0.630; b20 = 0.090; \), \(\(radii = Table[Abs[g[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]], { n, 1, Length[numbersList], 4}]; \)\), \(\(loxCircleNums = Table[gInv[ radii\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUD], {n, numberOfLoxCircles}]; \)\), \(\(loxCirclePointsList = Table[Transpose[{ Re[loxCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[loxCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[loxCircleNums]}]; \)\), \(\(loxFamilyOfCirclesPict = Graphics[Table[{ RGBColor[0, \(n\ g20 + \((numberOfLoxCircles - n)\)\ g10\)\/numberOfLoxCircles, \(n\ b20 + \((numberOfLoxCircles - n)\)\ b10\)\/numberOfLoxCircles], Line[loxCirclePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[loxCirclePointsList]}]]; \)\)}], "Input"], Cell[BoxData[ \(\(loxFamilyOfCirclesPict = Graphics[Table[{ RGBColor[0, \(n\ g20 + \((numberOfLoxCircles - n)\)\ g10\)\/numberOfLoxCircles, \(n\ b20 + \((numberOfLoxCircles - n)\)\ b10\)\/numberOfLoxCircles], Line[loxCirclePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[loxCirclePointsList]}]]; \)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[arcPict, LoxPointsPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; \)\)], "Input"], Cell[BoxData[ \(\(Show[arcPict, LoxPointsPict, loxFamilyOfCirclesPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Changing the setting (Integral Forms & Stereographic Projections)\ \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Transfer to an ", PageWidth->Infinity, Evaluatable->False], StyleBox["integral", PageWidth->Infinity, Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" tramsformation", PageWidth->Infinity, Evaluatable->False] }], "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[BoxData[{ \(\(beginningPointIndex = 6; \)\), \(\(num1 = Length[numbersList] - 2\ beginningPointIndex; \)\), \(\(steps = 5; \)\), \(\(newArcNums = Table[N[E\^\(\(n\ Log[kay]\)\/steps\)\ g[numbersList\[LeftDoubleBracket]beginningPointIndex \[RightDoubleBracket]]], {n, 0, num1\ steps}]; \)\), \(\(newArcPoint = Transpose[{Re[newArcNums], Im[newArcNums]}]; \)\), \(\(newArcPict = Graphics[{RGBColor[1.000, 0.403, 0.121], Line[newArcPoint]}]; \)\), \(\(newNumbersList = g[numbersList]; \)\), \(\(integralizedLoxPointsList = Table[{Re[newNumbersList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[newNumbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[newNumbersList]}]; \)\), \(\(integralizedLoxPointsPict = Graphics[Table[{ RGBColor[0, 0, \(Length[integralizedLoxPointsList] - n\/2\)\/Length[ integralizedLoxPointsList]], PointSize[ \( .02\ \((Length[integralizedLoxPointsList] - n\/2) \)\)\/Length[integralizedLoxPointsList]], Point[integralizedLoxPointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[integralizedLoxPointsList]}]]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[newArcPict, integralizedLoxPointsPict, AspectRatio \[Rule] Automatic, Axes \[Rule] Automatic, Ticks \[Rule] None]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (original \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPointIndex = 6; \)\), \(\(num1 = Length[numbersList] - 2\ beginningPointIndex; \)\), \(\(steps = 5; \)\), \(\(newArcNums = Table[N[gInv[ E\^\(\(n\ Log[kay]\)\/steps\)\ g[numbersList\[LeftDoubleBracket]beginningPointIndex \[RightDoubleBracket]]]], {n, 0, num1\ steps}]; \)\), \(\(newArcPoint3D = stereoProjToSphere[newArcNums]; \)\), \(\(sphereArcPict = Graphics3D[{RGBColor[1.000, 0.403, 0.121], Line[newArcPoint3D]}]; \)\)}], "Input"], Cell[BoxData[ \(spherePict\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Show[sphereArcPict, pointsOnSphere[numbersList, RGBColor[0, 0, 1], .02], Boxed \[Rule] False]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (normalized \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPointIndex = 6; \)\), \(\(num1 = Length[numbersList] - 2\ beginningPointIndex; \)\), \(\(steps = 5; \)\), \(\(newArcNums = Table[N[E\^\(\(n\ Log[kay]\)\/steps\)\ g[numbersList\[LeftDoubleBracket]beginningPointIndex \[RightDoubleBracket]]], {n, 0, num1\ steps}]; \)\), \(\(newArcPoint3D = stereoProjToSphere[newArcNums]; \)\), \(\(sphereArcPict = Graphics3D[{RGBColor[1.000, 0.403, 0.121], Line[newArcPoint3D]}]; \)\)}], "Input"], Cell[BoxData[ \(\(Show[sphereArcPict, pointsOnSphere[g[numbersList], RGBColor[0, 0, 1], .02], Boxed \[Rule] False]; \)\)], "Input"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Improper Hyperbolic transformations. ", "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell["Text", "Text", Evaluatable->False], Cell["\<\ Improper hyperbolic transformations are sometimes refered to as \ loxodromic transformations with theta equal to `. What distinguishes them \ from other loxodromic transformations is that improper hyperbolic \ transformations have invariant circles. \ \>", "SmallText", Evaluatable->False], Cell["\<\ There are two distinct fixed points for improper hyperbolic \ transformations. K is real and less than zero.\ \>", "SmallText", Evaluatable->False], Cell["As always, ad-bc=1. Thus d=(1+bc)/a.", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Code", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[a, b, c, d, fixedPoint1, fixedPoint2, kay]; \)\), \(\(fixedPoint1 = 1\/4; \)\), \(\(fixedPoint2 = \(-\(1\/4\)\); \)\), \(\(a = 2; \)\), \(\(kay = \(-1.5\); \)\), \(\(c = \(a\ \((1 - kay)\)\)\/\(fixedPoint1 - kay\ fixedPoint2\); \)\), \(\(b = \(-c\)\ fixedPoint1\ fixedPoint2; \)\), \(\(d = a - c\ \((fixedPoint1 + fixedPoint2)\); \)\), \(\({a, b, c, d} = {a, b, c, d}\/\@\(a\ d - b\ c\); \)\)}], "Input"], Cell[BoxData[ \({a, b, c, d}\)], "Input"] }, Open ]], Cell[BoxData[ \(a\ d - b\ c\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes: ", "Text", Evaluatable->False], Cell["\<\ One characterization of an improper hyperbolic transformations is \ that its trace, i.e., a+d is ???? with absolute value ?less? than 2.\ \>", "SmallText", Evaluatable->False], Cell[BoxData[ \(a + d\)], "Input"], Cell[BoxData[ \({a, b, c, d}\)], "Input"], Cell[CellGroupData[{ Cell["Invariant Circle Data", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(pole = \(center = \(-\(d\/c\)\)\); \)\), \(\(zero = \(-\(b\/a\)\); \)\), \(\(radius = Abs[1\/c]; \)\)}], "Input", CellHorizontalScrolling->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["The transformation operating on points and circles", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ Choose an initial point and its iterates under the transformation \ will be computed.\ \>", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(beginningPoint = {1.14528\/4, 0.194945\/4}; \)\), \(\(numbersList = NestList[mTrans, beginningPoint\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ beginningPoint\[LeftDoubleBracket]2\[RightDoubleBracket], 10]; \)\), \(\(pointsList = Table[{Re[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[numbersList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[numbersList]}]; \)\), \(\(pointsPict = Graphics[Table[{ RGBColor[0, 0, \(Length[pointsList] - n\/2\)\/Length[pointsList]], PointSize[ \( .03\ \((Length[pointsList] - n\/2)\)\)\/Length[pointsList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}, {n, Length[pointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Hyperbolic pencil of circles generated ", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 15 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(numberOfHyperbolicCircles = 10\), \(r10 = 1; g10 = 0.090; b10 = .118; r20 = 1; g20 = 0.630; b20 = 0.090; \), \(\(centers = Table[\(-\((fixedPoint1 - fixedPoint2)\)\)\ I\ y\ .2, {y, \(-Floor[numberOfHyperbolicCircles\/2]\), Ceiling[numberOfHyperbolicCircles\/2]}]; \)\), \(\(radii = Table[Abs[ fixedPoint1 - centers\[LeftDoubleBracket]n\[RightDoubleBracket]], { n, numberOfHyperbolicCircles}]; \)\), \(\(hypCircleNums = Table[N[radii\[LeftDoubleBracket]n\[RightDoubleBracket]\ numbersUDShort + centers\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, numberOfHyperbolicCircles}]; \)\), \(\(hypCirclePointsList = Table[Transpose[{ Re[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], Im[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[hypCircleNums]}]; \)\), \(\(hyperbolicFamilyOfCirclesPict = Graphics[Table[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hypCirclePointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[hypCirclePointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[pointsPict, hyperbolicFamilyOfCirclesPict, Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Elliptic pencil of circles generated", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 15 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(r1 = 0.043; g1 = 0.113; b1 = 1; r2 = 0.489; g2 = 0.740; b2 = 1; \), \(\(pointOnBeginningCircle = pointsList\[LeftDoubleBracket]1\[RightDoubleBracket]; \)\), \(\(begPtNum = pointOnBeginningCircle\[LeftDoubleBracket]1\[RightDoubleBracket] + I\ pointOnBeginningCircle\[LeftDoubleBracket]2\[RightDoubleBracket]; \)\), \(\(ellipticRadius = Abs[g[begPtNum]]; \)\), \(\(beginningCircleNums = gInv[ellipticRadius\ numbersUDShort]; \)\), \(\(numberOfEllipticCircles = Length[pointsList]; \)\), \(\(ellipticCircleFamilyNums = NestList[mTrans, beginningCircleNums, numberOfEllipticCircles - 1]; \)\), \(ellipticCircleFamilyPoints = Table[Transpose[{ Re[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCircleFamilyNums]}]; Null; \), \(\(ellipticFamilyOfCirclesPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCircleFamilyPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCircleFamilyPoints]}]; \)\), \(\(iteratedPointsPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((Length[pointsList] - n)\)\ r1\)\/Length[ pointsList], \(n\ g2 + \((Length[pointsList] - n)\)\ g1\)\/Length[ pointsList], \(n\ b2 + \((Length[pointsList] - n)\)\ b1\)\/Length[ pointsList]], PointSize[ \( .02\ \((Length[pointsList] - n\/2)\)\)\/Length[pointsList]], Point[pointsList\[LeftDoubleBracket]n\[RightDoubleBracket]]}], { n, Length[pointsList]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[ellipticFamilyOfCirclesPict, hyperbolicFamilyOfCirclesPict, iteratedPointsPict, Axes \[Rule] Automatic, Ticks \[Rule] None, AspectRatio \[Rule] Automatic]; \)\)], "Input"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Changing the setting (Integral Forms & Stereographic Projections)\ \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Picture of the ", Evaluatable->False], StyleBox["integral", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" transformation which is similar to this one. ", Evaluatable->False] }], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(iteratedPointsNormalized = Transpose[{Re[g[numbersList]], Im[g[numbersList]]}]; num = Length[numbersList]; \), \(\(iteratedPointsNormalizedPict = Table[Graphics[{ RGBColor[\(n\ r2 + \((num - n)\)\ r1\)\/num, \(n\ g2 + \((num - n)\)\ g1\)\/num, \(n\ b2 + \((num - n)\)\ b1\)\/num], PointSize[ .03], Point[iteratedPointsNormalized\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsNormalized]}]; \)\), \(\(ellipticCirclesNormalizedPoints = Table[Transpose[{ Re[g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], Im[g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]]}], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(ellipticCirclesNormalizedPict = Table[Graphics[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesNormalizedPoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesNormalizedPoints]}]; \)\), \(\(hypCirclesNormalizedNumsList = Table[g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[hypCircleNums]}]; \)\), \(\(hypCirclesNormalizedPointsList = Table[Transpose[{ Re[hypCirclesNormalizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]], Im[hypCirclesNormalizedNumsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hypCirclesNormalizedNumsList]}]; \)\), \(\(hyperbolicCirclesNormalizedPict = Graphics[Table[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hypCirclesNormalizedPointsList\[LeftDoubleBracket]n \[RightDoubleBracket]]}, {n, Length[hypCirclesNormalizedPointsList]}]]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[hyperbolicCirclesNormalizedPict, iteratedPointsNormalizedPict, ellipticCirclesNormalizedPict, AspectRatio \[Rule] Automatic]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes on the integral form.", "Text", Evaluatable->False], Cell["\<\ In the transfomation's integral form, the hyperbolic pencil of \ circles appear as straight lines through the origin and the elliptic pencil \ of circles as concentric circles about the origin.\ \>", "SmallText", Evaluatable->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (original \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[numbersList]; \)\), \(\(iteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(ellipticCirclesOnSpherePoints = Table[stereoProjToSphere[ ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(ellipticCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesOnSpherePoints]}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]], {n, Length[hypCircleNums]}]; \)\), \(\(hyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 50 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[iteratedPointsOnSpherePict, ellipticCirclesOnSpherePict, hyperbolicCirclesOnSpherePict, Boxed \[Rule] False, ViewPoint \[Rule] {2.300, 3.320, \(-8.000\)}]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Move the setting to the Riemann Sphere (integral \ transformation)\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Code (approx. 100 sec)", "SmallText", Evaluatable->False], Cell[BoxData[{ \(\(iteratedPointsOnSphere = stereoProjToSphere[g[numbersList]]; \)\), \(\(integralIteratedPointsOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((Length[numbersList] - n)\)\ r1\)\/Length[ numbersList], \(n\ g2 + \((Length[numbersList] - n)\)\ g1\)\/Length[ numbersList], \(n\ b2 + \((Length[numbersList] - n)\)\ b1\)\/Length[ numbersList]], PointSize[ .03], Point[iteratedPointsOnSphere\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[iteratedPointsOnSphere]}]; \)\), \(\(ellipticCirclesOnSpherePoints = Table[stereoProjToSphere[ g[ellipticCircleFamilyNums\[LeftDoubleBracket]n \[RightDoubleBracket]]], {n, Length[ellipticCircleFamilyNums]}]; \)\), \(\(integralEllipticCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r2 + \((numberOfEllipticCircles - n)\)\ r1 \)\/numberOfEllipticCircles, \(n\ g2 + \((numberOfEllipticCircles - n)\)\ g1 \)\/numberOfEllipticCircles, \(n\ b2 + \((numberOfEllipticCircles - n)\)\ b1 \)\/numberOfEllipticCircles], Line[ellipticCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[ellipticCirclesOnSpherePoints]}]; \)\), \(\(hyperbolicCirclesOnSpherePoints = Table[stereoProjToSphere[ g[hypCircleNums\[LeftDoubleBracket]n\[RightDoubleBracket]]], {n, Length[hypCircleNums]}]; \)\), \(\(integralHyperbolicCirclesOnSpherePict = Table[Graphics3D[{ RGBColor[ \(n\ r20 + \((numberOfHyperbolicCircles - n)\)\ r10 \)\/numberOfHyperbolicCircles, \(n\ g20 + \((numberOfHyperbolicCircles - n)\)\ g10 \)\/numberOfHyperbolicCircles, \(n\ b20 + \((numberOfHyperbolicCircles - n)\)\ b10 \)\/numberOfHyperbolicCircles], Line[hyperbolicCirclesOnSpherePoints\[LeftDoubleBracket]n \[RightDoubleBracket]]}], {n, Length[hyperbolicCirclesOnSpherePoints]}]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Picture (approx. 40 sec)", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Show[integralIteratedPointsOnSpherePict, integralEllipticCirclesOnSpherePict, integralHyperbolicCirclesOnSpherePict, Boxed \[Rule] False]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Notes on the integral transformation.", "Text", Evaluatable->False], Cell["\<\ Since the two fixed points in the integral form are antipodal on \ the Riemann Sphere, the hyperbolic pencil of circles are transformed into \ great circles through the fixed points while the elliptic pencil of circles \ are transformed into circles with their centers on the axis which passes \ through the two fixed points.\ \>", "SmallText", Evaluatable->False] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Animation of improper hyperbolic Mobius transformation operating on \ circle in pencil, creating other members of pencil.\ \>", "SmallText", Evaluatable->False], Cell[BoxData[ \(\(Do[ Show[circleFamilyPict\[LeftDoubleBracket]n\[RightDoubleBracket], pointsPict\[LeftDoubleBracket]n\[RightDoubleBracket], Axes \[Rule] Automatic, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}}, Ticks \[Rule] None], { n, Length[circleFamilyPict]}]; \)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Text on Mobius Transformation Examples", "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["Prelude", "SmallText", Evaluatable->False], Cell[TextData[{ StyleBox[ "I have seen many things on the computer scope which are exciting, even \ startling, and which could do much to restore the graphical image as a proper \ object of mathematical study. The computer scope has remarkable capabilities \ of animation, and the dynamic figures which emerge are - to steal a line from \ Descartes - as far removed from old-fasioned static figures as are the \ orations of Cicero from the simple ABC's. They cannot be exhibited in books, \ they must be experienced as movies.\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ \t\t-Philip J. Davis \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tin the \ Carus Mathematical Monograph #17:\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ \t", Evaluatable->False, FontFamily->"New York"], StyleBox["The Schwarz Function and Its Applications", Evaluatable->False, FontFamily->"New York", FontSlant->"Italic"] }], "SmallText", Evaluatable->False], Cell["\<\ This expository paper will describe in detail certain aspects of \ past work on Mobius transformations. Accompanying the descriptions will be \ still and animated computer graphics to illustrate the results.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["General Introduction", "Subsubsection", Evaluatable->False], Cell["\<\ Early in the ninteenth century, K. F. Gauss, N.I. Lobachevsky, and \ J. Bolyai each independently proved that noneuclidean geometries could exist. \ Later in the century Poincare constructed a model of the hyperbolic plane. \ Riemann, Klein and Poincare investigated the problems associated with \ uniformizing multivalued functions. Included in their studies were discrete \ groups of transformations. Among these were those associated with Poincare's \ Theta Function, Weierstrass' \"Pe\" Function, and the J-Function of the \ Modular Group. With each discrete group is associated a Riemann surface. These studies continue to be of interest for several reasons. 1) Insights \ into the more abstract topics of Riemannian Differential Geometry are still \ yielded by the more concrete examples involving the discrete groups studied \ in the 1800's. 2) These groups of Mobius transformations are elementary \ examples which share contact with three of the more important branches of \ modern mathematics, namely analysis, algebra, and topology. For this reason \ these objects can be used to highlight the interconnectedness of the \ branches. Further, since much of the material lends itself well to graphical \ representation, it provides a natural context for introducing computer \ technology into the mathematics curriculum. The materials associated with \ Mobius transformations provide many opportunities to introduce a variety of \ connected topics that are too often viewed as separate, their \ interconnections often hidden from view even in very advanced course work. Certainly, extensive study of all the related topics are required for a full \ understanding and appreciation of the material. At the same time, glimpses \ of the significance, interconnectedness, and, frankly, astonishing aspects of \ this mathematics can be made accessible to many different levels of teachers, \ preservice teaching professionals and students. This is especially true when \ extensive use is made of computer graphics. An intuitive introduction can be \ made, providing motivation for further explorations/studies. \ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Fixed Points", "Subsubsection", Evaluatable->False], Cell["\<\ All Mobius transformations excepting those that are parabolic have \ two fixed points. Parabolic Mobius transformations have a single fixed point \ which we sometimes view as a double point. The fixed points are easy to \ compute from the relation: \t\t\t\t\t\t\t\t\t\t\tz = (a z + b)/(c z + d), which is equivalent to the quadratic equation: \t\t\t\t\t\t\t\t\t\tc z\[Currency] + (-a + d) z - b = 0. Thus, the transformation is parabolic if and only if the discriminant, \t\t\t\t\t\t\t\t\t(-a + d)\[Currency] + 4 bc = 0. If the transformation is normalized so that ad - bc = 1, then the above \ expression is equivalent to: \t\t\t\t\t\t\t\t (a + d)\[Currency] - 4 = 0. We may note that the expression (a + d) is the trace of the matrix of \ coefficients of the Mobius transformation. Also every Mobius transformation \ that is not equivalent to the identitiy transformation, so that ad - bc \ \[CapitalAE] 0, may be normalized by multiplying each of the coefficients in \ turn by 1/Sqrt[ad - bc], without changing the effect the transformation has. \ This is why we may view all Mobius transformation, WLOG, as elements of \ SL[2,C]. Further the orientation preserving transformations have ad - bc > \ 0. \ \>", "SmallText", Evaluatable->False], Cell["\<\ If we view the single fixed point of the parabolic Mobius \ transformation as a double point, we may make the following general \ statement: For all Mobius transformations other than those that are \ elliptic, one of the fixed points is an attractor under iterations of the \ transformation and the other fixed point is a repulsor. The single fixed \ point of the parabolic transformation fills both roles of attractor and \ repulsor. For, if the transformation is parabolic, then iterations of the \ transformation move points in a single direction along invariant curves, all \ of which pass through the fixed point. Thus, when the invariant curves are \ viewed with one orientation the points are repulsed by the fixed point while \ they are attracted when the orientation is reversed. \ \>", "SmallText", Evaluatable->False], Cell["\<\ The nature of the fixed points is determined by the trace of the \ matrix associated with the normalized transformation. When the trace, (a+d), \ is equal to +/- 2, then there is a single fixed point. When the trace is \ real with absolute value greater than 2, then the fixed points are either \ both real or complex depending on c. The trace yields much more information \ than this about the character of the transformation, as will be described \ below.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["The Various Types of Mobius Transformations ", "Subsubsection", Evaluatable->False], Cell["\<\ With each mobius transformations is associated a matrix of its \ coefficients: \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t| a b | \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t| c d | \ \>", "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["The trace: (a + d).", "SmallText", Evaluatable->False], Cell["\<\ We will see that the trace of the matrix associated with the \ normalized transformation, i.e., (a + d), when ad - bc = 1, determines the character of the Mobius \ tramsformation: If (a + d) = 2, then the transformation is parabolic. If -2 < (a + d) < 2, then the transformation is elliptic. If (a + d) < -2 or 2 < (a + d), then the transformation is hyperbolic. If (a + d) is non real, then the transformation is loxodromic. If (a + d) is non real and the modulus is greater than 2, then the \ transformation is sometimes called improper hyperbolic. This is the only \ type of loxodromic transformation that has invariant circles. \ \>", "SmallText", Evaluatable->False] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Integral", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" Forms of Mobius Transformations", Evaluatable->False] }], "Subsubsection", Evaluatable->False], Cell[TextData[{ StyleBox[ "Consider the Mobius transformation, \n\t\t\t\t\t\t\t\t\tG(z) = (z - \ fixedPoint2)/(z - fixedPoint1) \nwhen there are two fixed points, i.e., when \ the transformation is ", Evaluatable->False], StyleBox["not", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[ " parabolic, and G(z) = 1/(z - fixedPoint) when the transformation is \ parabolic. Clearly G carries the first fixed point to infinity and the \ second fixed point, if there is a second fixed point, to zero. If a \ transformation M is conjugated by G the result is a Mobius transformation \ with infinity as its first fixed point and zero as its second fixed point.\n\n\ \t\t\tGInverse(M(G(z)))) = K G(z) + C, K and C complex constants.", Evaluatable->False] }], "SmallText", Evaluatable->False], Cell[TextData[{ StyleBox[ "\nEvery Mobius transformation, \n\t\t\t\t\t\t\t\t\t\t\tw(z) = (a z + b)/(c \ z + d) , is similar to an ", Evaluatable->False], StyleBox["integral", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[ " transformation, i. e., one of the form: \n\t\t\t\t\t\t\t\t\t\t\t\t\t\ \tW(Z) = K Z + C , where K and C are complex constants. One arrives at this \ form by applying the Mobius transformation, \n\t\t\t\t\t\t\t\t\tG(z) = (z - \ fixedPoint2)/(z - fixedPoint1) \nwhen there are two fixed points, i.e., when \ the transformation is ", Evaluatable->False], StyleBox["not", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[ " parabolic, and G(z) = 1/(z - fixedPoint) when the transformation is \ parabolic, to both variables z and w(z). An easy computation yields that \n\t\ \t\t\t\t\t\t\t\tK = (a - c fixedPoint2)/(a - c fixedPoint1) \n\t\t\t\t\t\t\ \t\t\t\t\t\t\t\t\t(We interpret this as = 1 when there is a \t\t\t\t\t\t\t\t\t\ \t\t\t\t\t\t\t\t\t\tsingle fixed point, i.e., when the \t\t\t\t\t\t\t\t\t\t\t\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ttransformation is parabolic.) \ and\n\t\t\t\t\t\t\t\t\tC = 0, c, or -c, \nwhen\t\t\t\tW = G(w) \tand\t\t\t\t\ Z = G(z).\nIt turns out that C will be non-zero only when the transformation \ is parabolic. Further, C = c when (a + d) = 2 and = -c when (a + d) = -2.", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["The Multiplier K of the ", Evaluatable->False], StyleBox["Integral", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" Forms. ", Evaluatable->False] }], "Subsubsection", Evaluatable->False], Cell[CellGroupData[{ Cell["General", "SmallText", Evaluatable->False], Cell[TextData[ "As seen in the section above, every Mobius transformation is similar to an \ integral transformation:\n\t\t\t\t\t\t\t\t\t\t\t\t\tW = K * Z + C.\n\tWe \ also saw that:\n\t\t\t\t\t\t\t\t K = (a - c fixedPoint2)/(a - c \ fixedPoint1), with fixedPoint1=fixedPoint2 when there is a single fixed \ point. \nFurther, we saw that the character of the transformation is \ determined by the trace, (a + d). In a similar manner K determines the \ character of the transformation.\nWhen K is real, > 0 and \[CapitalAE] 1, \ then the transformation is \t\t\t\t\t\t\t\t\t\thyperbolic. \nWhen Abs[K]=1, \ but K\[CapitalAE]1, i.e., K = e\[CapitalEGrave] \ \[CapitalEHat]\[CapitalOAcute]\:2030\[CapitalEHat]\[CapitalARing], theta\ \[CapitalAE]2n`,\n\t\t\t\t\tthen the transformation is elliptiic.\nWhen K = \ 1, then the transformation is parabolic.\nWhen K = A e\[CapitalEGrave] \ \[CapitalEHat]\[CapitalOAcute]\:2030\[CapitalEHat]\[CapitalARing], A>0, 0\ \[Ellipsis]theta\[Ellipsis]2` (but theta\[CapitalAE]`),\n\t\t\t\tthen the \ transformation is loxodromic.\nWhen K = A e\[CapitalEGrave]\:0178 = -A, A>0, \ then the transformation is \t\t\t\t\t\t\t\timproper hyperbolic.\n"], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Further Notes Regarding K", "SmallText", Evaluatable->False], Cell["\<\ K + 1/K = (a messy rational expression in terms of a, b, c, d, \ fixedPoint1 and fixedPoint2). Since fixedPoint1 and fixedPoint2 are foots of cz\[Currency]+(d-a)z-b=0, \ fixedPoint1 + fixedPoint2 = (a-d)/c, (fixedPoint1)*(fixedPoint2)=-b/c, and fixedPoint1\[Currency] + fixedPoint2\[Currency] = {(a-d)\[Currency] + 2bc}/c\ \[Currency]. Simplifying the (messy expression), we get: K + 1/K = {(a+d)\[Currency] - 2(ad-bc)}/(ad-bc), which, if the transformation \ is normalized so that ad-bc=1, may be simplified to: K + 1/K = (a+d)\ \[Currency] - 2. We also see that Sqrt[K] + 1/Sqrt[k] = a + d.\ \>", "SmallText", Evaluatable->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Invariant Curves", "Subsubsection", Evaluatable->False], Cell[TextData[{ StyleBox[ "For each Mobius transformation there are invariant curves, i.e., continua \ that are mapped onto themselves. In fact, for all transformations other than \ loxodromic, these curves are a family of circles. The invariant curves for \ loxodromic transformations are known as Bernoulli curves and are similar, via \ a normalizing Mobius transformation, to logarithmic spirals. Further these \ invariant circles are all orthogonal to the ", Evaluatable->False], StyleBox["isometric circles", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" of the transformation.", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Isometric Circles", "Subsubsection", Evaluatable->False], Cell[CellGroupData[{ Cell["Isometric circles of the transformations:", "SmallText", Evaluatable->False], Cell["\<\ We consider the locus of points at which the derivative of the \ transformation has modulus = 1. Since the derivative of a Mobius \ transformation = (ad - bc)/(c z + d)\[Currency], when the transformation is \ normalized so that ad-bc = 1, this locus is the circle determined by |z - \ d/c| = |1/c|. This circle is the locus of all point that experience no \ distortion under the transformation. In fact, this circle is mapped onto a \ circle of the same radius with no distortion of the arcs. The circle's image \ is the isometric circle of the transformation's inverse. No other continuum \ of the plane has the property of being distortion free under mapping by the \ transformation. \ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Relationship between K and isometric circles", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["Isometric circles of the transformations:", "SmallText", Evaluatable->False], Cell[TextData[{ StyleBox[ "We consider the locus of points at which the derivative of the \ transformation has modulus = 1. Since the derivative of a Mobius \ transformation = (ad - bc)/(c z + d)\[Currency], when the transformation is \ normalized so that ad-bc = 1, this locus is the circle determined by |z - \ d/c| = |1/c|. This circle is the locus of all point that experience no \ distortion under the transformation. This circle, called the ", Evaluatable->False], StyleBox["isometric", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[" ", Evaluatable->False], StyleBox["circle", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[ " is mapped onto a circle of the same radius with no distortion of the \ arcs. This is true of no other continuum in the plane. ", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]], Cell["\<\ Let \t\t\tM(z) = (a z + b)/(c z + d) and let the two fixed points \ of M be f1 & f2. Then \t\t\t\t\t\t\t\t\t\t\t\t\tM'(z) = K {(M(z) - f1)/(z-f1)}\[Currency] \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t=1/K {(M(z) - f2)/(z-f2)}\[Currency] \t\t\t\tand, so M'(f2) = K and M'(f1) = 1/K. \t\t\t\t If |K| \[CapitalAE] 1, then there is distortion at the fixed points, i.e., \ the derivative of the transformation at the fixed point is not a unimodular \ value. Thus, the fixed points of elliptic and parabolic transformations, \ having |K| = 1, must lie on the isometric circles of the transformations. \ The fixed points of all other transformations have one of their fixed points \ lying inside the isometric circle and the other lying without.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Relationship of fixed points and isometric circles ", "SmallText", Evaluatable->False], Cell[TextData[{ StyleBox[ "If the fixed points of the transformation lie on the isometric circle, \ then they also lie on the image of the isometric circle; the image of a \ transformation's isometric circle is the isometric circle of the \ transformation's inverse. Thus, the fixed points lie on the ", Evaluatable->False], StyleBox["intersection", Evaluatable->False, FontVariations->{"Underline"->True}], StyleBox[ " of these two circles. In particular, the isometric circles of elliptic \ and parabolic transformations do intersect each other while the isometric \ circles of hyperbolic transformations are disjoint.\n\tThis can be seen in a \ slightly different way refering to the trace, (a + d): The distance between \ the centers of the isometric circles of a transformation is |(a+d)/c| with \ the radius of each circle equal |1/c|. This the two circles will intersect, \ be tangent, or disjoint according to whether the absolute value of the trace, \ |a + d|, is < 2 (elliptic case), \n\t= 2 (parbolic case), or > 2 (hyperbolic \ case). If the transformation is loxadromic or improper hyperbolic, then the \ trace is complex and the isometric circles can have any of the above \ relations.", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Circle Inversions", "Subsubsection", Evaluatable->False], Cell["\<\ Elliptic, parabolic, and hyperbolic Mobius transformations are all \ equivalent to an inversion in a circle followed by a reflection in a line. \ If the straight line is viewed as a circle on the Riemann sphere, we may say \ that these transormations are all compositions of two reflections in two \ circles. Loxadromic and improper hyperbolic transformations are equivalent \ to the composition of two such circle inversions followed by a rotation about \ a point. Since a rotation is equivalent to reflections across two lines that \ intersect at the rotation's fixed point, we can say that the loxadromic \ transformation is equivalent to the composition of four circle \ inversions.\ \>", "SmallText", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Table of: the Various Types, the Trace, and K", "Subsubsection", Evaluatable->False], Cell[TextData[{ StyleBox[ "Type of Mobius \t\t\t\t\t Trace,\t\t\t\t\t\t\t\t\t\t\t K=\n\ Transformation\t\t\t \ti.e., (a+d) \t(a-c f2)/(a-c f1) \n", Evaluatable->False, FontWeight->"Bold"], StyleBox[ "\nHyperbolic\t\t\t\t\t\t\t\t\t\t\t\tReal & |a+d|>2\t\t\t\t\t\t\tReal & \ \[CapitalAE]1\n\nElliptic\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tReal & |a+d|<2\t\t\t\t\ \t\t\t= e\[CapitalEGrave]\[CapitalEHat], t\[CapitalAE]n`\n\nParabolic\t\t\t\t\ \t\t\t\t\t\t\t\t\t\t\t= +2 or -2\t\t\t\t\t\t\t\t= 1\n\nLoxodromic\t\t\t\t\t\t\ \t\t\t\t\t\t\t\tcomplex\t\t\t\t\t\t\t\t\t \ =Ae\[CapitalEGrave]\[CapitalEHat],A>0,t\[CapitalAE]n`\n\nImproper Hyperbolic\t\ \t\t\t\t\tpure imaginary\t\t\t\t=A e\[CapitalEGrave]\:0178,A>0\n\t\t\t\t\t\t\t\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t=-A", Evaluatable->False] }], "SmallText", Evaluatable->False] }, Closed]], Cell["\<\ Fuchsian Groups, i.e., Groups of Mobius Transformations That Fix a \ Circle\ \>", "Subsubsection", Evaluatable->False], Cell["Riemann Surfaces", "Subsubsection", Evaluatable->False], Cell["\<\ Galois Groups Represented By Aut{F[x1,x2,. . .,xn]} where F is the \ field of rational functions generated by the polynomial with {x1,x2,...,xn} \ as roots, which in turn is the polynomial which is associated with a certain \ Riemann surface on which automorphic functions can be defined.\ \>", "Subsubsection", Evaluatable->False] }, Closed]] }, Closed]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, AutoGeneratedPackage->Automatic, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{764, 577}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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