(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 31599, 1126]*) (*NotebookOutlinePosition[ 32297, 1150]*) (* CellTagsIndexPosition[ 32253, 1146]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Gallery of mathematical images", "Title"], Cell[CellGroupData[{ Cell["Author", "Subsection"], Cell["\<\ Bernd Thaller Institute for Mathematics and Scientific Computing University of Graz Austria\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Source", "Subsection"], Cell[TextData[{ " ", ButtonBox["http://www.uni-graz.at/imawww/vqm/", ButtonData:>{ URL[ "http://www.uni-graz.at/imawww/vqm/"], None}, ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Abstract", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " source code for the images at ", ButtonBox["http://www.uni-graz.at/imawww/vqm/pages/colorgallery/index.html", ButtonData:>{ URL[ "http://www.uni-graz.at/imawww/vqm/pages/colorgallery/index.html"], None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell["\<\ In this notebook, the value of the option PlotPoints has been \ reduced for performance reasons.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Load package", "Section"], Cell[BoxData[ \(\(<< VQM`ComplexPlot`;\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["01: alien1", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[Sin[ab], 1\/\(1 + ar\^2\), 1\/\(1 + ar\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Exp[Tan[y + I\ x]], {x, \(-\(\[Pi]\/3\)\), \[Pi]\/3}, {y, 0, \[Pi]}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["02: alien2", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(myColorMap = Function[{ab, ar}, Hue[Sin[ab], 1\/\(1 + ar\^2\), 1\/\(1 + ar\^2\)]]; QComplexDensityPlot[ Zeta[Zeta[y + I\ x]], {x, \(- .5\), .5}, {y, .5, 1.2}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["03: alien3", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[Sin[ab], 1\/\(1 + ar\^2\), 1\/\(1 + ar\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Exp[x + I\ y]\^Tan[y + I\ x], {x, \(-\(\[Pi]\/8\)\), \[Pi]\/4}, {y, \ \[Pi] + 0.0001, \(15\ \[Pi]\)\/8}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["04: alien4", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[Sin[\@ab], 1\/\(1 + ar\^2\), 1\/\(1 + ar\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Exp[x + I\ y]\^Tan[y + I\ x], {x, \(-\(\[Pi]\/9\)\), \(5\ \ \[Pi]\)\/12}, {y, \[Pi] + 0.0001, \(14\ \[Pi]\)\/8}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["05: butterfly", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[1\/\(1 + ab\^2\), 1\/\(1 + ar\^2\), 1\/\(1 + ar\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Sin[Sin[1\/\(y + I\ x\)]], {x, \(-1\), 1}, {y, \(- .5\), .5}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["06: crossing1", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[1\/\(1 + ab\^2\), 1\/\(1 + ar\^2\), 1\/\(1 + ar\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Cot[x + I\ y]\^Tan[y + I\ x], {x, \(-\(\(5\ \[Pi]\)\/4\)\), 3\ \[Pi] - 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1\/\((1 - Sin[x + I\ y])\)\^2)\)\^2, {x, \(-\(\[Pi]\/2\)\), \ \(3\ \[Pi]\)\/2}, {y, \(-\(\(3\ \[Pi]\)\/4\)\), \(3\ \[Pi]\)\/4}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["12: flowers2", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[1\/\(1 + ar\^2\), 1\/\(1 + ab\^2\), 1\/\(1 + ab\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ 1\/\((1 - 1\/\((1 - Sin[x + I\ y])\)\^2)\)\^2, {x, \(-\(\[Pi]\/2\)\), \ \(11\ \[Pi]\)\/2}, {y, \(-\[Pi]\), \[Pi]}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["13: flowers3", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[Mod[ar, 1], Mod[ab, 1], Mod[ar, 1]]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ 1\/\((1 - 1\/\((1 - Sin[x + I\ y])\)\^2)\)\^2, {x, \(-1.5\), 2.5}, {y, \(-1.5\), 2.5}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["14: flowers4", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, Hue[Mod[ab, 1], Mod[ab, 1], Mod[ar, 1]]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ 1\/\((1 - 1\/\((1 - Sin[x + I\ y])\)\^2)\)\^2, {x, \(-2\), 6}, {y, \(-2.2\), 3.3}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["15: flowers5", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, RGBColor[ .6\/\(1 + ar\^2\) + .4\/\(1 + ab\^2\), .4\/\(1 + \ ar\^2\) + .6\/\(1 + ab\^2\), .1\/\(1 + ar\^2\) + .9\/\(1 + \ ab\^2\)]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Sin[1\/\((1 - 1\/\((1 - Sin[x + I\ y])\)\^2)\)\^2]\^2, {x, \(-\(\[Pi]\ \/2\)\), \(3\ \[Pi]\)\/2}, {y, \(-\(\(3\ \[Pi]\)\/4\)\), \(3\ \[Pi]\)\/4}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["16: flowers6", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap[ab_, ar_] := Hue[1\/\(1 + ab\^2\), Sin[Log[ab]]\^2, 1\/\(1 + ar\^2\)];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Log[Sin[1\/\((y + I\ x)\)\^5]], {x, \(-1.5\), 1.5}, {y, \(-1.5\), 1.5}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["17: flowers7", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap[ab_, ar_] := Hue[1\/\(1 + ab\^2\), Sin[Log[ab]]\^2, .3\/\(1 + ar\^2\) + .7\ Cos[Log[Abs[ar]]]\^2];\)\ \)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Log[Sin[Log[\((x + I\ y)\)\^7]]], {x, \(-\[Pi]\), \[Pi]}, {y, \ \(-\[Pi]\), \[Pi]}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["18: flowers8", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap[ab_, ar_] := Hue[1\/\(1 + ab\^2\), Sin[Log[ab]]\^2, .1\/\(1 + ar\^2\) + .9\ Cos[Log[Abs[ar]]]\^2];\)\ \)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Log[Sin[1\/\((x + I\ y)\)\^3]], {x, \(-\(\[Pi]\/2\)\), \[Pi]\/2}, {y, \ \(-\(\[Pi]\/2\)\), \[Pi]\/2}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["19: flowers9", "Section"], Cell[BoxData[ \(\(Remove[myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap[ab_, ar_] := Hue[1\/\(1 + ab\^2\), Sin[Log[ab]]\^2, .1\/\(1 + ar\^2\) + .9\ Cos[Log[Abs[ar]]]\^2];\)\ \)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ Log[Sin[1\/\((x + I\ y)\)\^11]], {x, \(-\(\[Pi]\/2\)\), \[Pi]\/2}, \ {y, \(-\(\[Pi]\/2\)\), \[Pi]\/2}, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["20: fractal1", "Section"], Cell[BoxData[ \(\(Remove[mandelbrot];\)\)], "Input"], Cell[BoxData[ \(\(mandelbrot = Compile[{{k, _Complex}}, Module[{val = 0 + 0\ I, cnt = 0}, While[Re[val]\^2 + Im[val]\^2 < 15000 && cnt < 100, val = val\^2 + k; \(cnt++\)]; cnt\ Exp[I\ Arg[val]]]];\)\)], "Input"], Cell[BoxData[ \(Timing[ QComplexDensityPlot[ mandelbrot[x + I\ y], {x, \(-2.6\), 1.4}, {y, \(-2\), 2}, PlotPoints \[Rule] 100, Mesh \[Rule] False, Compiled \[Rule] False, QSphereRadius \[Rule] 10, QLightnessRange \[Rule] {0, 0.9}]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["21: fractal2", "Section"], Cell[BoxData[ \(\(Remove[mandelbrot];\)\)], "Input"], Cell[BoxData[ \(\(mandelbrot = Compile[{{k, _Complex}}, Module[{val = 0 + 0\ I, cnt = 0}, While[Re[val]\^2 + Im[val]\^2 < 15000 && cnt < 100, val = val\^2 + k; \(cnt++\)]; cnt\ Exp[I\ Arg[val]]]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ mandelbrot[x + I\ y], {x, \(-1.49\), \(-1.47\)}, {y, \(-0.01\), 0.01}, Compiled \[Rule] False, PlotPoints \[Rule] 100, Mesh \[Rule] False, QSphereRadius \[Rule] 50, QValueRange \[Rule] {15, 110}];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["22: kitchen1", "Section"], Cell[BoxData[ \(\(Remove[f, g, myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(f[x_] := 1 - x\^2;\)\)], "Input"], Cell[BoxData[ \(\(g = Compile[{{x, _Complex}}, Simplify[Nest[f, x, 5]]];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap = Function[{ab, ar}, RGBColor[Mod[Log[ab + 1. ], 1], Mod[Log[Log[ab + 1. ]], 1], Mod[ .8\ ar, 1]]];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[ g[x + I\ y], {x, \(-1.4\), 1.4}, {y, \(- .8\), .8}, Compiled \[Rule] False, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["23: kitchen2", "Section"], Cell[BoxData[ \(\(Remove[f, g, h, myColorMap];\)\)], "Input"], Cell[BoxData[ \(\(f[x_, y_] = 1\/\(\((x + I\ y)\)\^5 - 1\);\)\)], "Input"], Cell[BoxData[ \(\(g[x_, y_] = Simplify[f[f[x, y], 1\/f[x, y]]];\)\)], "Input"], Cell[BoxData[ \(\(h = Compile[{{x, _Complex}, y}, g[x, y]];\)\)], "Input"], Cell[BoxData[ \(\(myColorMap[ab_, ar_] = Hue[5\ ab, 1\/2\ \((1 + Sin[ar\/\[Pi]])\), 1\/2\ \((1 + Cos[ar\/\[Pi]])\)];\)\)], "Input"], Cell[BoxData[ \(\(QComplexDensityPlot[h[x, y], {x, \(-1.4\), 1.4}, {y, \(-1.4\), 1.4}, Compiled \[Rule] False, QComplexToColorMap \[Rule] myColorMap, PlotPoints \[Rule] 100, Mesh \[Rule] False, Frame \[Rule] False];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["24: quantum1", "Section"], Cell[BoxData[ \(\(Remove[Gauss, p0, f, doplot];\)\)], "Input"], Cell[BoxData[ \(\(Gauss[x_, t_, a_, x0_, p0_] := \(\@\(a\/\[Pi]\)\ Exp[\(-\(\(\(a\ x . x\)\/2 - 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