(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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). 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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[ 10605, 410]*) (*NotebookOutlinePosition[ 45888, 1621]*) (* CellTagsIndexPosition[ 45844, 1617]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Rhombic Helicoids", "Title"], Cell["\<\ by Russell Towle rustybel@foothill.net March, 2000\ \>", "Subsubtitle"], Cell[TextData[{ "The helicoid is the only ruled minimal surface, other than the plane. Its \ boundary is a helix. It is \[OpenCurlyDoubleQuote]minimal\ \[CloseCurlyDoubleQuote] because of all surfaces bounded by a given helix, \ the helicoid has least area. It seems that the helicoid surface was \ discovered by Meusnier in 1776. See Eric Weisstein's MathWorld at ", ButtonBox["http://mathworld.wolfram.com/MinimalSurface.html", ButtonStyle->"Hyperlink"], ".\n\nThe ", StyleBox["nHelicoid", FontWeight->"Bold"], " function accepts six arguments, of which the most important are ", StyleBox["n", FontWeight->"Bold"], " and ", StyleBox["pitch", FontWeight->"Bold"], ", where ", StyleBox["n", FontWeight->"Bold"], " dictates the fineness of the mesh of rhombs, while ", StyleBox["pitch", FontWeight->"Bold"], " determines the pitch or slope of the helicoid, 0<", StyleBox["pitch", FontWeight->"Bold"], "<90 degrees. Additional arguments are ", StyleBox["factor", FontWeight->"Bold"], ", ", StyleBox["ratio", FontWeight->"Bold"], ", ", StyleBox["ilim", FontWeight->"Bold"], ", and ", StyleBox["jlim", FontWeight->"Bold"], ". The examples below illustrate the roles of these variables in the ", StyleBox["nHelicoid", FontWeight->"Bold"], " function." }], "Text"], Cell[CellGroupData[{ Cell["Code", "Subsection"], Cell["\<\ vectors[n_Integer,pitch_,factor_Integer,ratio_]:= Block[{e}, Table[If[Mod[i,factor]==0,e=ratio,e=1]; \tN[{e Cos[Degree pitch] Cos[2Pi i/n], \te Cos[Degree pitch] Sin[2Pi i/n], \te Sin[Degree pitch]}], \t{i,0,n-1}]] vectors[n_:12,pitch_:35.2643,factor_:1,ratio_:1]:= vectors[n, pitch, factor, ratio] \t quad[x_,ref_]:={ref, ref+x[[1]], ref+x[[1]]+x[[2]], mark=ref+x[[2]]} nHelicoid[n_Integer, pitch_, factor_, ratio_, ilim_, jlim_]:= Block[{v=vectors[n,pitch,factor,ratio],h=1.5*(Plus @@ \ vectors[n,pitch,factor,ratio]),q}, \tmark=-h; \tq=Table[ \t\ta=Table[quad[{v[[ 1+Mod[j-1,n] ]],v[[ 1+Mod[i+j-1,n] ]]},mark],{i,ilim}]; \t\tmark=First[a][[3]];(*Print[\"mark=\",mark];*)a, \t\t{j,jlim}]; \tGraphics3D[ Polygon /@ Cases[q ,{{_,_,_}..},Infinity] ] ] nHelicoid[n_:12,pitch_:35.2643,factor_:1,ratio_:1, ilim_:11, jlim_:12]:= nHelicoid[n, pitch, factor, ratio, ilim, jlim]\ \>", "Input", InitializationCell->True], Cell["\<\ mag[v_]:=Sqrt[v.v](*magnitude of vector 'v'*) unit[v_]:=v/mag[v](*unit vector in direction of 'v'*) cent[v_List]:=Plus@@v/Length[v](*centroid of point list*) mycross[ {ax_, ay_, az_}, {bx_, by_, bz_} ] := \t{ay bz - az by, az bx - ax bz, ax by - ay bx}(*cross product*) tosegs[gon_]:=Transpose[ {gon, RotateLeft[gon]} ](*polygon to edges*) (*area of parallelogram*) sz[f_List] := mag[mycross[ f[[2]] - f[[1]], f[[3]] - f[[2]] ]] (* treat 4gons specially for efficiency *) totalarea[v_List] /; Length[v] == 4 := \tsz[v] (* treat triangles specially for efficiency *) totalarea[v_List] /; Length[v] == 3 := \tsz[v]/2 \t (* general case: use polygon center as additional vertex *) totalarea[v_List] := Module[{bary = cent[v], circ, t}, circ = tosegs[v]; t=Apply[ {#1, #2, bary}&, circ, {1} ]; Sum[sz[t[[i]]]/2,{i,Length[t]}] ] (*break Graphics3D down into Headless polygons*) parse[poly_] := Cases[Level[poly,{-3}],{{_,_,_}..}] colorSort[poly_]:=(*~30* faster than old version*) Block[{q1,q2,p}, p=parse[poly]; q1=Table[{p[[i]],totalarea[ p[[i]] ] },{i,Length[p]}]; q2=Split[Sort[q1,#1[[2]]<#2[[2]]&], TrueQ[(Abs[#1[[2]]-#2[[2]]]<.0001)]&]; Print[\"You will need at least \",Length[q2], \" colors in the colorlist below.\"];(*how many colors?*) Table[Cases[q2[[i]],{{_,_,_}..},Infinity],{i,Length[q2]}] ]\ \>", "Input", InitializationCell->True], Cell["\<\ trans[shape_, vec_List] := Block[{t = N[vec]}, shape /. { poly:Polygon[_] :> Map[(t + #)&, poly, {2}], ln:Line[_] :> Map[(t + #)&, ln, {2}], pt:Point[_] :> Map[(t + #)&, pt,{1}] } ]\ \>", "Input", InitializationCell->True], Cell["\<\ vunion::usage = \"vunion[v] returns the union of a list of \ n-vectors.\"; vunion[v_]:= Union[Chop[v], SameTest->((Sign[#1]==Sign[#2]) && (Chop[#1-#2]==Table[0,{Length[v[[1]]]}]) \ &)]\ \>", "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Examples", "Subsection"], Cell[TextData[{ "A somewhat coarse mesh of rhombs (", StyleBox["n", FontWeight->"Bold"], "=12) is used to compare low, medium, and high pitch angles. Since 12 is \ the default for ", StyleBox["n", FontWeight->"Bold"], ", the last four arguments to the ", StyleBox["nHelicoid", FontWeight->"Bold"], " function are not needed.", " " }], "Text"], Cell["\<\ (*low pitch: 10 degrees*) n=12; Show[nHelicoid[n, 10], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-2,0}]]; (*medium pitch: 35 degrees*) Show[nHelicoid[n, 35], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-2,0}]]; (*high pitch: 60 degrees*) Show[nHelicoid[n, 60], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-2,0}]];\ \>", "Input"], Cell[TextData[{ "The third and fourth arguments, ", StyleBox["factor", FontWeight->"Bold"], " and ", StyleBox["ratio", FontWeight->"Bold"], ", are set by default to unity. Every ", StyleBox["factor", FontWeight->"Bold"], "-th of the ", StyleBox["n", FontWeight->"Bold"], " unit vectors is multiplied by ", StyleBox["ratio", FontWeight->"Bold"], "." }], "Text"], Cell["\<\ (*every 4th vector multiplied by 3*) n=24; Show[nHelicoid[n, 35, 4, 3, n-1, n], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-1,2}]];\ \>", "Input"], Cell["\<\ (*every other vector multiplied by 3: lattice-like*) n=24; Show[nHelicoid[n, 35, 2, 3, n-1, n], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-1,1}]];\ \>", "Input"], Cell[TextData[{ "The fifth and sixth arguments, ", StyleBox["ilim", FontWeight->"Bold"], " and ", StyleBox["jlim", FontWeight->"Bold"], ", are set by default to ", StyleBox["n", FontWeight->"Bold"], "-1 and ", StyleBox["n", FontWeight->"Bold"], ". These limit the loops in which the helicoid surface is developed." }], "Text"], Cell["\<\ (*lengthen helicoid*) n=24; Show[nHelicoid[n, 35, 1, 1, n-1, 3n], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-1,2}]];\ \>", "Input"], Cell["\<\ (*shorten helicoid*) n=24; Show[nHelicoid[n, 35, 1, 1, n-1, 12], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-1,2}]];\ \>", "Input"], Cell["\<\ Here, we chop the helicoid so that it is confined within a range of \ z coordinates.\ \>", "Text"], Cell["\<\ (*before chopping*) n=24; Show[a=nHelicoid[n, 35, 1, 1, n-1, n], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-1,0}]];\ \>", "Input"], Cell["\<\ (*chop and display*) allpoly=Cases[a,Polygon[x_]:>x,Infinity]; subpoly=Select[allpoly, Abs[Max[Map[#[[3]]&, #]]]<14&]; Show[Graphics3D[Polygon/@subpoly], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,-1,0}]];\ \>", "Input"], Cell["\<\ Here, we translate the helicoid so that it is centered upon the \ origin.\ \>", "Text"], Cell["\<\ (*not centered*) n=12; Show[a=nHelicoid[n, 35, 1, 1, n-1, n], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,0,1}]];\ \>", "Input"], Cell["\<\ v=vunion[ Flatten[ Cases[a,Polygon[x_]:>x,Infinity], 1 ] ]; c=Plus@@v/Length[v]; Show[trans[a,-c], Axes->True, Boxed->True, ViewPoint->2.75*unit[{0,0,1}]];\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Colored Helicoids", "Subsection"], Cell[TextData[{ "The structure of the rhombic mesh is revealed if one colors the rhombs \ acording to surface area. This is done using the ", StyleBox["colorSort", FontWeight->"Bold"], " function. A list of random colors is then constructed, and the colored \ helicoid displayed. 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{Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->7, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageMargins->{{38, Inherited}, {Inherited, 0}}, Magnification->0.6], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a cell \ exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, 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FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "EnhancedPrintout"], StyleMenuListing->None, FontFamily->"Futura", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", 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The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { 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FontSize->11], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ 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FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder objects in \ palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "Condensed"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles of most \ cells. If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an inline cell \ within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display of certain \ types of objects in typeset expressions. For example, \"UnmatchedBracket\" \ style defines how unmatched bracket, curly bracket, and parenthesis \ characters are displayed (typically by coloring them to make them stand out).\ \ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 34, 0, 105, "Title"], Cell[1776, 53, 81, 5, 107, "Subsubtitle"], Cell[1860, 60, 1356, 42, 198, "Text"], Cell[CellGroupData[{ Cell[3241, 106, 26, 0, 47, "Subsection"], Cell[3270, 108, 926, 28, 480, "Input", InitializationCell->True], Cell[4199, 138, 1395, 45, 786, "Input", InitializationCell->True], Cell[5597, 185, 249, 9, 138, "Input", InitializationCell->True], Cell[5849, 196, 238, 9, 102, "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[6124, 210, 30, 0, 31, "Subsection"], Cell[6157, 212, 370, 13, 54, "Text"], Cell[6530, 227, 357, 19, 282, "Input"], Cell[6890, 248, 402, 17, 54, "Text"], Cell[7295, 267, 162, 7, 102, "Input"], Cell[7460, 276, 178, 7, 102, "Input"], Cell[7641, 285, 360, 14, 52, "Text"], Cell[8004, 301, 148, 7, 102, "Input"], Cell[8155, 310, 147, 7, 102, "Input"], Cell[8305, 319, 108, 3, 32, "Text"], Cell[8416, 324, 147, 7, 102, "Input"], Cell[8566, 333, 236, 8, 117, "Input"], Cell[8805, 343, 97, 3, 32, "Text"], Cell[8905, 348, 143, 7, 102, "Input"], Cell[9051, 357, 180, 7, 102, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[9268, 369, 39, 0, 31, "Subsection"], Cell[9310, 371, 303, 7, 55, "Text"], Cell[9616, 380, 180, 7, 120, "Input"], Cell[9799, 389, 92, 3, 48, "Input"], Cell[9894, 394, 683, 12, 426, "Input"] }, Closed]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)