(************** 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[ 229418, 9833]*) (*NotebookOutlinePosition[ 230141, 9858]*) (* CellTagsIndexPosition[ 230097, 9854]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Code", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "The main commands are GraphicsIFS[", StyleBox["system", FontSlant->"Italic"], ",", StyleBox["starting picture", FontSlant->"Italic"], ",", StyleBox["number of iterations", FontSlant->"Italic"], "] and RandomIFS[", StyleBox["system", FontSlant->"Italic"], ",", StyleBox["number of points", FontSlant->"Italic"], "] together with the \"try a system\" commands which are:\n\tCurlicue[]\n\t\ SierpinskiTriangle[]\n\tSierpinskiCarpet[]\n\tCantorSquare[]\nAlso one can \ generate a picture of your own system with AskASystem[", StyleBox["number of maps", FontSlant->"Italic"], "]", "\n\t" }], "Subsubsection"], Cell[BoxData[{ \(Off[General::spell]\), "\[IndentingNewLine]", \(Off[General::spell1]\)}], "Input", InitializationCell->True], Cell[BoxData[ \(<< LinearAlgebra`MatrixManipulation`\)], "Input", InitializationCell->True], Cell[BoxData[{ \(\(\(Clear[NMaps, Maps, Dim]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(NMaps[IFS[nmaps_Integer, As_List, bs_List]] := nmaps;\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(Maps[IFS[nmaps_Integer, As_List, bs_List], m_Integer, pos_List] := As\[LeftDoubleBracket]m\[RightDoubleBracket] . pos + bs\[LeftDoubleBracket]m\[RightDoubleBracket]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Dim[IFS[nmaps_Integer, As_List, bs_List]]\ := Length[bs\[LeftDoubleBracket]1\[RightDoubleBracket]]\), "\ \[IndentingNewLine]", \(\)}], "Input", InitializationCell->True], Cell[BoxData[{ \(\(\(Clear[DoRandomMap, RandomIFS, Converter, RandomWithDistribution]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Converter[probs_List]\ := \ FoldList[Plus, 0, probs]\), "\[IndentingNewLine]", \(RandomWithDistribution[dist_List]\ := \ Module[{rand}, rand = Random[]; Length[Select[dist, \((rand > #\ &)\)]]]\), "\[IndentingNewLine]", \(DoRandomMap[sys_IFS, pos_List]\ := Maps[sys, Random[Integer, {1, NMaps[sys]}], pos]\), "\[IndentingNewLine]", \(\(\(DoRandomMap[sys_IFS, dist_List, pos_List]\ := Maps[sys, RandomWithDistribution[dist], pos]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Options[RandomIFS]\ = \ {DropFirst \[Rule] 1, Probs \[Rule] {}, DropInterval \[Rule] {}};\)\), "\[IndentingNewLine]", \(RandomIFS[sys_IFS, steps_Integer, opts___]\ := \ Module[{probs, dropfirst, repeater, dropcoords, dropper}, dropfirst = \(DropFirst\ /. \ {opts}\)\ /. \ Options[RandomIFS]; probs = \(Probs /. {opts}\) /. Options[RandomIFS]; dropcoords\ = \ \(DropInterval\ /. \ {opts}\)\ /. \ Options[RandomIFS]; If[probs \[Equal] {}, repeater = \((DoRandomMap[sys, #]\ &)\), repeater = \((DoRandomMap[sys, Converter[probs], #] &)\)]; If[dropcoords \[Equal] {}, ShowStraight[ Graphics[ Point\ /@ \ Drop[NestList[repeater, Table[Random[], {2}], steps], dropfirst]]], dropper = \((Drop[#, dropcoords]\ &)\); ShowStraight[ Graphics[ Point\ /@ \ Drop[dropper\ /@ \ NestList[repeater, Table[Random[], {Dim[sys]}], steps], dropfirst]]]]]\)}], "Input", InitializationCell->True], Cell[BoxData[{ \(\(\(Clear[ApplyIFS, ApplyMap, DropFrom, GraphicsIFS, ShowStraight];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(DropFrom[Line[l_List], drop_]\ := \ Line[\((Drop[#, drop] &)\) /@ l]\), "\[IndentingNewLine]", \(DropFrom[Polygon[l_List], drop_] := Polygon[\((Drop[#, drop] &)\) /@ l]\), "\[IndentingNewLine]", \(\(\(DropFrom[Point[l_List], drop_] := Point[\((Drop[#, drop] &)\) /@ l]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(ApplyMap[sys_IFS, m_Integer, Line[poss_List]]\ := \ Line[\((Maps[sys, m, #] &)\) /@ poss]\), "\[IndentingNewLine]", \(ApplyMap[sys_IFS, m_Integer, Polygon[poss_List]] := Polygon[\((Maps[sys, m, #] &)\) /@ poss]\), "\[IndentingNewLine]", \(ApplyMap[sys_IFS, m_Integer, Point[pos_List]] := Point[Maps[sys, m, pos]]\), "\[IndentingNewLine]", \(\(\(ApplyMap[sys_IFS, m_Integer, l_List] := \((ApplyMap[sys, m, #]\ &)\) /@ \ l\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(ApplyIFS[sys_IFS, l_List]\ := \ Join\ @@ \ \(\((ApplyIFS[sys, #]\ &)\)\ /@ \ l\)\), "\[IndentingNewLine]", \(\(\(ApplyIFS[sys_IFS, item_]\ := \ Table[ApplyMap[sys, m, item], {m, NMaps[sys]}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Options[ GraphicsIFS] = {DropInterval \[Rule] {}};\)\), \ "\[IndentingNewLine]", \(\(\(GraphicsIFS[sys_IFS, init_, steps_Integer, opts___]\ := \ Module[{dropcoords}, dropcoords\ = \ \(DropInterval\ /. \ {opts}\)\ /. \ Options[RandomIFS]; If[dropcoords \[Equal] {}, Graphics[Nest[\((ApplyIFS[sys, #]\ &)\), init, steps]]\ // ShowStraight, dropper\ = \ \((DropFrom[#, dropcoords] &)\); Graphics[ dropper\ /@ \ Nest[\((ApplyIFS[sys, #]\ &)\), init, steps]]\ // ShowStraight]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(Graphics3DIFS[sys_IFS, init_, steps_Integer]\ := \ Graphics3D[Nest[\((ApplyIFS[sys, #]\ &)\), init, steps]]\ // \ ShowStraight\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(ShowStraight[args__]\ := \ Show[args, AspectRatio \[Rule] 1, PlotRange \[Rule] All]\)}], "Input", InitializationCell->True], Cell[BoxData[{ \(\(sq\ = \ Line[{{0, 0}, {1, 0}, {1, 1}, {0, 1}, {0, 0}}];\)\), "\[IndentingNewLine]", \(\(cross\ = \ {Line[{{0.5, 0}, {0.5, 1}}], Line[{{0, 0.5}, {1, 0.5}}]};\)\), "\[IndentingNewLine]", \(\(nhappy\ = \ 20;\)\), "\[IndentingNewLine]", \(\(happy\ = \ {Line[ Table[{0.5, 0.5} + 0.5 {Cos[2 Pi\ i/nhappy], Sin[2 Pi\ i/nhappy]}, {i, 0, nhappy}]], Line[Table[{0.5, 0.5} + 0.25 {Cos[2 Pi\ i/nhappy], Sin[2 Pi\ i/nhappy]}, {i, nhappy/2, nhappy}]], Line[Table[{0.3, 0.7} + 0.1 {Cos[2 Pi\ i/nhappy], Sin[2 Pi\ i/nhappy]}, {i, 0, nhappy}]], Line[Table[{0.7, 0.7} + 0.1 {Cos[2 Pi\ i/nhappy], Sin[2 Pi\ i/nhappy]}, {i, 0, nhappy}]]};\)\), "\[IndentingNewLine]", \(\(block\ = \ Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}, {0, 0}}];\)\), "\[IndentingNewLine]", \(\(horiz = Line[{{0, 0}, {1, 0}}];\)\), "\[IndentingNewLine]", \(\(spike = Line[{{0, 0}, {0.5, 1}, {1, 0}}];\)\), "\[IndentingNewLine]", \(\(diag = Line[{{0, 0}, {1, 1}}];\)\), "\[IndentingNewLine]", \(\(flatline = Line[{{0, 0, 0}, {1, 1, 0}}];\)\), "\[IndentingNewLine]", \(\(bentline = Line[{{0, 0, 0}, {0.5, 0.5, 1}, {1, 1, 0}}];\)\), "\[IndentingNewLine]", \(\(diagline = Line[{{0, 0, 0}, {1, 1, 1}}];\)\)}], "Input", InitializationCell->True], Cell[BoxData[{ \(\(\(Clear[Interpolating, Threed];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(Interpolating[x_List, vscales_List]\ := Module[{a, e, c, f, n, i}, n = Length[x]; a = Table[ x\[LeftDoubleBracket]i + 1, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]i, 1\[RightDoubleBracket], {i, n - 1}]/\((x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket])\); e = Table[ x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] x\[LeftDoubleBracket]i - 1, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket] x\[LeftDoubleBracket]i, 1\[RightDoubleBracket], {i, 2, n}]/\((x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket])\); c = Table[ x\[LeftDoubleBracket]i + 1, 2\[RightDoubleBracket] - x\[LeftDoubleBracket]i, 2\[RightDoubleBracket] - vscales\[LeftDoubleBracket] i\[RightDoubleBracket] \((x\[LeftDoubleBracket]n, 2\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 2\[RightDoubleBracket])\), {i, n - 1}]/\((x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket])\); f = Table[ x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] x\[LeftDoubleBracket]i - 1, 2\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket] x\[LeftDoubleBracket]i, 2\[RightDoubleBracket] - vscales\[LeftDoubleBracket] i - 1\[RightDoubleBracket] \((x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] x\[LeftDoubleBracket]1, 2\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket] x\[LeftDoubleBracket]n, 2\[RightDoubleBracket])\), {i, 2, n}]/\((x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket])\); Print[a, c, e, f]; IFS[n - 1, Table[{{a\[LeftDoubleBracket]i\[RightDoubleBracket], 0}, {c\[LeftDoubleBracket]i\[RightDoubleBracket], vscales\[LeftDoubleBracket]i\[RightDoubleBracket]}}, {i, n - 1}], Table[{e\[LeftDoubleBracket]i\[RightDoubleBracket], f\[LeftDoubleBracket]i\[RightDoubleBracket]}, {i, n - 1}]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Threed[x_List, H_List, vmaps_List]\ := Module[{a, c, k, e, f, g, n, xrange, Frange, Hrange}, n = Length[x]; xrange = x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket]; Frange = x\[LeftDoubleBracket]n, 2\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 2\[RightDoubleBracket]; Hrange = H\[LeftDoubleBracket]n\[RightDoubleBracket] - H\[LeftDoubleBracket]1\[RightDoubleBracket]; a = Table[ x\[LeftDoubleBracket]i + 1, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]i, 1\[RightDoubleBracket], {i, n - 1}]/ xrange; c = Table[x\[LeftDoubleBracket]i + 1, 2\[RightDoubleBracket] - x\[LeftDoubleBracket]i, 2\[RightDoubleBracket] - vmaps\[LeftDoubleBracket]i, 1, 1\[RightDoubleBracket]* Frange - vmaps\[LeftDoubleBracket]i, 1, 2\[RightDoubleBracket]* Hrange, {i, n - 1}]/xrange; k = Table[ H\[LeftDoubleBracket]i + 1\[RightDoubleBracket] - H\[LeftDoubleBracket]i\[RightDoubleBracket] - vmaps\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket]* Frange - vmaps\[LeftDoubleBracket]i, 2, 2\[RightDoubleBracket]* Hrange, {i, n - 1}]/xrange; \ e = Table[ x\[LeftDoubleBracket]n, 1\[RightDoubleBracket] x\[LeftDoubleBracket]i, 1\[RightDoubleBracket] - x\[LeftDoubleBracket]1, 1\[RightDoubleBracket] x\[LeftDoubleBracket]i + 1, 1\[RightDoubleBracket], {i, n - 1}]/xrange; f = Table[ x\[LeftDoubleBracket]i, 2\[RightDoubleBracket] - vmaps\[LeftDoubleBracket]i, 1, 1\[RightDoubleBracket] x\[LeftDoubleBracket]1, 2\[RightDoubleBracket] - vmaps\[LeftDoubleBracket]i, 1, 2\[RightDoubleBracket] H\[LeftDoubleBracket]1\[RightDoubleBracket] - c\[LeftDoubleBracket]i\[RightDoubleBracket] x\[LeftDoubleBracket]1, 1\[RightDoubleBracket], {i, n - 1}]; g = Table[ H\[LeftDoubleBracket]i\[RightDoubleBracket] - vmaps\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket] x\[LeftDoubleBracket]1, 2\[RightDoubleBracket] - vmaps\[LeftDoubleBracket]i, 2, 2\[RightDoubleBracket] H\[LeftDoubleBracket]1\[RightDoubleBracket] - k\[LeftDoubleBracket]i\[RightDoubleBracket] x\[LeftDoubleBracket]1, 1\[RightDoubleBracket], {i, n - 1}]; IFS[n - 1, Table[{{a\[LeftDoubleBracket]i\[RightDoubleBracket], 0, 0}, {c\[LeftDoubleBracket]i\[RightDoubleBracket], vmaps\[LeftDoubleBracket]i, 1, 1\[RightDoubleBracket], vmaps\[LeftDoubleBracket]i, 1, 2\[RightDoubleBracket]}, {k\[LeftDoubleBracket]i\ \[RightDoubleBracket], vmaps\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket], vmaps\[LeftDoubleBracket]i, 2, 2\[RightDoubleBracket]}}, {i, n - 1}], Table[{e\[LeftDoubleBracket]i\[RightDoubleBracket], f\[LeftDoubleBracket]i\[RightDoubleBracket], g\[LeftDoubleBracket]i\[RightDoubleBracket]}, {i, n - 1}]]]\)}], "Input", InitializationCell->True], Cell[BoxData[{ \(Clear[AskASystem, CantorSquare, SierpinskiCarpet, SierpinskiTriangle, Curlicue, ExtractConstant]\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(strings\ = \ {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", \ "\"};\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(ExtractConstant[poly_]\ := \ Module[{coeffs}, coeffs\ = \ CoefficientList[poly, {x, y}]; If[Length[coeffs] == 0, Return[0], If[Length[ coeffs\[LeftDoubleBracket]1\[RightDoubleBracket]] \[Equal] 0, 0, coeffs\[LeftDoubleBracket]1, 1\[RightDoubleBracket]]]]\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(AskASystem[n_]\ := \ Module[{inputs, coeffs, A, b, sys}, inputs\ = \ Table[0, {n}, {2}]; coeffs = Table[0, {2}]; A = Table[0, {n}]; b = Table[0, {n}]; For[i = 1, i \[LessEqual] n, \(i++\), For[j = 1, j \[LessEqual] 2, \(j++\), inputs\[LeftDoubleBracket]i, j\[RightDoubleBracket] = Input["\" <> strings\[LeftDoubleBracket] j\[RightDoubleBracket] <> "\< component of the \>" <> strings\[LeftDoubleBracket] i\[RightDoubleBracket] <> "\< map?\>"]]; \ \[IndentingNewLine]A\[LeftDoubleBracket] i\[RightDoubleBracket] = {{Coefficient[ inputs\[LeftDoubleBracket]i, 1\[RightDoubleBracket], x], Coefficient[ inputs\[LeftDoubleBracket]i, 1\[RightDoubleBracket], y]}, {Coefficient[ inputs\[LeftDoubleBracket]i, 2\[RightDoubleBracket], x], Coefficient[ inputs\[LeftDoubleBracket]i, 2\[RightDoubleBracket], y]}}; b\[LeftDoubleBracket]i\[RightDoubleBracket] = {ExtractConstant[ inputs\[LeftDoubleBracket]i, 1\[RightDoubleBracket]], ExtractConstant[ inputs\[LeftDoubleBracket]i, 2\[RightDoubleBracket]]}]; sys = IFS[n, A, b]; For[i = 1, i \[LessEqual] 5, \(i++\), GraphicsIFS[sys, block, i]; Pause[3]]; GraphicsIFS[sys, block, 6]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(SierpinskiTriangle[]\ := \ AskASystem[3]\), "\[IndentingNewLine]", \(CantorSquare[] := AskASystem[4]\), "\[IndentingNewLine]", \(SierpinskiCarpet[] := AskASystem[8]\), "\[IndentingNewLine]", \(Curlicue[] := AskASystem[3]\), "\[IndentingNewLine]", \(\)}], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(ExtractConstant[0]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Curlicue[]\)], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind 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