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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[ 1236691, 45204]*) (*NotebookOutlinePosition[ 1237632, 45233]*) (* CellTagsIndexPosition[ 1237588, 45229]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ \(usage : \ \(call : \ \ prog[{{2, 1}, {3, 1}, {1}}, 8]\);\)\ \ *) \)\(\ \)\(\[IndentingNewLine]\)\( \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \)\(by\)\)\ \ Jarek\ Kwapisz, \ 8\ Feb\ 2006\ ** ** ** ** *\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)\ *****) \)\(\[IndentingNewLine]\)\( (*\(\(\(**\)\(**\)\)*\ \ \(NOTE : \ this\ is\ very\ inefficient\)\)\(,\)\ \ ********************************************) \)\( (*\(\(**\)\(**\)\)*\ \ \ adapted\ from\ a\ larger\ program\ and\ never\ optimized\ ** ** ** ** *\(\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*****) \)\(\[IndentingNewLine]\)\( (*\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)*****) \)\(\[IndentingNewLine]\)\ \( (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*\ PRELIMINARY\ LIN\ ALG\ ** ** ** ** ** ** ** ** ** ** \ ** *****) \)\(\[IndentingNewLine]\)\( \ (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)*****) \)\(\ \[IndentingNewLine]\)\(<< LinearAlgebra`; \ << Graphics`;\ \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\) ** \ ** *\ matrix\ of\ the\ subst\ ** ** ** ** ** ** ** ** ** ** ** ** \ ** ** *****) \[IndentingNewLine] \(matrix[subst_]\ := Table[\ \ \ \ \ \ \ \ \ \ \ \[IndentingNewLine]Count[\ subst[\([\ \ i\ ]\)], j\ ], \ {j, 1, Length[subst]}, \ {i, 1, Length[subst]}];\)\[IndentingNewLine] \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \)\(preliminary\)\)\ info\ \ on\ matrix\ M\ and\ reated\ global\ definitions\ \ ** ** \ ** ******) \[IndentingNewLine] \(prelimlinalg[M_]\ := \ Module[{}, \ \ \ \[IndentingNewLine]Print["\<********************** \ prelim lin alg info: **********************\>"]; \[IndentingNewLine]d\ = \ Length[M]; \ \ \ (*\ global\ variable\ set\ *) \[IndentingNewLine]Zero = Table[\ 0, {j, 1, d}]; \ (*\ global\ variable\ set\ *) \[IndentingNewLine]Ident\ = Table[\ If[i == j, 1, 0], {i, 1, d}, {j, 1, d}]; (*\ global\ variable\ set\ *) \[IndentingNewLine] (*\(\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*\ BASIC\ STATES\ ** ** ** ** ** ** ** ** ** ** ** ** \ ** *****) \[IndentingNewLine]\ e[i_] := Table[\ If[j == i, \ 1, \ 0], {j, 1, d}]; \ (*\ basis\ vectors\ *) \ \[IndentingNewLine] (*\(\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*\ ABSTRACT\ DUAL\ STATE\ \((as\ list)\)\ ** ** ** ** ** ** \ ** ** ** ** ** *****) \ \[IndentingNewLine]\ dual[i_]\ := \ Delete[Table[e[j], \ {j, 1, d}], {i}]; \ \ (*\ list\ of\ e[j]' s\ except\ for\ e[i]\ *) \[IndentingNewLine]B\ = \ Inverse[M]; \ \ (*\ global\ variable\ set\ *) \[IndentingNewLine]AA[0] = Ident; \ \[IndentingNewLine]AA[n_] := \ \(AA[n]\ = \ M . AA[n - 1]\); \[IndentingNewLine]BB[0] = \ Ident; \[IndentingNewLine]\ BB[n_] := \ \(BB[n]\ = \ B . BB[n - 1]\); \[IndentingNewLine]{eigenval, \ eigenvect}\ = \ N[\ Eigensystem[M]\ ]; \[IndentingNewLine]lambda\ = \ Max[Abs[eigenval]]\ ; \ (*\ leading\ Perron\ eigenvalue\ *) \[IndentingNewLine]omega\ = \ For[k = 1, k \[LessEqual] Length[eigenvect], \ \(k++\), \ If[eigenval[\([k]\)] == lambda, Break[\ Abs[ eigenvect[\ \([k]\)]\ ]\ \ \ \ ]\ ]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ ]; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ eigenvector\ of\ lambda; \ positive\ *) \[IndentingNewLine]Print["\", N[lambda], "\< omega =\>", \ N[omega]\ ];\ \[IndentingNewLine]];\)\ (*\ end\ module\ of\ preliminary\ linear\ algebra\ function\ \ *) \[IndentingNewLine] (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\ *****) \[IndentingNewLine] (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(\ \)\(SUBST\)\)\ SPEC\ \(DEF : \ \(\(**\)\(**\)\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*\(\(**\)\(**\)\(**\)\(**\)\(**\)\ \)\)******) \[IndentingNewLine] \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \ \ \)\(iterations\)\)\ of\ subst\ ****************) \[IndentingNewLine] \(ss[subst_, 0]\ := Table[{ii}, {ii, 1, d}];\)\[IndentingNewLine] \(ss[subst_, n_]\ := \ \ \ \ (*\ n - th\ iteration\ of\ s\ as\ a\ list\ of\ lists\ representing\ \ \\phi^n \((i)\)\ *) \ \(ss[subst, n]\ = \ \ \ \ (*\ this\ form\ of\ recursion\ remembers\ intermediate\ \(values\ !\)\ \ per\ 2.4 .9\ in\ the\ book\ *) \[IndentingNewLine]Module[\ \ \[IndentingNewLine]{x, y, z, \ k, j, L}, \[IndentingNewLine]z = {}; \ \ \ (*\ null\ initialize\ *) \[IndentingNewLine]For[\ \ \ \ \ \ \ \ \ \ (*\ loop\ over\ alphabet\ *) \[IndentingNewLine]j = 1, \ j\ < \ d + 1, \ \(j++\), \ \ \[IndentingNewLine]x\ = \ \(ss[subst, n - 1]\)[\([j]\)]; \ \[IndentingNewLine]L\ = \ Length[x]; \ \ (*\ the\ lenght\ of\ \(s[ n - 1]\)[\([j]\)]\ *) \[IndentingNewLine]\ (*\ Print[L]; \ \ test\ *) \[IndentingNewLine]y = {}\ ; \ \ \ \ (*\ null\ initialize\ *) \[IndentingNewLine]\ \ For[\ \[IndentingNewLine]\ \ \ k = 1, k\ < \ L\ + 1, \(k++\), \[IndentingNewLine]\ \ y = \ Join\ [\ y, subst[\([\ x[\([k]\)]\ \ \ ]\)]\ ]; \ \ \ (*\ substitute\ the\ first\ element\ *) \[IndentingNewLine] (*\ Print[y]; \ test\ *) \ \[IndentingNewLine]\ ]; \ \[IndentingNewLine]z = Append[z, y];\[IndentingNewLine]]; \[IndentingNewLine]z]\);\)\ (*\ end\ ss[ n]\ module\ *) \[IndentingNewLine] (*\(**\)\(**\)\(**\)\(**\)\(**\)\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\ \(**\)\(**\)\(**\)*****) \[IndentingNewLine] (*\(\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(\ \)\(PRELIM\)\)\ ANALYSIS\ OF\ SUBST\ \ ** ** \ ** ** ** ** ** ** ** *****) \[IndentingNewLine] (*\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)*****) \[IndentingNewLine] \(prelim\ [subst_]\ := Module[{}, \ \[IndentingNewLine]Print["\<********************** \ PRELIM INFO: **********************\>"]; \[IndentingNewLine]A\ = \ matrix[subst]; \ \ \ (*\ global\ variable\ set\ *) \[IndentingNewLine]prelimlinalg[ A];\[IndentingNewLine]];\)\ (*\ end\ prelim\ module\ *) \[IndentingNewLine] (*\(\(**\)\(**\)\) ** ** ** \ ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** \ ** ******************************) \[IndentingNewLine] (*\(\(**\)\(**\)\(**\)\ \(**\)\)*\ STATE\ LIST\ DISPLAY\ PANNEL\ \((GOOD\ in\ 3 D\ only)\)\ *****************) \ \[IndentingNewLine] (*\(\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*\ uses\ global\ variable\ d\ as\ argument\ ** ** ** ** **\ *****) \[IndentingNewLine] (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\ *****) \[IndentingNewLine] (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\)*\ DUAL\ FACE\ AS\ POLYGON\ \((with\ offset\ )\) ** ** ** ** ** ** \ ** *****) \[IndentingNewLine] \(face[{position_, i_}]\ := \ \ \ (*\ dual\ to\ e[ i]\ face\ at\ position\ as\ Polygon\ type\ \ *) \[IndentingNewLine]Module[{x, ff, tip}, \ \[IndentingNewLine]tip\ = \ position\ + e[i]; \ \ (*\ MOD\ from\ tip\ = \ position\ *) \ (*\ this\ is\ the\ offset\ *) \[IndentingNewLine]x\ = \ dual[i]; \[IndentingNewLine]If[ d \[Equal] 3, \[IndentingNewLine]ff\ = \ Polygon[\ {tip, \ tip + x[\([1]\)], tip + x[\([1]\)] + x[\([2]\)], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tip + x[\([2]\)], \ tip}]\ ]; \[IndentingNewLine]If[d \[Equal] 2, ff\ = \ Line[\ {tip, \ tip + x[\([1]\)]}\ ]\ ]; \[IndentingNewLine]ff];\)\ (*\ end\ module\ *) \[IndentingNewLine] \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*\ DUAL\ FACE\ AS\ POLYGON\ \((no\ offset)\)\ ** ** ** ** \(\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*****) \[IndentingNewLine] \(rauzyface[{position_, i_}]\ := \ \ \ (*\ dual\ to\ e[ i]\ face\ at\ position\ as\ Polygon\ type\ \ *) \[IndentingNewLine]Module[{x, ff, tip}, \ \[IndentingNewLine]tip\ = \ position\ ; \ \ (*\ MOD\ from\ tip\ = \ position\ *) \[IndentingNewLine]x\ = \ dual[i]; \[IndentingNewLine]If[ d \[Equal] 3, \[IndentingNewLine]ff\ = \ Polygon[\ {tip, \ tip + x[\([1]\)], tip + x[\([1]\)] + x[\([2]\)], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tip + x[\([2]\)], \ tip}]\ ]; \[IndentingNewLine]If[d \[Equal] 2, ff\ = \ Line[\ {tip, \ tip + x[\([1]\)]}\ ]\ ]; \[IndentingNewLine]ff];\)\ (*\ end\ module\ *) \[IndentingNewLine] (*\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)***********************************) \[IndentingNewLine] (*\(\(**\)\(**\)\ \(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \)\(RAUZY\)\)\ PICTURE\ \ VIA\ FORWARD\ TRANSLATING\ *********************) \[IndentingNewLine] \ (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)*****) \[IndentingNewLine]\ \(forwardtranslates[subst_, 0, i_]\ := \ {face[{Zero, i}]};\)\[IndentingNewLine] \(forwardtranslates[subst_, n_, i_]\ := \ \(forwardtranslates[subst, n, i]\ = \ (*\ list\ of\ dual\ to\ e[ i]\ faces\ translated\ along\ A^{\(-n\)}\\phi^ n \((i)\)\ *) \ \[IndentingNewLine]Module[\ \[IndentingNewLine]{x, image, \ position, \ k, L}, \[IndentingNewLine]image\ = \ {}; \ \[IndentingNewLine]position\ = \ {0, \ 0, 0}; \ \ (*\ MOD\ to\ something\ beyond\ d\ = 2\ *) \ \[IndentingNewLine]x\ = \ \(ss[subst, n]\)[\([i]\)]; L\ = \ Length[x]; \[IndentingNewLine]For[\[IndentingNewLine]k = 1, \ k\ < \ L + 1, \ \(k++\), \[IndentingNewLine]image\ = \ Append[image, \ rauzyface[{BB[n] . position, i}]]\ ; \[IndentingNewLine]position\ = \ position\ + \ e[\ x[\([k]\)]\ ]\ \ \ ;\ \[IndentingNewLine]]; \ \[IndentingNewLine]image\[IndentingNewLine]]\);\)\ \ \ (*\ end\ module\ *) \[IndentingNewLine] \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(\ \)\(Approximation\)\)\ to\ Rauzy\ fractal\ ** ** ** ** ** ** ** \ ** ** ** ** ** *****) \[IndentingNewLine] \(rauzyfigure[subst_, n_]\ := Module[{}, \[IndentingNewLine]\(If[ d == 3, \[IndentingNewLine]Show[ Graphics3D[\ {forwardtranslates[subst, n, 1], \ forwardtranslates[subst, \ n, 2], \ forwardtranslates[subst, \ n, 3]}, \ ViewPoint\ \[Rule] 100*omega, \ Boxed \[Rule] \ False\ ]], \[IndentingNewLine]\(Print["\"];\)\[IndentingNewLine]];\)\ (*\ end\ if\ *) \[IndentingNewLine]];\)\ (*\ end\ module\ *) \[IndentingNewLine] (*\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)***********************************) \ (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \ \ \ \)\(MAIN\)\)\ PROG\ \ \ ** ** ** ** \ \(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\)*****) \ \ \[IndentingNewLine] \ (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)*****) \ \ \[IndentingNewLine] \(prog[substitution_, maxm_, maxrauzylevel_]\ := Module[{subst}, \[IndentingNewLine]subst\ = \ substitution; \[IndentingNewLine]prelim[subst]; \ (*\ defines\ functions\ and\ computes\ and\ prints\ matrix\ of\ dubst\ \ etc\ *) \[IndentingNewLine]Print["\"]; For[jj = 1, \ jj\ < \ maxrauzylevel + 1, \ \(jj++\), \ rauzyfigure[subst, jj]\ ]; \[IndentingNewLine]Print["\<****************************\ **************************************\>"];\[IndentingNewLine]];\)\ (*\ end\ prog\ *) \[IndentingNewLine] \ (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)******) \[IndentingNewLine] \ (*\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)******) \[IndentingNewLine] \)\)\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(Ident\)\" is similar to \ existing symbol \"\!\(Indent\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(maxrauzylevel = 4;\)\), "\[IndentingNewLine]", \(\(Trib = \ {{1, 2}, {1, 3}, {1}};\)\), 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