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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[ 255614, 4618]*) (*NotebookOutlinePosition[ 256374, 4644]*) (* CellTagsIndexPosition[ 256330, 4640]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Asmus Schmidt's Complex Continued Fraction Algorithm", "Title"], Cell["\<\ Implementing the Schmidt algorithm for approximation of complex numbers by \ elements of Q(\[ImaginaryI])\ \>", "Subtitle"], Cell["by Doug Hensley", "Text"], Cell[CellGroupData[{ Cell["Explanation", "Section"], Cell[CellGroupData[{ Cell["\<\ This implementation is restricted to the case of Gaussian rational numbers w \ inside the square with corners at 0,1,1+\[ImaginaryI], and \[ImaginaryI]. It \ returns a list containing the approximation target w, the first Schmidt \ matrix, global and zoomed-in graphics structures, and the sequence of T's \ needed to build the successive M's. \ \>", "Subsection"], Cell[TextData[{ "Asmus Schmidt's Complex Continued Fraction algorithm takes as input a \ complex number which we shall call w, and returns a sequence of matrices (", Cell[BoxData[ \(TraditionalForm\`T\_n\)]], ") that encode a sequence of essentially best-possible Gaussian rational \ approximations (", Cell[BoxData[ \(TraditionalForm\`r\_n\)]], ") to w, each of the form (a+\[ImaginaryI] b)/(c+\[ImaginaryI] d). \n\nIn \ this implementation, we restrict attention to w drawn from a basic square, \ with opposite corners at 0 and 1+\[ImaginaryI]. The T's are drawn from a list \ of 7 choices detailed below. Each of these 7 has Gaussian integer entries \ and determinant 1,-1,\[ImaginaryI], or -\[ImaginaryI], and thus has an \ inverse with Gaussian integer entries. With these T's, one constructs a \ further sequence (", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], "), where ", Cell[BoxData[ \(TraditionalForm\`M\_n = \(\(T\_1\) T\_2 ... \) \(\(T\_n\)\(.\)\)\)]], " From each ", Cell[BoxData[ FormBox[ RowBox[{\(M\_n\), "=", RowBox[{"(", GridBox[{ {\(p\^\((1)\)\), \(p\^\((2)\)\)}, {\(q\^\((1)\)\), \(q\^\((2)\)\)} }], ")"}]}], TraditionalForm]]], ", one constructs three Gaussian rationals ", Cell[BoxData[ \(TraditionalForm\`\(\(r\^\((1)\)\)\(=\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(p\^\((1)\)\)\(/\)\)\)]], Cell[BoxData[ \(TraditionalForm\`q\^\((1)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\) = p\^\((2)\)/q\^\((2)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\) = \(\((p\^\((1)\) + p\^\((2)\))\)/\((q\^\((1)\) + q\^\((2)\))\) = p\^\((3)\)/q\^\((3)\)\)\)]], ", say. Equivalently, {{p1,p2,p3},{q1,q2,q3}}=", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], ".{{1,0,1},{0,1,1}}. Whichever of these three candidate Gaussian rational \ approximations is nearest w is our ", Cell[BoxData[ \(TraditionalForm\`r\_n\)]], ". \n\nThe T's are chosen so that certain regions ", Cell[BoxData[ \(TraditionalForm\`F\_n\)]], " shall contain w. If det(", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], ")=1 or -1, the region is the {Schmidt disk} of ", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], ", the solid disk bounded by the circle through ", Cell[BoxData[ \(TraditionalForm\`r\^\((1)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)\)]], ". If instead, det(", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], ")=\[ImaginaryI] or -\[ImaginaryI], the region is a {Schmidt triangle} of \ ", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], " , a triangle-like region bounded by arcs, and with vertices ", Cell[BoxData[ \(TraditionalForm\`r\^\((1)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)\)]], ". \n\nThe Schmidt triangle of ", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], " is the closed, bounded region contained within three arcs, from ", Cell[BoxData[ \(TraditionalForm\`r\^\((1)\)\)]], " to ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)\)]], "to ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)\)]], "to ", Cell[BoxData[ \(TraditionalForm\`r\^\((1)\)\)]], ". These arcs can be either arcs of a circle, or, exceptionally, segments \ of a straight line. Each arc must be perpendicular at both ends to the circle \ through ", Cell[BoxData[ \(TraditionalForm\`r\^\((1)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)\)]], ", and each lies within the disk rather than through its complement. \n\n\ Each Schmidt disk has six distinguished points: the first three are ", Cell[BoxData[ \(TraditionalForm\`r\^\((1)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)\)]], ", and the next three are termed ", Cell[BoxData[ \(TraditionalForm\`\(r\^\((1)\)\)'\)]], ", ", Cell[BoxData[ \(TraditionalForm\`r\^\((2)\)'\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`r\^\((3)\)'\)]], ". These last three are given by {{p1',p2',p3'},{q1',q2',q3'}}=", Cell[BoxData[ \(TraditionalForm\`M\_n\)]], ".c.{{1,0,1},{0,1,1}}, (c={{1,-1+\[ImaginaryI]},{1-\[ImaginaryI],1}}), and \ are inside the Schmidt disk and on the edges of the Schmidt triangle, \ depending on whether M has real or imaginary determinant. \n\nEach Schmidt \ disk D is partitioned into 7 regions, consisting of 3 smaller Schmidt disks \ D1, D2, D3 tangent to D at r1, r2, r3 respectively, and tangent to each other \ as well, and 4 Schmidt triangles, one occupying the central region bounded by \ arcs of D1, D2, and D3.\n\nD1 and D2 are tangent at r3', D2 and D3 at r1', \ and D3 and D1 tangent at r2'. The four Schmidt triangles G1,G2,G3, and GC, \ say, have vertices r3r1'r2, r1r2'r3, r2r3'r1, and r1'r2'r3' respectively. \n\n\ \nThe figure immediately below shows the situation after the third step in \ the execution of the Schmidt algorithm when w=E^\[ImaginaryI] =0.540302\ \[InvisibleSpace]+0.841471 \[ImaginaryI]. We have \n{T1,T2,T3}={c,c,v3}, so \ that ", Cell[BoxData[ \(TraditionalForm\`M\_3 = \(\({{3 + 3\ \[ImaginaryI], \(-4\) - \[ImaginaryI]}, {4 - \ \[ImaginaryI], \(-3\) + 3\ \[ImaginaryI]}}\)\(.\)\(\ \)\)\)]], "Thus ", Cell[BoxData[ \(TraditionalForm\`F\_3\)]], "is the disk shown. Now w sits inside the central triangle of the picture, \ so that T4=c, and ", Cell[BoxData[ \(TraditionalForm\`M\_4\)]], "={{-2+6 \[ImaginaryI],-5-4 \[ImaginaryI]},{4+5 \[ImaginaryI],-6+2 \ \[ImaginaryI]}}. The figures next below show the situation at the sixth \ generation. Now, the T-sequence is {c,c,v3,c,c,v3}. The left picture shows \ the overall layout of the nested zones bracketing w, the middle picture \ zooms in to show the top picture and further iterations, and the right \ picture shows a tighter zoom on the sixth generation situation. The figures \ at bottom show the situation as of the 20th generation; the window sizes vary \ from 10^(-4)*2^(-4), to 10^(-4)*2^(-9) by powers of 2. \n\nIt should be noted \ that E^\[ImaginaryI], our approximation target in this series of examples, \ has a highly atypical Schmidt expansion. Schmidt was able to show that the \ T-sequence for this number is ", Cell[BoxData[ \(TraditionalForm\`\(C\^2\) \(VC\^2\) \(V\^3\) \(C\^2\) \(V\^5\) \(C\^2\ \) V\^7 ... \)]], "\n\nThe algorithm makes somewhat slower progress when it encounters a \ string of ", Cell[BoxData[ \(TraditionalForm\`E\_j\)]], "'s, with fixed j. This can happen if w finds itself nestled nearly at the \ vertex of a Schmidt triangle at iteration n; several iterations of a \ particular ", Cell[BoxData[ \(TraditionalForm\`E\_j\)]], " will be needed. Happily, the area of such regions is small so the average \ performance of the algorithm is not much affected. " }], "Text"], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell["\<\ Notation tracks closely that used by Schmidt where possible.The matrices \ v1,v2,v3,e1,e2,e3,and c are the same as in[Diophantine Approximation of \ Complex Numbers].Other notation tries to be helpful.frax[a,b] gives \ \[Infinity] if b=0,else a/b.ctrnradsqrd gives a list,{center,square of the \ radius} for the center and radius-squared of the circle through three complex \ numbers.Degenerate cases return {\[Infinity],\[Infinity]}.The function \ \"inside\" takes as input a complex number w and a circle defined by three \ complex numbers on its perimeter,and returns either 1,0,or-1 depending on \ whether w lies inside,on the rim,or outside the circle.It should not be given \ a degenerate circle.\ \>", "Text"], Cell[BoxData[ \(Off[General::spell]\)], "Input", InitializationCell->True], Cell["\<\ (Many of the names assigned in this notebook are variations on a theme. )\ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(v1 = {{1, I}, {0, 1}}; v2 = {{1, 0}, {\(-I\), 1}}; v3 = {{1 - I, I}, {\(-I\), 1 + I}}\)], "Input", InitializationCell->True], Cell[BoxData[ \({{1 - \[ImaginaryI], \[ImaginaryI]}, {\(-\[ImaginaryI]\), 1 + \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(e1 = {{1, 0}, {1 - I, I}}; e2 = {{1, \(-1\) + I}, {0, I}}; e3 = {{I, 0}, {0, 1}}; c = {{1, \(-1\) + I}, {1 - I, I}}\)], "Input", InitializationCell->True], Cell[BoxData[ \({{1, \(-1\) + \[ImaginaryI]}, {1 - \[ImaginaryI], \[ImaginaryI]}}\)], \ "Output"] }, Open ]], Cell[BoxData[{ \(schmidtmatrix[1] = v1; schmidtmatrix[2] = v2; schmidtmatrix[3] = v3; schmidtmatrix[4] = e1; schmidtmatrix[5] = e2; schmidtmatrix[6] = e3; schmidtmatrix[7] = c;\), "\[IndentingNewLine]", \(\(schmidtmatrix[8] = v1;\)\), "\[IndentingNewLine]", \(\(schmidtmatrix[9] = v2;\)\), "\[IndentingNewLine]", \(\(schmidtmatrix[10] = v3;\)\), "\[IndentingNewLine]", \(\(schmidtmatrix[11] = c;\)\)}], "Input", InitializationCell->True], Cell["\<\ The T's of the Schmidt algorithm are drawn from this list of choices, \ v1-v3,e1-e3, and c. \ \>", "Text"], Cell[BoxData[ \(frax[a_, b_] := If[b \[Equal] 0, ComplexInfinity, a/b]\)], "Input", InitializationCell->True], Cell["\<\ Now we give the code which got the pictures. The function \"rvals\" converts \ a matrix M into the list {{r1,r2,r3},{r1',r2',r3'}}. \ \>", "Text"], Cell[BoxData[ \(rvals[m_] := Module[{p1, q1, p2, q2, p3, q3, p1prime, q1prime, p2prime, q2prime, p3prime, q3prime, r1, r2, r3, r1prime, r2prime, r3prime}, \[IndentingNewLine]{{p1, p2, p3}, {q1, q2, q3}} = m . {{1, 0, 1}, {0, 1, 1}}; \[IndentingNewLine]{{p1prime, p2prime, p3prime}, {q1prime, q2prime, q3prime}} = m . c . {{1, 0, 1}, {0, 1, 1}}; \[IndentingNewLine]r1 = frax[p1, q1]; r2 = frax[p2, q2]; r3 = frax[p3, q3]; \[IndentingNewLine]r1prime = frax[p1prime, q1prime]; r2prime = frax[p2prime, q2prime]; r3prime = frax[p3prime, q3prime]; \[IndentingNewLine]{{r1, r2, r3}, {r1prime, r2prime, r3prime}}]\)], "Input", InitializationCell->True], Cell["\<\ The functions cleanup and cleanupb remove text and point elements from \ graphics, for cleaner display. \ \>", "Text"], Cell[BoxData[ \(cleanup[pix_] := Select[pix, Head[#] \[Equal] Circle || Head[#] \[Equal] Line || Head[#] \[Equal] Point || Head[#] \[Equal] PointSize &]\)], "Input", InitializationCell->True], Cell[BoxData[ \(cleanupb[pix_] := Select[pix, Head[#1] \[Equal] Circle || Head[#1] \[Equal] Line &]\)], "Input", InitializationCell->True], Cell[BoxData[ \(nearest[z_] := Floor[z + 1/2]\)], "Input", InitializationCell->True], Cell["\<\ The function \"linedup\" determines whether or not three points are \ collinear. \ \>", "Text"], Cell[BoxData[ \(linedup[zs_] := Module[{z1, z2, z3, s}, \[IndentingNewLine]{z1, z2, z3} = zs; \[IndentingNewLine]s = \((z3 - z1)\)/\((z2 - z1)\); \[IndentingNewLine]\((Im[s] \[Equal] 0 && Re[s] \[GreaterEqual] 1)\)]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(reim[z_] := {Re[z], Im[z]};\)\)], "Input", InitializationCell->True], Cell[BoxData[{ \(\(midpt[u_, v_] := \((u + v)\)/2;\)\), "\[IndentingNewLine]", \(diff[v1_, v2_] := v2 - v1\)}], "Input", InitializationCell->True], Cell[BoxData[ \(centernradsqrd[{z1_, z2_, z3_}] := Module[{x1, y1, x2, y2, x3, y3, s, t, d, u, w, v1, v2, ctr, rr, ck}, {{x1, y1}, {x2, y2}, {x3, y3}} = Map[reim, {z1, z2, z3}]; \[IndentingNewLine] (*\(Print["\", {z1, z2, z3}];\)*) \[IndentingNewLine]v1 = {x2, y2} - {x1, y1}; v2 = {x3, y3} - {x2, y2}; \[IndentingNewLine]d = Det[{v1, v2}]; \[IndentingNewLine]If[d \[Equal] 0, Return[{ComplexInfinity, ComplexInfinity (*\(,\)\(True\)*) }]]; \[IndentingNewLine]u = \ {x3, y3} - {x1, y1}; \[IndentingNewLine]s = v2 . u/2; s = s/d; t = v1 . u/2; t = t/d; \[IndentingNewLine]ctr = \((z1 + z2)\)/2 + s\ I \((z2 - z1)\); \[IndentingNewLine]rr = FullSimplify[ Abs[z1 - ctr]^2]; \[IndentingNewLine] (*\(ck = \((FullSimplify[ rr - Abs[z2 - ctr]^2] \[Equal] 0 && FullSimplify[rr - Abs[z3 - ctr]^2] \[Equal] 0)\);\)*) \[IndentingNewLine]{ctr, rr (*\(,\)\(ck\)*) }]\)], "Input", InitializationCell->True], Cell["\<\ The function \"inside\" returns 1 if the point is in the interior of the \ circle through the given points, 0 if it lies on the boundary, and -1 if it \ lies outside. \ \>", "Text"], Cell[BoxData[ \(cutup[m_] := Module[{t}, t = Det[m]; \[IndentingNewLine]If[ Re[t] \[NotEqual] 0, {m . c, m . v1, m . v2, m . v3, m . e1, m . e2, m . e3}, {m . c, m . v1, m . v2, m . v3}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(showcircle[m_] := Module[{c0, r1, r2, r3}, If[Im[Det[m]] \[NotEqual] 0, Return[Point[{0, 0}]]]; {r1, r2, r3} = \(rvals[m]\)[\([1]\)]; If[r1 \[Equal] ComplexInfinity || r2 \[Equal] ComplexInfinity || r3 \[Equal] ComplexInfinity, Print["\"]; Return[Point[{0, 0}]]]; \[IndentingNewLine]If[ collinear[{r1, r2, r3}], Return[Line[{r1, r2, r3}]]]; c0 = centernradsqrd[{r1, r2, r3}]; \[IndentingNewLine]c0 = {reim[ c0[\([1]\)]], Sqrt[c0[\([2]\)]]}; \[IndentingNewLine]Circle[ c0[\([1]\)], c0[\([2]\)]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(collinear[{r1_, r2_, r3_}] := Module[{u, v}, If[r1 \[Equal] ComplexInfinity || r2 \[Equal] ComplexInfinity || r3 \[Equal] ComplexInfinity, Print["\"]; Return[True]]; u = reim[r2 - r1]; v = reim[r3 - r1]; Det[{u, v}] \[Equal] 0]\)], "Input", InitializationCell->True], Cell[BoxData[ \(bigenough[m_, eps_] := Module[{c0, r1, r2, r3, mc, u, v, w}, mc = If[Im[Det[m]] \[NotEqual] 0, m . c, m]; {r1, r2, r3} = \(rvals[mc]\)[\([1]\)]; \[IndentingNewLine]If[ collinear[{r1, r2, r3}], Print["\", m, "\"]; Return[False]]; \[IndentingNewLine]c0 = centernradsqrd[{r1, r2, r3}]; \[IndentingNewLine]c0[\([2]\)] \[GreaterEqual] eps^2]\)], "Input", InitializationCell->True], Cell[BoxData[ \(nextgeneration[mlist_, eps_] := Module[{newms, goodms}, newms = Flatten[Map[cutup, mlist], 1]; goodms = Select[newms, bigenough[#, eps] &]; goodms]\)], "Input", InitializationCell->True], Cell[BoxData[ \(bigpicture[g0_, eps_] := Module[{gg, hh}, hh = g0; gg = g0; \[IndentingNewLine]While[gg \[NotEqual] {}, gg = nextgeneration[gg, eps]; hh = Union[gg, hh]]; hh]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Visualization - Schmidt Circle", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(g0 = {v3}\)], "Input", InitializationCell->True], Cell[BoxData[ \({{{1 - \[ImaginaryI], \[ImaginaryI]}, {\(-\[ImaginaryI]\), 1 + \[ImaginaryI]}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g1 = nextgeneration[g0, 0]\)], "Input"], Cell[BoxData[ \({{{2, \(-1\) + 2\ \[ImaginaryI]}, {2 - \[ImaginaryI], 2\ \[ImaginaryI]}}, {{1 - \[ImaginaryI], 1 + 2\ \[ImaginaryI]}, {\(-\[ImaginaryI]\), 2 + \[ImaginaryI]}}, {{2 - \[ImaginaryI], \[ImaginaryI]}, {1 - 2\ \[ImaginaryI], 1 + \[ImaginaryI]}}, {{1 - 2\ \[ImaginaryI], 2\ \[ImaginaryI]}, {\(-2\)\ \[ImaginaryI], 1 + 2\ \[ImaginaryI]}}, {{2, \(-1\)}, {2 - \[ImaginaryI], \(-1\) + \ \[ImaginaryI]}}, {{1 - \[ImaginaryI], \(-1\) + 2\ \[ImaginaryI]}, {\(-\[ImaginaryI]\), 2\ \[ImaginaryI]}}, {{1 + \[ImaginaryI], \[ImaginaryI]}, {1, 1 + \[ImaginaryI]}}}\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(g2 = nextgeneration[g1, 0];\)\), "\[IndentingNewLine]", \(\(g3 = nextgeneration[g2, 0];\)\), "\[IndentingNewLine]", \(\(g4 = nextgeneration[g3, 0];\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Length /@ \ {g0, \ g1, \ g2, \ g3, \ g4}\)], "Input"], Cell[BoxData[ \({1, 7, 37, 187, 937}\)], "Output"] }, Open ]], Cell["\<\ To keep successive generations from exploding, we make the smallest circle \ radius 1/60\ \>", "Text"], Cell[BoxData[{ \(\(cutoff\ = \ 1/300;\)\), "\[IndentingNewLine]", \(\(g2 = nextgeneration[g1, cutoff];\)\), "\[IndentingNewLine]", \(\(g3 = nextgeneration[g2, cutoff];\)\), "\[IndentingNewLine]", \(\(g4 = nextgeneration[g3, cutoff];\)\), "\[IndentingNewLine]", \(\(g5 = nextgeneration[g4, cutoff];\)\), "\[IndentingNewLine]", \(\(g6 = nextgeneration[g5, cutoff];\)\), "\[IndentingNewLine]", \(\(g7 = nextgeneration[g6, cutoff];\)\), "\[IndentingNewLine]", \(\(g8 = nextgeneration[g7, cutoff];\)\), "\[IndentingNewLine]", \(\(g9 = nextgeneration[g8, cutoff];\)\), "\[IndentingNewLine]", \(\(g10 = nextgeneration[g9, cutoff];\)\), "\[IndentingNewLine]", \(\(g11 = nextgeneration[g10, cutoff];\)\), "\[IndentingNewLine]", \(\(g12 = nextgeneration[g11, cutoff];\)\), "\[IndentingNewLine]", \(\(g13 = nextgeneration[g12, cutoff];\)\), "\[IndentingNewLine]", \(\(g14 = nextgeneration[g13, cutoff];\)\), "\[IndentingNewLine]", \(\(g15 = nextgeneration[g14, cutoff];\)\), "\[IndentingNewLine]", \(\(g16 = nextgeneration[g15, cutoff];\)\), "\[IndentingNewLine]", \(\(g17 = nextgeneration[g16, cutoff];\)\), "\[IndentingNewLine]", \(\(g18 = nextgeneration[g17, cutoff];\)\), "\[IndentingNewLine]", \(\(g19 = nextgeneration[g18, cutoff];\)\), "\[IndentingNewLine]", \(\(g20 = nextgeneration[g19, cutoff];\)\), "\[IndentingNewLine]", \(\(g21 = nextgeneration[g20, cutoff];\)\), "\[IndentingNewLine]", \(\(g22 = nextgeneration[g21, cutoff];\)\), "\[IndentingNewLine]", \(\(g23 = nextgeneration[g22, cutoff];\)\), "\[IndentingNewLine]", \(\(g24 = nextgeneration[g23, cutoff];\)\), "\[IndentingNewLine]", \(\(g25 = nextgeneration[g24, cutoff];\)\), "\[IndentingNewLine]", \(\(g26 = nextgeneration[g25, cutoff];\)\), "\[IndentingNewLine]", \(\(g27 = nextgeneration[g26, cutoff];\)\)}], "Input"], Cell[BoxData[ \(\(allgenerations\ = \ {g0, \ g1, \ g2, \ g3, \ g4, \ g5, \ g6, \ g7, \ g8, \ g9, \ g10, \ g11, \ g12, \ g13, \ g14, \ g15, \ g16, \ g17, \ g18, \ g19, \ g20, g21, \ g22, \ g23, \ g24, \ g25, \ g26, \ g27};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Length /@ \ allgenerations\)], "Input"], Cell[BoxData[ \({1, 7, 37, 186, 760, 2055, 3044, 2459, 1320, 728, 485, 355, 232, 166, 132, 90, 60, 44, 38, 30, 26, 22, 18, 18, 18, 18, 18, 18}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(pixygen = Drop[Union[ Map[showcircle, Union[Flatten[allgenerations, 1]]]], \(-1\)];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[Graphics[pixygen], AspectRatio \[Rule] Automatic]\)], "Input"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0?ooo`00B`3oool4 00000080oooo0P0000020?ooo`800000203oool2000000P0oooo00<00000 0?ooo`3oool00`3oool01@000000oooo0?ooo`3oool0000000<0oooo00L0 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00001`0000080?ooo`/0000000@0oooo0000003oool000000`3oool20000 00<0oooo0P0000080?ooo`@00000I@3oool006P0oooo0`0000070?ooo`<0 00000P3oool2000000<0oooo3@0000080?ooo`h000000P3oool200000080 oooo0`0000070?ooo`<00000J@3oool006/0oooo1@0000050?ooo`<00000 00<0oooo0000000000000P3oool3000000030?ooo`000000000000H00000 203oool:000000040?ooo`000000oooo0?ooo`80000000@0oooo00000000 000000001@3oool5000006`0oooo001`0?ooo`D000000`3ooolA000000L0 oooo4P0000030?ooo`D00000L@3oool007D0oooo100000020?ooo`l00000 1P3oool?00000080oooo1000001f0?ooo`00N@3ooolC00000080oooo4`00 001j0?ooo`00P`3ooolD000008@0oooo003o0?oooa`0oooo003o0?oooa`0 oooo003o0?oooa`0oooo003o0?oooa`0oooo003o0?oooa`0oooo003o0?oo oa`0oooo0000\ \>"], "Graphics", Evaluatable->False, ImageSize->{283, 283}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Code - deeper triangles", "Section"], Cell[BoxData[ \(inside[w_, zs_] := Module[{ctr, rr, s}, If[linedup[zs], \[IndentingNewLine]s = Im[\((w - zs[\([1]\)])\)/\((zs[\([2]\)] - zs[\([1]\)])\)]; \[IndentingNewLine]Return[ Sign[s]]]; \[IndentingNewLine]{ctr, rr (*\(,\)\(tf\)*) } = centernradsqrd[zs]; \[IndentingNewLine]Sign[ FullSimplify[rr - Abs[w - ctr]^2]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(schmidttriangle[w_, m_] := Module[{r, rp, ck0, ck1, ck2, ck3}, {r, rp} = rvals[m]; \[IndentingNewLine]ck0 = inside[w, r]; \[IndentingNewLine]ck1 = inside[w, {r[\([1]\)], rp[\([2]\)], r[\([3]\)]}]; \[IndentingNewLine]ck2 = inside[w, {r[\([2]\)], rp[\([3]\)], r[\([1]\)]}]; \[IndentingNewLine]ck3 = inside[w, {r[\([3]\)], rp[\([2]\)]\ , r[\([1]\)]}]; \[IndentingNewLine]\((ck0 \[GreaterEqual] 0 && ck1 \[LessEqual] 0 && ck2 \[LessEqual] 0 && ck3 \[LessEqual] 0)\)]\)], "Input", InitializationCell->True], Cell["\<\ This isn't used. But it's the core of the next bit of code. The second \ version of schmidttriangle says whether w lies inside a specific triangle \ zone associated with the matrix m. \ \>", "Text"], Cell[BoxData[ \(schmidttriangle[w_, m_, k_] := Module[{r, rp, ck0, ck1, ck2, ck3, c1, c2, c3, sk1, sk2, sk3, z1, z2, z3, z4, z5, z6, sk4, sk5, sk6, ck4, ck5, ck6, type}, \[IndentingNewLine]type = \((Re[Det[m]] \[NotEqual] 0)\); \[IndentingNewLine]If[type && k > 7, Print["\"]; Return[False]]; \[IndentingNewLine]If[ Not[type] && k \[LessEqual] 7, Print["\"]; \ \[IndentingNewLine]Return[False]]; \[IndentingNewLine]{r, rp} = rvals[m]; \[IndentingNewLine]ck0 = inside[w, r]; If[ck0 < 0, Return[False]]; \[IndentingNewLine]ck1 = inside[w, {r[\([1]\)], rp[\([3]\)], rp[\([2]\)]}]; \[IndentingNewLine]If[k \[Equal] 1, Return[\((ck1 \[GreaterEqual] 0)\)]]; \[IndentingNewLine]ck2 = inside[w, {r[\([2]\)], rp[\([1]\)], rp[\([3]\)]}]; \[IndentingNewLine]If[k \[Equal] 2, Return[\((ck2 \[GreaterEqual] 0)\)]]; \[IndentingNewLine]ck3 = inside[w, {r[\([3]\)], rp[\([2]\)], rp[\([1]\)]}]; \[IndentingNewLine]If[k \[Equal] 3, Return[\((ck3 \[GreaterEqual] 0)\)]]; \[IndentingNewLine]z1 = \(centernradsqrd[{r[\([1]\)], rp[\([3]\)], rp[\([2]\)]}]\)[\([1]\)]; \[IndentingNewLine]z2 = \ \(centernradsqrd[{r[\([2]\)], rp[\([1]\)], rp[\([3]\)]}]\)[\([1]\)]; \[IndentingNewLine]z3 = \ \(centernradsqrd[{r[\([3]\)], rp[\([2]\)], rp[\([1]\)]}]\)[\([1]\)]; \[IndentingNewLine]sk1 = Sign[Im[\((w - z3)\)/\((z2 - z3)\)]]; \[IndentingNewLine]sk2 = Sign[Im[\((w - z1)\)/\((z3 - z1)\)]]; \[IndentingNewLine]sk3 = Sign[Im[\((w - z2)\)/\((z1 - z2)\)]]; \[IndentingNewLine]If[ k \[Equal] 4, Return[\((ck2 \[LessEqual] 0 && ck3 \[LessEqual] 0 && sk1 \[GreaterEqual] 0)\)]]; \[IndentingNewLine]If[ k \[Equal] 5, Return[\((ck1 \[LessEqual] 0 && ck3 \[LessEqual] 0 && sk2 \[GreaterEqual] 0)\)]]; \[IndentingNewLine]If[ k \[Equal] 6, Return[\((ck1 \[LessEqual] 0 && ck2 \[LessEqual] 0 && sk3 \[GreaterEqual] 0)\)]]; \[IndentingNewLine]If[ k \[Equal] 7, Return[True]]; \[IndentingNewLine]ck4 = inside[w, {r[\([3]\)], rp[\([1]\)], r[\([2]\)]}]; \[IndentingNewLine]If[ck4 > 0, Return[False]]; \[IndentingNewLine]ck5 = inside[w, {r[\([1]\)], rp[\([2]\)], r[\([3]\)]}]; \[IndentingNewLine]If[ck5 > 0, Return[False]]; \[IndentingNewLine]ck6 = inside[w, {r[\([2]\)], rp[\([3]\)], r[\([1]\)]}]; \[IndentingNewLine]If[ck6 > 0, Return[False]]; \[IndentingNewLine]If[k \[Equal] 11, Return[inside[w, rp] \[GreaterEqual] 0]]; If[k < 7 && inside[w, rp] > 0, Return[False]]; \[IndentingNewLine]If[ k \[Equal] 8, Return[Im[\((w - rp[\([3]\)])\)/\((rp[\([2]\)] - rp[\([3]\)])\)] \[GreaterEqual] 0]]; \[IndentingNewLine]If[k \[Equal] 9, Return[Im[\((w - rp[\([1]\)])\)/\((rp[\([3]\)] - rp[\([1]\)])\)] \[GreaterEqual] 0]]; \[IndentingNewLine]If[k \[Equal] 10, Return[Im[\((w - rp[\([2]\)])\)/\((rp[\([1]\)] - rp[\([2]\)])\)] \[GreaterEqual] 0]]; \[IndentingNewLine]Print["\"]; Return[False]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(circlepix[w_, m_] := Module[{rs, u, v, r1, r2, r3, r1p, r2p, r3p, c0, c1, c2, c3, d1, d2, d3, det}, \[IndentingNewLine]det = Det[m]; If[Re[det] \[Equal] 0, (*\(Print["\"];\)*) Return[Null]]; \[IndentingNewLine]rs = rvals[m]; \[IndentingNewLine] (*\(Print["\", rs];\)*) \[IndentingNewLine]{{r1, r2, r3}, {r1p, r2p, r3p}} = rs; \[IndentingNewLine]If[linedup[{r1, r2, r3}], c0 = Line[{reim[r1], reim[r2], reim[r3]}]]; \[IndentingNewLine]c0 = centernradsqrd[{r1, r2, r3}]; \[IndentingNewLine]c0 = {reim[ c0[\([1]\)]], Sqrt[c0[\([2]\)]]}; \[IndentingNewLine]c0 = Circle[c0[\([1]\)], c0[\([2]\)]]; \[IndentingNewLine]If[ linedup[{r1, r3p, r2p}], c1 = Line[{reim[r1], reim[r3p], reim[r2p]}]]; \[IndentingNewLine]c1 = centernradsqrd[{r1, r3p, r2p}]; \[IndentingNewLine]c1 = {reim[ c1[\([1]\)]], Sqrt[c1[\([2]\)]]}; d1 = c1; \[IndentingNewLine]c1 = Circle[c1[\([1]\)], c1[\([2]\)]]; \[IndentingNewLine]If[ linedup[{r2, r1p, r3p}], c2 = Line[{reim[r2], reim[r1p], reim[r3p]}]]; \[IndentingNewLine]c2 = centernradsqrd[{r2, r1p, r3p}]; \[IndentingNewLine]c2 = {reim[ c2[\([1]\)]], Sqrt[c2[\([2]\)]]}; d2 = c2; \[IndentingNewLine]c2 = Circle[c2[\([1]\)], c2[\([2]\)]]; \[IndentingNewLine]If[ linedup[{r3, r2p, r1p}], c3 = Line[{reim[r3], reim[r2p], reim[r1p]}]]; \[IndentingNewLine]c3 = centernradsqrd[{r3, r2p, r1p}]; \[IndentingNewLine]c3 = {reim[ c3[\([1]\)]], Sqrt[c3[\([2]\)]]}; d3 = c3; \[IndentingNewLine]c3 = Circle[c3[\([1]\)], c3[\([2]\)]]; \[IndentingNewLine] (*Show[ Graphics[{PointSize[ .05\ Sqrt[c0[\([2]\)]]], Point[reim[r1]], Point[reim[r2]], Point[reim[r3]], Text["\", reim[w]], \[IndentingNewLine]Text["\<1\>", d1[\([1]\)]], Text["\<2\>", d2[\([1]\)]], \[IndentingNewLine]Text["\<3\>", d3[\([1]\)]], c0, c1, c2, c3}], AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]*) ]\)], "Input", InitializationCell->True], Cell[BoxData[ \(circlepix2[w_, m_] := Module[{rs, u, v, r1, r2, r3, r1p, r2p, r3p, c0, c1, c2, c3, d1, d2, d3, det}, \[IndentingNewLine]det = Det[m]; If[Re[det] \[Equal] 0, (*\(Print["\"];\)*) Return[Null]]; \[IndentingNewLine]rs = rvals[m]; \[IndentingNewLine] (*\(Print["\", rs];\)*) \[IndentingNewLine]{{r1, r2, r3}, {r1p, r2p, r3p}} = rs; \[IndentingNewLine]If[linedup[{r1, r2, r3}], c0 = Line[{reim[r1], reim[r2], reim[r3]}]]; \[IndentingNewLine]c0 = centernradsqrd[{r1, r2, r3}]; \[IndentingNewLine]c0 = {reim[ c0[\([1]\)]], Sqrt[c0[\([2]\)]]}; \[IndentingNewLine]c0 = Circle[c0[\([1]\)], c0[\([2]\)]]; \[IndentingNewLine]If[ linedup[{r1, r3p, r2p}], c1 = Line[{reim[r1], reim[r3p], reim[r2p]}]]; \[IndentingNewLine]c1 = centernradsqrd[{r1, r3p, r2p}]; \[IndentingNewLine]c1 = {reim[ c1[\([1]\)]], Sqrt[c1[\([2]\)]]}; d1 = c1; \[IndentingNewLine]c1 = Circle[c1[\([1]\)], c1[\([2]\)]]; \[IndentingNewLine]If[ linedup[{r2, r1p, r3p}], c2 = Line[{reim[r2], reim[r1p], reim[r3p]}]]; \[IndentingNewLine]c2 = centernradsqrd[{r2, r1p, r3p}]; \[IndentingNewLine]c2 = {reim[ c2[\([1]\)]], Sqrt[c2[\([2]\)]]}; d2 = c2; \[IndentingNewLine]c2 = Circle[c2[\([1]\)], c2[\([2]\)]]; \[IndentingNewLine]If[ linedup[{r3, r2p, r1p}], c3 = Line[{reim[r3], reim[r2p], reim[r1p]}]]; \[IndentingNewLine]c3 = centernradsqrd[{r3, r2p, r1p}]; \[IndentingNewLine]c3 = {reim[ c3[\([1]\)]], Sqrt[c3[\([2]\)]]}; d3 = c3; \[IndentingNewLine]c3 = Circle[c3[\([1]\)], c3[\([2]\)]]; \[IndentingNewLine]{PointSize[ .05\ Sqrt[ c0[\([2]\)]]], Point[reim[r1]], Point[reim[r2]], Point[reim[r3]], Text["\", reim[w]], \[IndentingNewLine]Text["\<1\>", d1[\([1]\)]], Text["\<2\>", d2[\([1]\)]], \[IndentingNewLine]Text["\<3\>", d3[\([1]\)]], c0, c1, c2, c3}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(ark[zs_] := Module[{z1, z2, z3, theta1, theta2, ctr, rad, cr}, \[IndentingNewLine]If[ linedup[zs], (*\(Print["\", reim[zs[\([1]\)]], "\<,\>", reim[zs[\([2]\)]], "\<}\>"];\)*) \[IndentingNewLine]Return[\ \[IndentingNewLine]Line[\[IndentingNewLine]{reim[zs[\([1]\)]], reim[zs[\([2]\)]], reim[zs[\([3]\)]]}\[IndentingNewLine]\t\ ]\[IndentingNewLine]]]; \[IndentingNewLine]cr = centernradsqrd[zs]; \[IndentingNewLine]ctr = reim[cr[\([1]\)]]; \[IndentingNewLine]rad = Sqrt[cr[\([2]\)]]; \[IndentingNewLine]theta1 = Arg[zs[\([1]\)] - cr[\([1]\)]]; \[IndentingNewLine]theta2 = Arg[zs[\([3]\)] - cr[\([1]\)]]; \[IndentingNewLine]If[ theta2 < theta1, theta2 += 2\ Pi]; \[IndentingNewLine] (*Print["\<{theta1,theta2}=\>", \ {theta1, theta2}]; \[IndentingNewLine]Print["\", \[IndentingNewLine]{ctr, rad, {theta1, theta2}}];*) \[IndentingNewLine]Circle[ctr, rad, {theta1, theta2}]]\)\(\[IndentingNewLine]\) \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(arc[zs_] := Module[{u, v, x, y, z}, \[IndentingNewLine]If[ linedup[zs], (*\(Print["\", zs];\)*) \[IndentingNewLine]Return[ Line[Map[reim, zs]]]]; \[IndentingNewLine]{x, y, z} = zs; \[IndentingNewLine]u = y - x; v = z - x; \[IndentingNewLine]v = v/u; \[IndentingNewLine]If[ Im[v] > 0, (*\(Print["\"];\)*) Return[ ark[zs]]]; ark[Reverse[zs]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(insideschmidttriangle[w_, {z1_, z2_, z3_}] := Module[{u, s, t}, \[IndentingNewLine]If[ w \[Equal] z1 || w \[Equal] z2 || w \[Equal] z3, Return[True]]; \[IndentingNewLine]s = \((z2 - z1)\)/\((w - z1)\); \[IndentingNewLine]u = \((z2 - z1)\)/\((z3 - z1)\); \[IndentingNewLine]t = \((s - 1)\)/\((u - 1)\); \[IndentingNewLine]t = reim[t]; \[IndentingNewLine]\((t[\([1]\)] - 1/2)\)^2 + t[\([2]\)]^2 \[GreaterEqual] 1/4 && t[\([2]\)] \[GreaterEqual] 0 && t[\([1]\)] \[GreaterEqual] 0 && t[\([1]\)] \[LessEqual] 1]\)], "Input", InitializationCell->True], Cell[BoxData[ \(trianglepix3[w_, zs_] := Module[{z1, z2, z3, u, v, a1, a2, a3}, \[IndentingNewLine]{z1, z2, z3} = zs; \[IndentingNewLine]If[ linedup[zs], Print["\"]; Return[False]]; \[IndentingNewLine]u = \((z3 - z1)\)/\((z2 - z1)\); \[IndentingNewLine]v = 1/u; \[IndentingNewLine]a1 = {1, 1/\((1 - I \((1 - v)\))\), 0}; \[IndentingNewLine]a2 = {u, 1/\((v - I \((1 - v)\))\), 0}; \[IndentingNewLine]a3 = {u, 1/\((\((1 + v)\)/2 - I \((1 - v)\)/2)\), 1}; \[IndentingNewLine]{a1, a2, a3} = \((z2 - z1)\) {a1, a2, a3}; \[IndentingNewLine]{a1, a2, a3} = z1 + {a1, a2, a3}; \[IndentingNewLine]{Text["\", reim[w]], arc[a1], arc[a2], arc[a3]}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(trianglepix4[w_, m_] := Module[{rs, circ, triangle}, rs = rvals[m]; \[IndentingNewLine]triangle = trianglepix3[w, rs[\([1]\)]]; \[IndentingNewLine]circ = centernradsqrd[ rs[\([2]\)]]; \[IndentingNewLine]circ = {reim[circ[\([1]\)]], Sqrt[circ[\([2]\)]]}; \[IndentingNewLine]circ = Circle[circ[\([1]\)], circ[\([2]\)]]; \[IndentingNewLine] (*\(Print["\", circ];\)*) \[IndentingNewLine] (*\(Show\)\([\)\(\(Graphics\)\([\ \)\)*) {Text["\", reim[w]], triangle, circ} (*\(\(]\)\(,\)\(AspectRatio \[Rule] Automatic\)\(,\)\(PlotRange \[Rule] All\)\)\(]\)*) ]\)], "Input", InitializationCell->True], Cell[BoxData[ \(trianglepix5[w_, m_] := Module[{rs, circ, triangle}, rs = rvals[m]; \[IndentingNewLine]triangle = trianglepix3[w, rs[\([1]\)]]; \[IndentingNewLine]circ = centernradsqrd[ rs[\([2]\)]]; \[IndentingNewLine]circ = {reim[circ[\([1]\)]], Sqrt[circ[\([2]\)]]}; \[IndentingNewLine]circ = Circle[circ[\([1]\)], circ[\([2]\)]]; \[IndentingNewLine] (*\(Print["\", circ];\)*) \[IndentingNewLine]{Text["\", reim[w]], triangle, circ}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(zset[m_, k_] := Module[{rs, zs, r1, r2, r3, r1p, r2p, r3p, type}, \[IndentingNewLine]type = \((Re[Det[m]] \[NotEqual] 0)\); rs = rvals[m]; \[IndentingNewLine]{{r1, r2, r3}, {r1p, r2p, r3p}} = rs; \[IndentingNewLine]If[ type, \[IndentingNewLine]If[k > 7, Print["\"]; Return[False]]; \[IndentingNewLine]If[k \[Equal] 1, Return[{r1, r3p, r2p}]]; \[IndentingNewLine]If[k \[Equal] 2, Return[{r2, r1p, r3p}]]; \[IndentingNewLine]If[k \[Equal] 3, Return[{r3, r2p, r1p}]]; \[IndentingNewLine]If[k \[Equal] 4, Return[{r3, r1p, r2}]]; \[IndentingNewLine]If[k \[Equal] 5, Return[{r1, r2p, r3}]]; \[IndentingNewLine]If[k \[Equal] 6, Return[{r2, r3p, r1}]]; \[IndentingNewLine]If[k \[Equal] 7, Return[{r1p, r2p, r3p}]]\[IndentingNewLine]]; \[IndentingNewLine]If[ Not[type], \[IndentingNewLine]If[k \[LessEqual] 7 || k > 11, Print["\"]; Return[False]]; \[IndentingNewLine]If[k \[Equal] 8, Return[{r1, r3p, r2p}]]; \[IndentingNewLine]If[k \[Equal] 9, Return[{r2, r1p, r3p}]]; \[IndentingNewLine]If[k \[Equal] 10, Return[{r3, r2p, r1p}]]; \[IndentingNewLine]If[k \[Equal] 11, Return[{r1p, r2p, r3p}]]]]\)\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(ei = N[E^I, 100]\)], "Input"], Cell[BoxData[ \(0.5403023058681397174009366074429766037323104206179222276700972553811003\ 947744717645179518560871830893435717311600301`99.88315181882976 + 0.8414709848078965066525023216302989996225630607983710656727517099919104\ 043912396689486397435430526958543490379079207`100.07555414327149\ \ \[ImaginaryI]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ei = 10^\((\(-100\))\)\ nearest[10^100\ \ ei]\)], "Input"], Cell[BoxData[ \(540302305868139717400936607442976603732310420617922227670097255381100394\ 7744717645179518560871830893\/\ 100000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000 + \ \(4207354924039482533262511608151494998112815303991855328363758549959552021956\ 198344743198717715263479\ \ \[ImaginaryI]\)\/\ 500000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[ei]\)], "Input"], Cell[BoxData[ \(\(\(0.5403023058681398`\)\(\[InvisibleSpace]\)\) + 0.8414709848078965`\ \[ImaginaryI]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c\)], "Input"], Cell[BoxData[ \({{1, \(-1\) + \[ImaginaryI]}, {1 - \[ImaginaryI], \[ImaginaryI]}}\)], \ "Output"] }, Open ]], Cell[BoxData[ \(m = c; w = ei;\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(inside[2 + I/2, {1, 1 + I, \((1 + I)\)/2}]\)], "Input"], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell[BoxData[ \(rangy[ww_, eps_] := Module[{x, y}, {x, y} = reim[ww]; {{x - eps, x + eps}, {y - eps, y + eps}}]\)], "Input", InitializationCell->True], Cell["\<\ The function schmidtstart should be given a Gaussian rational within the \ starting box. It then returns a list with 5 elements, suitable for \ continuation of the algorithm by iterating the next function, schmidtstep.\ \>", "Text"], Cell[BoxData[ \(\(schmidtstart[z_] := Module[{w, x, y, m, slist}, \[IndentingNewLine]If[ Re[z] > 1 || Re[z] < 0 || Im[z] > 1 || Im[z] < 0, \[IndentingNewLine]Print["\"]; Return[False]]; \[IndentingNewLine]If[ inside[z, {0, \((1 + I)\)/2, I}] \[GreaterEqual] 0, Return[{z, v2, {}, {}, {{"\", v2}}}]]; \[IndentingNewLine]If[ inside[z, {1, 1 + I, \((1 + I)\)/2}] \[GreaterEqual] 0, Return[{z, v3, {}, {}, {{"\", v3}}}]]; \[IndentingNewLine]If[ Im[w] \[LessEqual] 1/2, Return[{z, e1, {}, {}, {{"\", e1}}}]]; \[IndentingNewLine]{z, c, {}, {}, {{"\", c}}}];\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(schmidtstart[w2]\)], "Input"], Cell[BoxData[ \({w2, {{1, \(-1\) + \[ImaginaryI]}, {1 - \[ImaginaryI], \[ImaginaryI]}}, \ {}, {}, {{"c", {{1, \(-1\) + \[ImaginaryI]}, {1 - \[ImaginaryI], \ \[ImaginaryI]}}}}}\)], "Output"] }, Open ]], Cell[BoxData[ \(schmidtstep[{w_, m_, pictures_, localpix_, schmidtlist_}] := Module[{type, k, zs, rs, pix, zoom, tmatrix, tlist}, zoom = {}; tlist = schmidtlist; \[IndentingNewLine]pix = pictures; type = \((Re[Det[m]] \[NotEqual] 0)\); If[type, zoom = circlepix2[w, m]; pix = Join[pix, circlepix2[w, m]], \[IndentingNewLine] (*\(zoom = trianglepix3[ w, \(rvals[m]\)[\([1]\)]];\)*) \[IndentingNewLine]zoom = trianglepix5[w, m]; \[IndentingNewLine]pix = Join[pix, trianglepix3[ w, \(rvals[ m]\)[\([1]\)]]]]; \[IndentingNewLine] (*\(Print["\", Length[pix]];\)*) \[IndentingNewLine]Show[ Graphics[cleanupb[pix]], AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; \[IndentingNewLine]circlepix[w, m]; rs = rvals[ m]; \[IndentingNewLine]If[\(rs[\([1]\)]\)[\([1]\)] \[Equal] w, Print["\", {w, \(rs[\([1]\)]\)[\ \([1]\)]}]; Return[rs]]; \[IndentingNewLine]If[\(rs[\([1]\)]\)[\([2]\)] \ \[Equal] w, Print["\", {w, \ \(rs[\([1]\)]\)[\([2]\)]}]; Return[rs]]; \[IndentingNewLine]If[\(rs[\([1]\)]\)[\([3]\)] \ \[Equal] w, Print["\", {w, \ \(rs[\([1]\)]\)[\([3]\)]}]; \[IndentingNewLine]Return[ rs]]; \[IndentingNewLine] (*\(Print["\", type];\)*) \[IndentingNewLine]If[ type, \[IndentingNewLine] (*\(Print["\"];\)*) \[IndentingNewLine]k = 1; \[IndentingNewLine]tf = schmidttriangle[w, m, 1]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . v1];\)*) AppendTo[ tlist, {"\", v1}]; \[IndentingNewLine] (*\(circlepix[w, m . v1];\)*) Return[{w, m . v1, pix, zoom, tlist}]]; \[IndentingNewLine]k = 2; tf = schmidttriangle[w, m, 2]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . v2];\)*) AppendTo[ tlist, {"\", v2}]; \[IndentingNewLine] (*\(circlepix[w, m . v2];\)*) \[IndentingNewLine]Return[{w, m . v2, pix, zoom, tlist}]]; \[IndentingNewLine]k = 3; tf = schmidttriangle[w, m, 3]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . v3];\)*) AppendTo[ tlist, {"\", v3}]; \[IndentingNewLine] (*\(circlepix[w, m . v3];\)*) \[IndentingNewLine]Return[{w, m . v3, pix, zoom, tlist}]]; \[IndentingNewLine]k = 4; zs = zset[m, k]; tf = insideschmidttriangle[w, zs]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . schmidtmatrix[4]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", schmidtmatrix[4]}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[4], pix, zoom, tlist}]]; \[IndentingNewLine]k = 5; zs = zset[m, k]; \[IndentingNewLine]tf = insideschmidttriangle[w, zs]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . schmidtmatrix[5]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", schmidtmatrix[5]}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[5], pix, zoom, tlist}]]; \[IndentingNewLine]k = 6; zs = zset[m, k]; \[IndentingNewLine]tf = insideschmidttriangle[w, zs]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . schmidtmatrix[6]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", schmidtmatrix[6]}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[6], pix, zoom, tlist}]]; \[IndentingNewLine]k = 7; zs = zset[m, k]; \[IndentingNewLine]tf = insideschmidttriangle[w, zs]; \[IndentingNewLine]If[ tf, (*\(Print["\", m . schmidtmatrix[7]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", schmidtmatrix[7]}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[7], pix, zoom, tlist}]]; \[IndentingNewLine]Print["\", Return[False]]]; \[IndentingNewLine]If[ Not[ type], \[IndentingNewLine] (*\(Print["\", m];\)*) \[IndentingNewLine]rs = rvals[m]; \[IndentingNewLine] (*\(trianglepix4[w, m];\)*) \[IndentingNewLine]k = 8; zs = zset[m, k]; (*\(Print["\", zs];\)*) \[IndentingNewLine]tf = insideschmidttriangle[w, zs]; (*\(Print["\", tf];\)*) \[IndentingNewLine]If[ tf, \[IndentingNewLine] (*\(Print["\", m . schmidtmatrix[8]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", v1}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[8], pix, zoom, tlist}]]; \[IndentingNewLine]k = 9; zs = zset[m, k]; \[IndentingNewLine]tf = insideschmidttriangle[w, zs]; \[IndentingNewLine]If[ tf, \[IndentingNewLine] (*\(Print["\", m . schmidtmatrix[9]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", v2}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[9], pix, zoom, tlist}]]; \[IndentingNewLine]k = 10; zs = zset[m, k]; \[IndentingNewLine]tf = insideschmidttriangle[w, zs]; \[IndentingNewLine]If[ tf, \[IndentingNewLine] (*\(Print["\", m . schmidtmatrix[10]];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", v3}]; \[IndentingNewLine] (*\(trianglepix3[w, zs];\)*) \[IndentingNewLine]Return[{w, m . schmidtmatrix[10], pix, zoom, tlist}]]; \[IndentingNewLine]k = 11; \[IndentingNewLine]zs = \(tf = \((inside[w, rs[\([2]\)]] \[GreaterEqual] 0)\)\); \[IndentingNewLine]If[ tf, \[IndentingNewLine] (*\(Print["\", m . c];\)*) \[IndentingNewLine]AppendTo[ tlist, {"\", c}]; \[IndentingNewLine] (*\(circlepix[w, m . c];\)*) \[IndentingNewLine]Return[{w, m . c, pix, zoom, tlist}]]; \[IndentingNewLine]Print["\"]; Return[False]]]\)], "Input", InitializationCell->True], Cell["\<\ And now, we execute the algorithm on some examples and see what we get. \ Outputs have been suppressed, as they tend to be graphics-intensive and thus \ fatten the file into several extra megs of space. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(w = 10^\((\(-100\))\) Floor[10^\((100)\) N[E^I, 105]]\)], "Input"], Cell[BoxData[ \(540302305868139717400936607442976603732310420617922227670097255381100394\ 7744717645179518560871830893\/\ 100000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000 + \ \(4207354924039482533262511608151494998112815303991855328363758549959552021956\ 198344743198717715263479\ \ \[ImaginaryI]\)\/\ 500000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000\)], "Output"] }, Open ]], Cell[CellGroupData[{ 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