(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 1098601, 48199] NotebookOptionsPosition[ 1094213, 48082] NotebookOutlinePosition[ 1094584, 48099] CellTagsIndexPosition[ 1094541, 48096] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Hilbert Curve", "Title",ExpressionUUID->"100bdea1-055a-45ae-88a9-d6d9b8cf1e63"], Cell[CellGroupData[{ Cell["Author", "Subsection",ExpressionUUID->"1c8accb0-b855-416a-9836-0ab20410b156"], Cell["\<\ Eric W. Weisstein January 20, 2017\ \>", "Text",ExpressionUUID->"899e10ca-44bb-4171-9174-7d2b0cfa3c00"], Cell[TextData[{ "This notebook downloaded from ", ButtonBox["http://mathworld.wolfram.com/notebooks/Fractals/HilbertCurve.nb", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/notebooks/Fractals/HilbertCurve.nb"], None}], "." }], "Text",ExpressionUUID->"991f8eec-abe5-4f4c-97a2-132ec9bf26e6"], Cell[TextData[{ "For more information, see Eric's ", StyleBox["MathWorld", FontSlant->"Italic"], " entry ", ButtonBox["http://mathworld.wolfram.com/HilbertCurve.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/HilbertCurve.html"], None}], "." }], "Text",ExpressionUUID->"e0480919-d5d0-4229-ae66-58c06f7c32be"], Cell["\<\ \[Copyright]2017 Wolfram Research, Inc. except for portions noted otherwise\ \>", "Text",ExpressionUUID->"7aec4b80-c4ac-4ce1-92ee-09479f5d9883"] }, Open ]], Cell[CellGroupData[{ Cell["Additional Authors", 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32], Rational[25, 32]}, { Rational[9, 32], Rational[3, 4]}, { Rational[5, 16], Rational[3, 4]}, { Rational[11, 32], Rational[3, 4]}, { Rational[11, 32], Rational[25, 32]}, { Rational[5, 16], Rational[25, 32]}, { Rational[5, 16], Rational[13, 16]}, { Rational[11, 32], Rational[13, 16]}, { Rational[11, 32], Rational[27, 32]}, { Rational[5, 16], Rational[27, 32]}, { Rational[9, 32], Rational[27, 32]}, { Rational[9, 32], Rational[13, 16]}, { Rational[1, 4], Rational[13, 16]}, { Rational[1, 4], Rational[27, 32]}, { Rational[1, 4], Rational[7, 8]}, { Rational[9, 32], Rational[7, 8]}, { Rational[9, 32], Rational[29, 32]}, { Rational[1, 4], Rational[29, 32]}, { Rational[1, 4], Rational[15, 16]}, { Rational[1, 4], Rational[31, 32]}, { Rational[9, 32], Rational[31, 32]}, { Rational[9, 32], Rational[15, 16]}, { Rational[5, 16], Rational[15, 16]}, { Rational[5, 16], Rational[31, 32]}, { Rational[11, 32], Rational[31, 32]}, { Rational[11, 32], Rational[15, 16]}, { Rational[11, 32], Rational[29, 32]}, { Rational[5, 16], Rational[29, 32]}, { Rational[5, 16], Rational[7, 8]}, { Rational[11, 32], Rational[7, 8]}, { Rational[3, 8], Rational[7, 8]}, { Rational[13, 32], Rational[7, 8]}, { Rational[13, 32], Rational[29, 32]}, { Rational[3, 8], Rational[29, 32]}, { Rational[3, 8], Rational[15, 16]}, { Rational[3, 8], Rational[31, 32]}, { Rational[13, 32], Rational[31, 32]}, { Rational[13, 32], Rational[15, 16]}, { Rational[7, 16], Rational[15, 16]}, { Rational[7, 16], Rational[31, 32]}, { Rational[15, 32], Rational[31, 32]}, { Rational[15, 32], Rational[15, 16]}, { Rational[15, 32], Rational[29, 32]}, { Rational[7, 16], Rational[29, 32]}, { Rational[7, 16], Rational[7, 8]}, { Rational[15, 32], Rational[7, 8]}, { Rational[15, 32], Rational[27, 32]}, { Rational[15, 32], Rational[13, 16]}, { Rational[7, 16], Rational[13, 16]}, { Rational[7, 16], Rational[27, 32]}, { Rational[13, 32], Rational[27, 32]}, { Rational[3, 8], Rational[27, 32]}, { Rational[3, 8], Rational[13, 16]}, { Rational[13, 32], Rational[13, 16]}, { Rational[13, 32], Rational[25, 32]}, { Rational[3, 8], Rational[25, 32]}, { Rational[3, 8], Rational[3, 4]}, { Rational[13, 32], Rational[3, 4]}, { Rational[7, 16], Rational[3, 4]}, { Rational[7, 16], Rational[25, 32]}, { Rational[15, 32], Rational[25, 32]}, { Rational[15, 32], Rational[3, 4]}, { Rational[15, 32], Rational[23, 32]}, { Rational[7, 16], Rational[23, 32]}, { Rational[7, 16], Rational[11, 16]}, { Rational[15, 32], Rational[11, 16]}, { Rational[15, 32], Rational[21, 32]}, { Rational[15, 32], Rational[5, 8]}, { Rational[7, 16], Rational[5, 8]}, { Rational[7, 16], Rational[21, 32]}, { Rational[13, 32], Rational[21, 32]}, { Rational[13, 32], Rational[5, 8]}, { Rational[3, 8], Rational[5, 8]}, { Rational[3, 8], Rational[21, 32]}, { Rational[3, 8], Rational[11, 16]}, { Rational[13, 32], Rational[11, 16]}, { Rational[13, 32], Rational[23, 32]}, { Rational[3, 8], Rational[23, 32]}, { Rational[11, 32], Rational[23, 32]}, { Rational[11, 32], Rational[11, 16]}, { Rational[5, 16], Rational[11, 16]}, { Rational[5, 16], Rational[23, 32]}, { Rational[9, 32], Rational[23, 32]}, { Rational[1, 4], Rational[23, 32]}, { Rational[1, 4], Rational[11, 16]}, { Rational[9, 32], Rational[11, 16]}, { Rational[9, 32], Rational[21, 32]}, { Rational[1, 4], Rational[21, 32]}, { Rational[1, 4], Rational[5, 8]}, { Rational[9, 32], Rational[5, 8]}, { Rational[5, 16], Rational[5, 8]}, { Rational[5, 16], Rational[21, 32]}, { Rational[11, 32], Rational[21, 32]}, { Rational[11, 32], Rational[5, 8]}, { Rational[11, 32], Rational[19, 32]}, { Rational[11, 32], Rational[9, 16]}, { Rational[5, 16], Rational[9, 16]}, { Rational[5, 16], Rational[19, 32]}, { Rational[9, 32], Rational[19, 32]}, { Rational[1, 4], Rational[19, 32]}, { Rational[1, 4], Rational[9, 16]}, { Rational[9, 32], Rational[9, 16]}, { Rational[9, 32], Rational[17, 32]}, { Rational[1, 4], Rational[17, 32]}, { Rational[1, 4], Rational[1, 2]}, { Rational[9, 32], Rational[1, 2]}, { Rational[5, 16], Rational[1, 2]}, { Rational[5, 16], Rational[17, 32]}, { Rational[11, 32], Rational[17, 32]}, { Rational[11, 32], Rational[1, 2]}, { Rational[3, 8], Rational[1, 2]}, { Rational[13, 32], Rational[1, 2]}, { Rational[13, 32], Rational[17, 32]}, { Rational[3, 8], Rational[17, 32]}, { Rational[3, 8], Rational[9, 16]}, { Rational[3, 8], Rational[19, 32]}, { Rational[13, 32], Rational[19, 32]}, { Rational[13, 32], Rational[9, 16]}, { Rational[7, 16], Rational[9, 16]}, { Rational[7, 16], Rational[19, 32]}, { Rational[15, 32], Rational[19, 32]}, { Rational[15, 32], Rational[9, 16]}, { Rational[15, 32], Rational[17, 32]}, { Rational[7, 16], Rational[17, 32]}, { Rational[7, 16], Rational[1, 2]}, { Rational[15, 32], Rational[1, 2]}, { Rational[1, 2], Rational[1, 2]}, { Rational[17, 32], Rational[1, 2]}, { Rational[17, 32], Rational[17, 32]}, { Rational[1, 2], Rational[17, 32]}, { Rational[1, 2], Rational[9, 16]}, { Rational[1, 2], Rational[19, 32]}, { Rational[17, 32], Rational[19, 32]}, { Rational[17, 32], Rational[9, 16]}, { Rational[9, 16], Rational[9, 16]}, { Rational[9, 16], Rational[19, 32]}, { Rational[19, 32], Rational[19, 32]}, { Rational[19, 32], Rational[9, 16]}, { Rational[19, 32], Rational[17, 32]}, { Rational[9, 16], Rational[17, 32]}, { Rational[9, 16], Rational[1, 2]}, { Rational[19, 32], Rational[1, 2]}, { Rational[5, 8], Rational[1, 2]}, { Rational[5, 8], Rational[17, 32]}, { Rational[21, 32], Rational[17, 32]}, { Rational[21, 32], Rational[1, 2]}, { Rational[11, 16], Rational[1, 2]}, { Rational[23, 32], Rational[1, 2]}, { Rational[23, 32], Rational[17, 32]}, { Rational[11, 16], Rational[17, 32]}, { Rational[11, 16], Rational[9, 16]}, { Rational[23, 32], Rational[9, 16]}, { Rational[23, 32], Rational[19, 32]}, { Rational[11, 16], Rational[19, 32]}, { Rational[21, 32], Rational[19, 32]}, { Rational[21, 32], Rational[9, 16]}, { Rational[5, 8], Rational[9, 16]}, { Rational[5, 8], Rational[19, 32]}, { Rational[5, 8], Rational[5, 8]}, { Rational[5, 8], Rational[21, 32]}, { Rational[21, 32], Rational[21, 32]}, { Rational[21, 32], Rational[5, 8]}, { Rational[11, 16], Rational[5, 8]}, { Rational[23, 32], Rational[5, 8]}, { Rational[23, 32], Rational[21, 32]}, { Rational[11, 16], Rational[21, 32]}, { Rational[11, 16], Rational[11, 16]}, { Rational[23, 32], Rational[11, 16]}, { Rational[23, 32], Rational[23, 32]}, { Rational[11, 16], Rational[23, 32]}, { Rational[21, 32], Rational[23, 32]}, { Rational[21, 32], Rational[11, 16]}, { Rational[5, 8], Rational[11, 16]}, { Rational[5, 8], Rational[23, 32]}, { Rational[19, 32], Rational[23, 32]}, { Rational[9, 16], Rational[23, 32]}, { Rational[9, 16], Rational[11, 16]}, { Rational[19, 32], Rational[11, 16]}, { Rational[19, 32], Rational[21, 32]}, { Rational[19, 32], Rational[5, 8]}, { Rational[9, 16], Rational[5, 8]}, { Rational[9, 16], Rational[21, 32]}, { Rational[17, 32], Rational[21, 32]}, { Rational[17, 32], Rational[5, 8]}, { Rational[1, 2], Rational[5, 8]}, { Rational[1, 2], Rational[21, 32]}, { Rational[1, 2], Rational[11, 16]}, { Rational[17, 32], Rational[11, 16]}, { Rational[17, 32], Rational[23, 32]}, { Rational[1, 2], Rational[23, 32]}, { Rational[1, 2], Rational[3, 4]}, { Rational[1, 2], Rational[25, 32]}, { Rational[17, 32], Rational[25, 32]}, { Rational[17, 32], Rational[3, 4]}, { Rational[9, 16], Rational[3, 4]}, { Rational[19, 32], Rational[3, 4]}, { Rational[19, 32], Rational[25, 32]}, { Rational[9, 16], Rational[25, 32]}, { Rational[9, 16], Rational[13, 16]}, { Rational[19, 32], Rational[13, 16]}, { Rational[19, 32], Rational[27, 32]}, { Rational[9, 16], Rational[27, 32]}, { Rational[17, 32], Rational[27, 32]}, { Rational[17, 32], Rational[13, 16]}, { Rational[1, 2], Rational[13, 16]}, { Rational[1, 2], Rational[27, 32]}, { Rational[1, 2], Rational[7, 8]}, { Rational[17, 32], Rational[7, 8]}, { Rational[17, 32], Rational[29, 32]}, { Rational[1, 2], Rational[29, 32]}, { Rational[1, 2], Rational[15, 16]}, { Rational[1, 2], Rational[31, 32]}, { Rational[17, 32], Rational[31, 32]}, { Rational[17, 32], Rational[15, 16]}, { Rational[9, 16], Rational[15, 16]}, { Rational[9, 16], Rational[31, 32]}, { Rational[19, 32], Rational[31, 32]}, { Rational[19, 32], Rational[15, 16]}, { Rational[19, 32], Rational[29, 32]}, { Rational[9, 16], Rational[29, 32]}, { Rational[9, 16], Rational[7, 8]}, { Rational[19, 32], Rational[7, 8]}, { Rational[5, 8], Rational[7, 8]}, { Rational[21, 32], Rational[7, 8]}, { Rational[21, 32], Rational[29, 32]}, { Rational[5, 8], Rational[29, 32]}, { Rational[5, 8], Rational[15, 16]}, { Rational[5, 8], Rational[31, 32]}, { Rational[21, 32], Rational[31, 32]}, { Rational[21, 32], Rational[15, 16]}, { Rational[11, 16], Rational[15, 16]}, { Rational[11, 16], Rational[31, 32]}, { Rational[23, 32], Rational[31, 32]}, { Rational[23, 32], Rational[15, 16]}, { Rational[23, 32], Rational[29, 32]}, { Rational[11, 16], Rational[29, 32]}, { Rational[11, 16], Rational[7, 8]}, { Rational[23, 32], Rational[7, 8]}, { Rational[23, 32], Rational[27, 32]}, { Rational[23, 32], Rational[13, 16]}, { Rational[11, 16], Rational[13, 16]}, { Rational[11, 16], Rational[27, 32]}, { Rational[21, 32], Rational[27, 32]}, { Rational[5, 8], Rational[27, 32]}, { Rational[5, 8], Rational[13, 16]}, { Rational[21, 32], Rational[13, 16]}, { Rational[21, 32], Rational[25, 32]}, { Rational[5, 8], Rational[25, 32]}, { Rational[5, 8], Rational[3, 4]}, { Rational[21, 32], Rational[3, 4]}, { Rational[11, 16], Rational[3, 4]}, { Rational[11, 16], Rational[25, 32]}, { Rational[23, 32], Rational[25, 32]}, { Rational[23, 32], Rational[3, 4]}, { Rational[3, 4], Rational[3, 4]}, { Rational[3, 4], Rational[25, 32]}, { Rational[25, 32], Rational[25, 32]}, { Rational[25, 32], Rational[3, 4]}, { Rational[13, 16], Rational[3, 4]}, { Rational[27, 32], Rational[3, 4]}, { Rational[27, 32], Rational[25, 32]}, { Rational[13, 16], Rational[25, 32]}, { Rational[13, 16], Rational[13, 16]}, { Rational[27, 32], Rational[13, 16]}, { Rational[27, 32], Rational[27, 32]}, { Rational[13, 16], Rational[27, 32]}, { Rational[25, 32], Rational[27, 32]}, { Rational[25, 32], Rational[13, 16]}, { Rational[3, 4], Rational[13, 16]}, { Rational[3, 4], Rational[27, 32]}, { Rational[3, 4], Rational[7, 8]}, { Rational[25, 32], Rational[7, 8]}, { Rational[25, 32], Rational[29, 32]}, { Rational[3, 4], Rational[29, 32]}, { Rational[3, 4], Rational[15, 16]}, { Rational[3, 4], Rational[31, 32]}, { Rational[25, 32], Rational[31, 32]}, { Rational[25, 32], Rational[15, 16]}, { Rational[13, 16], Rational[15, 16]}, { Rational[13, 16], Rational[31, 32]}, { Rational[27, 32], Rational[31, 32]}, { Rational[27, 32], Rational[15, 16]}, { Rational[27, 32], Rational[29, 32]}, { Rational[13, 16], Rational[29, 32]}, { Rational[13, 16], Rational[7, 8]}, { Rational[27, 32], Rational[7, 8]}, { Rational[7, 8], Rational[7, 8]}, { Rational[29, 32], Rational[7, 8]}, { Rational[29, 32], Rational[29, 32]}, { Rational[7, 8], Rational[29, 32]}, { Rational[7, 8], Rational[15, 16]}, { Rational[7, 8], Rational[31, 32]}, { Rational[29, 32], Rational[31, 32]}, { Rational[29, 32], Rational[15, 16]}, { Rational[15, 16], Rational[15, 16]}, { Rational[15, 16], Rational[31, 32]}, { Rational[31, 32], Rational[31, 32]}, { Rational[31, 32], Rational[15, 16]}, { Rational[31, 32], Rational[29, 32]}, { Rational[15, 16], Rational[29, 32]}, { Rational[15, 16], Rational[7, 8]}, { Rational[31, 32], Rational[7, 8]}, { Rational[31, 32], Rational[27, 32]}, { Rational[31, 32], Rational[13, 16]}, { Rational[15, 16], Rational[13, 16]}, { Rational[15, 16], Rational[27, 32]}, { Rational[29, 32], Rational[27, 32]}, { Rational[7, 8], Rational[27, 32]}, { Rational[7, 8], Rational[13, 16]}, { Rational[29, 32], Rational[13, 16]}, { Rational[29, 32], Rational[25, 32]}, { Rational[7, 8], Rational[25, 32]}, { Rational[7, 8], Rational[3, 4]}, { Rational[29, 32], Rational[3, 4]}, { Rational[15, 16], Rational[3, 4]}, { Rational[15, 16], Rational[25, 32]}, { Rational[31, 32], Rational[25, 32]}, { Rational[31, 32], Rational[3, 4]}, { Rational[31, 32], Rational[23, 32]}, { Rational[15, 16], Rational[23, 32]}, { Rational[15, 16], Rational[11, 16]}, { Rational[31, 32], Rational[11, 16]}, { Rational[31, 32], Rational[21, 32]}, { Rational[31, 32], Rational[5, 8]}, { Rational[15, 16], Rational[5, 8]}, { Rational[15, 16], Rational[21, 32]}, { Rational[29, 32], Rational[21, 32]}, { Rational[29, 32], Rational[5, 8]}, { Rational[7, 8], Rational[5, 8]}, { Rational[7, 8], Rational[21, 32]}, { Rational[7, 8], Rational[11, 16]}, { Rational[29, 32], Rational[11, 16]}, { Rational[29, 32], Rational[23, 32]}, { Rational[7, 8], Rational[23, 32]}, { Rational[27, 32], Rational[23, 32]}, { Rational[27, 32], Rational[11, 16]}, { Rational[13, 16], Rational[11, 16]}, { Rational[13, 16], Rational[23, 32]}, { Rational[25, 32], Rational[23, 32]}, { Rational[3, 4], Rational[23, 32]}, { Rational[3, 4], Rational[11, 16]}, { Rational[25, 32], Rational[11, 16]}, { Rational[25, 32], Rational[21, 32]}, { Rational[3, 4], Rational[21, 32]}, { Rational[3, 4], Rational[5, 8]}, { Rational[25, 32], Rational[5, 8]}, { Rational[13, 16], Rational[5, 8]}, { Rational[13, 16], Rational[21, 32]}, { Rational[27, 32], Rational[21, 32]}, { Rational[27, 32], Rational[5, 8]}, { Rational[27, 32], Rational[19, 32]}, { Rational[27, 32], Rational[9, 16]}, { Rational[13, 16], Rational[9, 16]}, { Rational[13, 16], Rational[19, 32]}, { Rational[25, 32], Rational[19, 32]}, { Rational[3, 4], Rational[19, 32]}, { Rational[3, 4], Rational[9, 16]}, { Rational[25, 32], Rational[9, 16]}, { Rational[25, 32], Rational[17, 32]}, { Rational[3, 4], Rational[17, 32]}, { Rational[3, 4], Rational[1, 2]}, { Rational[25, 32], Rational[1, 2]}, { Rational[13, 16], Rational[1, 2]}, { Rational[13, 16], Rational[17, 32]}, { Rational[27, 32], Rational[17, 32]}, { Rational[27, 32], Rational[1, 2]}, { Rational[7, 8], Rational[1, 2]}, { Rational[29, 32], Rational[1, 2]}, { Rational[29, 32], Rational[17, 32]}, { Rational[7, 8], Rational[17, 32]}, { Rational[7, 8], Rational[9, 16]}, { Rational[7, 8], Rational[19, 32]}, { Rational[29, 32], Rational[19, 32]}, { Rational[29, 32], Rational[9, 16]}, { Rational[15, 16], Rational[9, 16]}, { Rational[15, 16], Rational[19, 32]}, { Rational[31, 32], Rational[19, 32]}, { Rational[31, 32], Rational[9, 16]}, { Rational[31, 32], Rational[17, 32]}, { Rational[15, 16], Rational[17, 32]}, { Rational[15, 16], Rational[1, 2]}, { Rational[31, 32], Rational[1, 2]}, { Rational[31, 32], Rational[15, 32]}, { Rational[31, 32], Rational[7, 16]}, { Rational[15, 16], Rational[7, 16]}, { Rational[15, 16], Rational[15, 32]}, { Rational[29, 32], Rational[15, 32]}, { Rational[7, 8], Rational[15, 32]}, { Rational[7, 8], Rational[7, 16]}, { Rational[29, 32], Rational[7, 16]}, { Rational[29, 32], Rational[13, 32]}, { Rational[7, 8], Rational[13, 32]}, { Rational[7, 8], Rational[3, 8]}, { Rational[29, 32], Rational[3, 8]}, { Rational[15, 16], Rational[3, 8]}, { Rational[15, 16], Rational[13, 32]}, { Rational[31, 32], Rational[13, 32]}, { Rational[31, 32], Rational[3, 8]}, { Rational[31, 32], Rational[11, 32]}, { Rational[15, 16], Rational[11, 32]}, { Rational[15, 16], Rational[5, 16]}, { Rational[31, 32], Rational[5, 16]}, { Rational[31, 32], Rational[9, 32]}, { Rational[31, 32], Rational[1, 4]}, { Rational[15, 16], Rational[1, 4]}, { Rational[15, 16], Rational[9, 32]}, { Rational[29, 32], Rational[9, 32]}, { Rational[29, 32], Rational[1, 4]}, { Rational[7, 8], Rational[1, 4]}, { Rational[7, 8], Rational[9, 32]}, { Rational[7, 8], Rational[5, 16]}, { Rational[29, 32], Rational[5, 16]}, { Rational[29, 32], Rational[11, 32]}, { Rational[7, 8], Rational[11, 32]}, { Rational[27, 32], Rational[11, 32]}, { Rational[13, 16], Rational[11, 32]}, { Rational[13, 16], Rational[5, 16]}, { Rational[27, 32], Rational[5, 16]}, { Rational[27, 32], Rational[9, 32]}, { Rational[27, 32], Rational[1, 4]}, { Rational[13, 16], Rational[1, 4]}, { Rational[13, 16], Rational[9, 32]}, { Rational[25, 32], Rational[9, 32]}, { Rational[25, 32], Rational[1, 4]}, { Rational[3, 4], Rational[1, 4]}, { Rational[3, 4], Rational[9, 32]}, { Rational[3, 4], Rational[5, 16]}, { Rational[25, 32], Rational[5, 16]}, { Rational[25, 32], Rational[11, 32]}, { Rational[3, 4], Rational[11, 32]}, { Rational[3, 4], Rational[3, 8]}, { Rational[3, 4], Rational[13, 32]}, { Rational[25, 32], Rational[13, 32]}, { Rational[25, 32], Rational[3, 8]}, { Rational[13, 16], Rational[3, 8]}, { Rational[27, 32], Rational[3, 8]}, { Rational[27, 32], Rational[13, 32]}, { Rational[13, 16], Rational[13, 32]}, { Rational[13, 16], Rational[7, 16]}, { Rational[27, 32], Rational[7, 16]}, { Rational[27, 32], Rational[15, 32]}, { Rational[13, 16], Rational[15, 32]}, { Rational[25, 32], Rational[15, 32]}, { Rational[25, 32], Rational[7, 16]}, { Rational[3, 4], Rational[7, 16]}, { Rational[3, 4], Rational[15, 32]}, { Rational[23, 32], Rational[15, 32]}, { Rational[11, 16], Rational[15, 32]}, { Rational[11, 16], Rational[7, 16]}, 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Rational[14, 27]}, { Rational[7, 9], Rational[14, 27]}, { Rational[22, 27], Rational[14, 27]}, { Rational[23, 27], Rational[14, 27]}, { Rational[23, 27], Rational[13, 27]}, { Rational[22, 27], Rational[13, 27]}, { Rational[7, 9], Rational[13, 27]}, { Rational[7, 9], Rational[4, 9]}, { Rational[22, 27], Rational[4, 9]}, { Rational[23, 27], Rational[4, 9]}, { Rational[8, 9], Rational[4, 9]}, { Rational[25, 27], Rational[4, 9]}, { Rational[26, 27], Rational[4, 9]}, { Rational[26, 27], Rational[13, 27]}, { Rational[25, 27], Rational[13, 27]}, { Rational[8, 9], Rational[13, 27]}, { Rational[8, 9], Rational[14, 27]}, { Rational[25, 27], Rational[14, 27]}, { Rational[26, 27], Rational[14, 27]}, { Rational[26, 27], Rational[5, 9]}, { Rational[25, 27], Rational[5, 9]}, { Rational[8, 9], Rational[5, 9]}, { Rational[8, 9], Rational[16, 27]}, { Rational[25, 27], Rational[16, 27]}, { Rational[26, 27], Rational[16, 27]}, { Rational[26, 27], Rational[17, 27]}, { Rational[25, 27], Rational[17, 27]}, { Rational[8, 9], Rational[17, 27]}, { Rational[23, 27], Rational[17, 27]}, { Rational[22, 27], Rational[17, 27]}, { Rational[7, 9], Rational[17, 27]}, { Rational[7, 9], Rational[16, 27]}, { Rational[22, 27], Rational[16, 27]}, { Rational[23, 27], Rational[16, 27]}, { Rational[23, 27], Rational[5, 9]}, { Rational[22, 27], Rational[5, 9]}, { Rational[7, 9], Rational[5, 9]}, { Rational[20, 27], Rational[5, 9]}, { Rational[19, 27], Rational[5, 9]}, { Rational[2, 3], Rational[5, 9]}, { Rational[2, 3], Rational[16, 27]}, { Rational[19, 27], Rational[16, 27]}, { Rational[20, 27], Rational[16, 27]}, { Rational[20, 27], Rational[17, 27]}, { Rational[19, 27], Rational[17, 27]}, { Rational[2, 3], Rational[17, 27]}, { Rational[17, 27], Rational[17, 27]}, { Rational[16, 27], Rational[17, 27]}, { Rational[5, 9], Rational[17, 27]}, { Rational[5, 9], Rational[16, 27]}, { Rational[16, 27], Rational[16, 27]}, { Rational[17, 27], Rational[16, 27]}, { Rational[17, 27], Rational[5, 9]}, { Rational[16, 27], Rational[5, 9]}, { Rational[5, 9], Rational[5, 9]}, { Rational[14, 27], Rational[5, 9]}, { Rational[13, 27], Rational[5, 9]}, { Rational[4, 9], Rational[5, 9]}, { Rational[4, 9], Rational[16, 27]}, { Rational[13, 27], Rational[16, 27]}, { Rational[14, 27], Rational[16, 27]}, { Rational[14, 27], Rational[17, 27]}, { Rational[13, 27], Rational[17, 27]}, { Rational[4, 9], Rational[17, 27]}, { Rational[11, 27], Rational[17, 27]}, { Rational[10, 27], Rational[17, 27]}, { Rational[1, 3], Rational[17, 27]}, { Rational[1, 3], Rational[16, 27]}, { Rational[10, 27], Rational[16, 27]}, { Rational[11, 27], Rational[16, 27]}, { Rational[11, 27], Rational[5, 9]}, { Rational[10, 27], Rational[5, 9]}, { Rational[1, 3], Rational[5, 9]}, { Rational[1, 3], Rational[14, 27]}, { Rational[10, 27], Rational[14, 27]}, { Rational[11, 27], Rational[14, 27]}, { Rational[11, 27], Rational[13, 27]}, { Rational[10, 27], Rational[13, 27]}, { Rational[1, 3], Rational[13, 27]}, { Rational[1, 3], Rational[4, 9]}, { Rational[10, 27], Rational[4, 9]}, { Rational[11, 27], Rational[4, 9]}, { Rational[4, 9], Rational[4, 9]}, { Rational[13, 27], Rational[4, 9]}, { Rational[14, 27], Rational[4, 9]}, { Rational[14, 27], Rational[13, 27]}, { Rational[13, 27], Rational[13, 27]}, { Rational[4, 9], Rational[13, 27]}, { Rational[4, 9], Rational[14, 27]}, { Rational[13, 27], Rational[14, 27]}, { Rational[14, 27], Rational[14, 27]}, { Rational[5, 9], Rational[14, 27]}, { Rational[16, 27], Rational[14, 27]}, { Rational[17, 27], Rational[14, 27]}, { Rational[17, 27], Rational[13, 27]}, { Rational[16, 27], Rational[13, 27]}, { Rational[5, 9], Rational[13, 27]}, { Rational[5, 9], Rational[4, 9]}, { Rational[16, 27], Rational[4, 9]}, { Rational[17, 27], Rational[4, 9]}, { Rational[17, 27], Rational[11, 27]}, { Rational[16, 27], Rational[11, 27]}, { Rational[5, 9], Rational[11, 27]}, { Rational[5, 9], Rational[10, 27]}, { Rational[16, 27], Rational[10, 27]}, { Rational[17, 27], Rational[10, 27]}, { Rational[17, 27], Rational[1, 3]}, { Rational[16, 27], Rational[1, 3]}, { Rational[5, 9], Rational[1, 3]}, { Rational[14, 27], Rational[1, 3]}, { Rational[13, 27], Rational[1, 3]}, { Rational[4, 9], Rational[1, 3]}, { Rational[4, 9], Rational[10, 27]}, { Rational[13, 27], Rational[10, 27]}, { Rational[14, 27], Rational[10, 27]}, { Rational[14, 27], Rational[11, 27]}, { Rational[13, 27], Rational[11, 27]}, { Rational[4, 9], Rational[11, 27]}, { Rational[11, 27], Rational[11, 27]}, { Rational[10, 27], Rational[11, 27]}, { Rational[1, 3], Rational[11, 27]}, { Rational[1, 3], Rational[10, 27]}, { Rational[10, 27], Rational[10, 27]}, { Rational[11, 27], Rational[10, 27]}, { Rational[11, 27], Rational[1, 3]}, { Rational[10, 27], Rational[1, 3]}, { Rational[1, 3], Rational[1, 3]}, { Rational[8, 27], Rational[1, 3]}, { Rational[7, 27], Rational[1, 3]}, { Rational[2, 9], Rational[1, 3]}, { Rational[2, 9], Rational[10, 27]}, { Rational[7, 27], Rational[10, 27]}, { Rational[8, 27], Rational[10, 27]}, { Rational[8, 27], Rational[11, 27]}, { Rational[7, 27], Rational[11, 27]}, { Rational[2, 9], Rational[11, 27]}, { Rational[5, 27], Rational[11, 27]}, { Rational[4, 27], Rational[11, 27]}, { Rational[1, 9], Rational[11, 27]}, { Rational[1, 9], Rational[10, 27]}, { Rational[4, 27], Rational[10, 27]}, { Rational[5, 27], Rational[10, 27]}, { Rational[5, 27], Rational[1, 3]}, { Rational[4, 27], Rational[1, 3]}, { Rational[1, 9], Rational[1, 3]}, { Rational[2, 27], Rational[1, 3]}, { Rational[1, 27], Rational[1, 3]}, {0, Rational[1, 3]}, { 0, Rational[10, 27]}, {Rational[1, 27], Rational[10, 27]}, { Rational[2, 27], Rational[10, 27]}, { Rational[2, 27], Rational[11, 27]}, { Rational[1, 27], Rational[11, 27]}, {0, Rational[11, 27]}, { 0, Rational[4, 9]}, {Rational[1, 27], Rational[4, 9]}, { Rational[2, 27], Rational[4, 9]}, { Rational[2, 27], Rational[13, 27]}, { Rational[1, 27], Rational[13, 27]}, {0, Rational[13, 27]}, { 0, Rational[14, 27]}, {Rational[1, 27], Rational[14, 27]}, { Rational[2, 27], Rational[14, 27]}, { Rational[1, 9], Rational[14, 27]}, { Rational[4, 27], Rational[14, 27]}, { Rational[5, 27], Rational[14, 27]}, { Rational[5, 27], Rational[13, 27]}, { Rational[4, 27], Rational[13, 27]}, { Rational[1, 9], Rational[13, 27]}, { Rational[1, 9], Rational[4, 9]}, { Rational[4, 27], Rational[4, 9]}, { Rational[5, 27], Rational[4, 9]}, { Rational[2, 9], Rational[4, 9]}, { Rational[7, 27], Rational[4, 9]}, { Rational[8, 27], Rational[4, 9]}, { Rational[8, 27], Rational[13, 27]}, { Rational[7, 27], Rational[13, 27]}, { Rational[2, 9], Rational[13, 27]}, { Rational[2, 9], Rational[14, 27]}, { Rational[7, 27], Rational[14, 27]}, { Rational[8, 27], Rational[14, 27]}, { Rational[8, 27], Rational[5, 9]}, { Rational[7, 27], Rational[5, 9]}, { Rational[2, 9], Rational[5, 9]}, { Rational[2, 9], Rational[16, 27]}, { Rational[7, 27], Rational[16, 27]}, { Rational[8, 27], Rational[16, 27]}, { Rational[8, 27], Rational[17, 27]}, { Rational[7, 27], Rational[17, 27]}, { Rational[2, 9], Rational[17, 27]}, { Rational[5, 27], Rational[17, 27]}, { Rational[4, 27], Rational[17, 27]}, { Rational[1, 9], Rational[17, 27]}, { Rational[1, 9], Rational[16, 27]}, { Rational[4, 27], Rational[16, 27]}, { Rational[5, 27], Rational[16, 27]}, { Rational[5, 27], Rational[5, 9]}, { Rational[4, 27], Rational[5, 9]}, { Rational[1, 9], Rational[5, 9]}, { Rational[2, 27], Rational[5, 9]}, { Rational[1, 27], Rational[5, 9]}, {0, Rational[5, 9]}, { 0, Rational[16, 27]}, {Rational[1, 27], Rational[16, 27]}, { Rational[2, 27], Rational[16, 27]}, { Rational[2, 27], Rational[17, 27]}, { Rational[1, 27], Rational[17, 27]}, {0, Rational[17, 27]}, { 0, Rational[2, 3]}, {Rational[1, 27], Rational[2, 3]}, { Rational[2, 27], Rational[2, 3]}, { Rational[2, 27], Rational[19, 27]}, { Rational[1, 27], Rational[19, 27]}, {0, Rational[19, 27]}, { 0, Rational[20, 27]}, {Rational[1, 27], Rational[20, 27]}, { Rational[2, 27], Rational[20, 27]}, { Rational[1, 9], Rational[20, 27]}, { Rational[4, 27], Rational[20, 27]}, { Rational[5, 27], Rational[20, 27]}, { Rational[5, 27], Rational[19, 27]}, { Rational[4, 27], Rational[19, 27]}, { Rational[1, 9], Rational[19, 27]}, { Rational[1, 9], Rational[2, 3]}, { Rational[4, 27], Rational[2, 3]}, { Rational[5, 27], Rational[2, 3]}, { Rational[2, 9], Rational[2, 3]}, { Rational[7, 27], Rational[2, 3]}, { Rational[8, 27], Rational[2, 3]}, { Rational[8, 27], Rational[19, 27]}, { Rational[7, 27], Rational[19, 27]}, { Rational[2, 9], Rational[19, 27]}, { Rational[2, 9], Rational[20, 27]}, { Rational[7, 27], Rational[20, 27]}, { Rational[8, 27], Rational[20, 27]}, { Rational[8, 27], Rational[7, 9]}, { Rational[7, 27], Rational[7, 9]}, { Rational[2, 9], Rational[7, 9]}, { Rational[2, 9], Rational[22, 27]}, { Rational[7, 27], Rational[22, 27]}, { Rational[8, 27], Rational[22, 27]}, { Rational[8, 27], Rational[23, 27]}, { Rational[7, 27], Rational[23, 27]}, { Rational[2, 9], Rational[23, 27]}, { Rational[5, 27], Rational[23, 27]}, { Rational[4, 27], Rational[23, 27]}, { Rational[1, 9], Rational[23, 27]}, { Rational[1, 9], Rational[22, 27]}, { Rational[4, 27], Rational[22, 27]}, { Rational[5, 27], Rational[22, 27]}, { Rational[5, 27], Rational[7, 9]}, { Rational[4, 27], Rational[7, 9]}, { Rational[1, 9], Rational[7, 9]}, { Rational[2, 27], Rational[7, 9]}, { Rational[1, 27], Rational[7, 9]}, {0, Rational[7, 9]}, { 0, Rational[22, 27]}, {Rational[1, 27], Rational[22, 27]}, { Rational[2, 27], Rational[22, 27]}, { Rational[2, 27], Rational[23, 27]}, { Rational[1, 27], Rational[23, 27]}, {0, Rational[23, 27]}, { 0, Rational[8, 9]}, {Rational[1, 27], Rational[8, 9]}, { Rational[2, 27], Rational[8, 9]}, { Rational[2, 27], Rational[25, 27]}, { Rational[1, 27], Rational[25, 27]}, {0, Rational[25, 27]}, { 0, Rational[26, 27]}, {Rational[1, 27], Rational[26, 27]}, { Rational[2, 27], Rational[26, 27]}, { Rational[1, 9], Rational[26, 27]}, { Rational[4, 27], Rational[26, 27]}, { Rational[5, 27], Rational[26, 27]}, { Rational[5, 27], Rational[25, 27]}, { Rational[4, 27], Rational[25, 27]}, { Rational[1, 9], Rational[25, 27]}, { Rational[1, 9], Rational[8, 9]}, { Rational[4, 27], Rational[8, 9]}, { Rational[5, 27], Rational[8, 9]}, { Rational[2, 9], Rational[8, 9]}, { Rational[7, 27], Rational[8, 9]}, { Rational[8, 27], Rational[8, 9]}, { Rational[8, 27], Rational[25, 27]}, { Rational[7, 27], Rational[25, 27]}, { Rational[2, 9], Rational[25, 27]}, { Rational[2, 9], Rational[26, 27]}, { Rational[7, 27], Rational[26, 27]}, { Rational[8, 27], Rational[26, 27]}, { Rational[1, 3], Rational[26, 27]}, { Rational[10, 27], Rational[26, 27]}, { Rational[11, 27], Rational[26, 27]}, { Rational[11, 27], Rational[25, 27]}, { Rational[10, 27], Rational[25, 27]}, { Rational[1, 3], Rational[25, 27]}, { Rational[1, 3], Rational[8, 9]}, { Rational[10, 27], Rational[8, 9]}, { Rational[11, 27], Rational[8, 9]}, { Rational[4, 9], Rational[8, 9]}, { Rational[13, 27], Rational[8, 9]}, { Rational[14, 27], Rational[8, 9]}, { Rational[14, 27], Rational[25, 27]}, { Rational[13, 27], Rational[25, 27]}, { Rational[4, 9], Rational[25, 27]}, { Rational[4, 9], Rational[26, 27]}, { Rational[13, 27], Rational[26, 27]}, { Rational[14, 27], Rational[26, 27]}, { Rational[5, 9], Rational[26, 27]}, { Rational[16, 27], Rational[26, 27]}, { Rational[17, 27], Rational[26, 27]}, { Rational[17, 27], Rational[25, 27]}, { Rational[16, 27], Rational[25, 27]}, { Rational[5, 9], Rational[25, 27]}, { Rational[5, 9], Rational[8, 9]}, { Rational[16, 27], Rational[8, 9]}, { Rational[17, 27], Rational[8, 9]}, { Rational[17, 27], Rational[23, 27]}, { Rational[16, 27], Rational[23, 27]}, { Rational[5, 9], Rational[23, 27]}, { Rational[5, 9], Rational[22, 27]}, { Rational[16, 27], Rational[22, 27]}, { Rational[17, 27], Rational[22, 27]}, { Rational[17, 27], Rational[7, 9]}, { Rational[16, 27], Rational[7, 9]}, { Rational[5, 9], Rational[7, 9]}, { Rational[14, 27], Rational[7, 9]}, { Rational[13, 27], Rational[7, 9]}, { Rational[4, 9], Rational[7, 9]}, { Rational[4, 9], Rational[22, 27]}, { Rational[13, 27], Rational[22, 27]}, { Rational[14, 27], Rational[22, 27]}, { Rational[14, 27], Rational[23, 27]}, { Rational[13, 27], Rational[23, 27]}, { Rational[4, 9], Rational[23, 27]}, { Rational[11, 27], Rational[23, 27]}, { Rational[10, 27], Rational[23, 27]}, { Rational[1, 3], Rational[23, 27]}, { Rational[1, 3], Rational[22, 27]}, { Rational[10, 27], Rational[22, 27]}, { Rational[11, 27], Rational[22, 27]}, { Rational[11, 27], Rational[7, 9]}, { Rational[10, 27], Rational[7, 9]}, { Rational[1, 3], Rational[7, 9]}, { Rational[1, 3], Rational[20, 27]}, { Rational[10, 27], Rational[20, 27]}, { Rational[11, 27], Rational[20, 27]}, { Rational[11, 27], Rational[19, 27]}, { Rational[10, 27], Rational[19, 27]}, { Rational[1, 3], Rational[19, 27]}, { Rational[1, 3], Rational[2, 3]}, { Rational[10, 27], Rational[2, 3]}, { Rational[11, 27], Rational[2, 3]}, { Rational[4, 9], Rational[2, 3]}, { Rational[13, 27], Rational[2, 3]}, { Rational[14, 27], Rational[2, 3]}, { Rational[14, 27], Rational[19, 27]}, { Rational[13, 27], Rational[19, 27]}, { Rational[4, 9], Rational[19, 27]}, { Rational[4, 9], Rational[20, 27]}, { Rational[13, 27], Rational[20, 27]}, { Rational[14, 27], Rational[20, 27]}, { Rational[5, 9], Rational[20, 27]}, { Rational[16, 27], Rational[20, 27]}, { Rational[17, 27], Rational[20, 27]}, { Rational[17, 27], Rational[19, 27]}, { Rational[16, 27], Rational[19, 27]}, { Rational[5, 9], Rational[19, 27]}, { Rational[5, 9], Rational[2, 3]}, { Rational[16, 27], Rational[2, 3]}, { Rational[17, 27], Rational[2, 3]}, { Rational[2, 3], Rational[2, 3]}, { Rational[19, 27], Rational[2, 3]}, { Rational[20, 27], Rational[2, 3]}, { Rational[20, 27], Rational[19, 27]}, { Rational[19, 27], Rational[19, 27]}, { Rational[2, 3], Rational[19, 27]}, { Rational[2, 3], Rational[20, 27]}, { Rational[19, 27], Rational[20, 27]}, { Rational[20, 27], Rational[20, 27]}, { Rational[7, 9], Rational[20, 27]}, { Rational[22, 27], Rational[20, 27]}, { Rational[23, 27], Rational[20, 27]}, { Rational[23, 27], Rational[19, 27]}, { Rational[22, 27], Rational[19, 27]}, { Rational[7, 9], Rational[19, 27]}, { Rational[7, 9], Rational[2, 3]}, { Rational[22, 27], Rational[2, 3]}, { Rational[23, 27], Rational[2, 3]}, { Rational[8, 9], Rational[2, 3]}, { Rational[25, 27], Rational[2, 3]}, { Rational[26, 27], Rational[2, 3]}, { Rational[26, 27], Rational[19, 27]}, { Rational[25, 27], Rational[19, 27]}, { Rational[8, 9], Rational[19, 27]}, { Rational[8, 9], Rational[20, 27]}, { Rational[25, 27], Rational[20, 27]}, { Rational[26, 27], Rational[20, 27]}, { Rational[26, 27], Rational[7, 9]}, { Rational[25, 27], Rational[7, 9]}, { Rational[8, 9], Rational[7, 9]}, { Rational[8, 9], Rational[22, 27]}, { Rational[25, 27], Rational[22, 27]}, { Rational[26, 27], Rational[22, 27]}, { Rational[26, 27], Rational[23, 27]}, { Rational[25, 27], Rational[23, 27]}, { Rational[8, 9], Rational[23, 27]}, { Rational[23, 27], Rational[23, 27]}, { Rational[22, 27], Rational[23, 27]}, { Rational[7, 9], Rational[23, 27]}, { Rational[7, 9], Rational[22, 27]}, { Rational[22, 27], Rational[22, 27]}, { Rational[23, 27], Rational[22, 27]}, { Rational[23, 27], Rational[7, 9]}, { Rational[22, 27], Rational[7, 9]}, { Rational[7, 9], Rational[7, 9]}, { Rational[20, 27], Rational[7, 9]}, { Rational[19, 27], Rational[7, 9]}, { Rational[2, 3], Rational[7, 9]}, { Rational[2, 3], Rational[22, 27]}, { Rational[19, 27], Rational[22, 27]}, { Rational[20, 27], Rational[22, 27]}, { Rational[20, 27], Rational[23, 27]}, { Rational[19, 27], Rational[23, 27]}, { Rational[2, 3], Rational[23, 27]}, { Rational[2, 3], Rational[8, 9]}, { Rational[19, 27], Rational[8, 9]}, { Rational[20, 27], Rational[8, 9]}, { Rational[20, 27], Rational[25, 27]}, { Rational[19, 27], Rational[25, 27]}, { Rational[2, 3], Rational[25, 27]}, { Rational[2, 3], Rational[26, 27]}, { Rational[19, 27], Rational[26, 27]}, { Rational[20, 27], Rational[26, 27]}, { Rational[7, 9], Rational[26, 27]}, 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