(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 132697, 3036] NotebookOptionsPosition[ 125687, 2871] NotebookOutlinePosition[ 127454, 2923] CellTagsIndexPosition[ 127411, 2920] WindowTitle->HexagonalSpiralPoints | Example Notebook WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Basic Examples", "\[NonBreakingSpace]", Cell["(4)", "ExampleCount"], "\[NonBreakingSpace]" }], "Subsection", TaggingRules->{}, CellID->462042388], Cell["Here are the points on the first four legs of the spiral:", "Text", TaggingRules->{}, CellChangeTimes->{{3.758294603410687*^9, 3.758294606202675*^9}, { 3.7583064174114933`*^9, 3.7583064332259207`*^9}, {3.758309373849429*^9, 3.758309376532813*^9}}, CellID->13078159], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "[", "4", "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.758294572868843*^9, 3.758294589685771*^9}, { 3.758306108060484*^9, 3.7583061083655787`*^9}, 3.7583089011954947`*^9, 3.758309812193181*^9, {3.758403500944913*^9, 3.758403502469702*^9}, { 3.7584035417353277`*^9, 3.7584035418689547`*^9}, 3.759008498070698*^9}, CellLabel->"In[47]:=", CellID->443871870], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox["1", "2"], ",", FractionBox[ SqrtBox["3"], "2"]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", SqrtBox["3"]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", SqrtBox["3"]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["3", "2"]}], ",", FractionBox[ SqrtBox["3"], "2"]}], "}"}]}], "}"}]], "Output", TaggingRules->{}, CellChangeTimes->{{3.758294583720296*^9, 3.7582945905008698`*^9}, 3.758306109400671*^9, 3.758308903069747*^9, 3.758309463092802*^9, 3.7583756973081713`*^9, {3.7584034941210546`*^9, 3.758403504750532*^9}, 3.758403542741929*^9, 3.759008511533997*^9}, CellLabel->"Out[47]=", CellID->876657973] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[Cell["\t", "ExampleDelimiter"], $Line = 0; Null]], "ExampleDelimiter", TaggingRules->{}, CellID->64435967], Cell["This shows the sequence of points in order for six sides:", "Text", TaggingRules->{}, CellChangeTimes->{{3.758294807060244*^9, 3.7582948264117727`*^9}, { 3.758375708403635*^9, 3.758375721691144*^9}}, CellID->697960947], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"s", "=", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "[", "6", "]"}]}], "}"}], ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"Pink", ",", RowBox[{"PointSize", "[", ".03", "]"}], ",", RowBox[{"Point", "[", "s", "]"}]}], "}"}], ",", RowBox[{"Arrow", "@", RowBox[{"Partition", "[", RowBox[{"s", ",", "2", ",", "1"}], "]"}]}]}], "}"}], "]"}]}], "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.7582946129978724`*^9, 3.758294769246249*^9}, 3.75830892474617*^9, {3.758309483842639*^9, 3.758309513117497*^9}, 3.758309812198606*^9, {3.7584035208566017`*^9, 3.758403549148116*^9}, 3.759008498075408*^9}, CellLabel->"In[48]:=", CellID->1710216102], Cell[BoxData[ GraphicsBox[{ {RGBColor[1, 0.5, 0.5], PointSize[0.03], PointBox[NCache[{{0, 0}, { Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { 0, 3^Rational[1, 2]}, {-1, 3^Rational[1, 2]}, { Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}, {-1, 0}, { Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2] 3^Rational[1, 2]}}, {{0, 0}, {0.5, 0.8660254037844386}, { 0, 1.7320508075688772`}, {-1, 1.7320508075688772`}, {-1.5, 0.8660254037844386}, {-1, 0}, {-0.5, -0.8660254037844386}, { 0.5, -0.8660254037844386}}]]}, ArrowBox[NCache[{{{0, 0}, { Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}}, {{ Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { 0, 3^Rational[1, 2]}}, {{ 0, 3^Rational[1, 2]}, {-1, 3^Rational[1, 2]}}, {{-1, 3^ Rational[1, 2]}, { Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}}, {{ Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}, {-1, 0}}, {{-1, 0}, {Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}}, {{ Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2] 3^Rational[1, 2]}}}, {{{0, 0}, {0.5, 0.8660254037844386}}, {{0.5, 0.8660254037844386}, { 0, 1.7320508075688772`}}, {{ 0, 1.7320508075688772`}, {-1, 1.7320508075688772`}}, {{-1, 1.7320508075688772`}, {-1.5, 0.8660254037844386}}, {{-1.5, 0.8660254037844386}, {-1, 0}}, {{-1, 0}, {-0.5, -0.8660254037844386}}, {{-0.5, -0.8660254037844386}, { 0.5, -0.8660254037844386}}}]]}]], "Output", TaggingRules->{}, CellChangeTimes->{{3.758309505709717*^9, 3.7583095147234383`*^9}, 3.75837573199726*^9, 3.7583813758990793`*^9, {3.7583814430312977`*^9, 3.7583814659436703`*^9}, 3.758381669988155*^9, {3.758403508948303*^9, 3.758403550128961*^9}, 3.759008514748505*^9}, CellLabel->"Out[48]=", CellID->1560128727] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[Cell["\t", "ExampleDelimiter"], $Line = 0; Null]], "ExampleDelimiter", TaggingRules->{}, CellID->643539281], Cell["40 black sides with 20 red sides overlaid:", "Text", TaggingRules->{}, CellChangeTimes->{{3.7582948913614197`*^9, 3.7582948958539753`*^9}, { 3.758403960906299*^9, 3.758403962286571*^9}, {3.758405931354084*^9, 3.7584059436661863`*^9}}, CellID->1925797651], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Line", "@", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "[", "40", "]"}]}], ",", "Red", ",", RowBox[{"Line", "@", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "[", "20", "]"}]}]}], "}"}], "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.758294852948044*^9, 3.758294882009931*^9}, { 3.7583061345745792`*^9, 3.7583061351011066`*^9}, {3.7583061709626017`*^9, 3.7583062164984818`*^9}, {3.758308720401732*^9, 3.758308724991219*^9}, { 3.7583095646285057`*^9, 3.7583095646303167`*^9}, {3.758309812202683*^9, 3.758309812203953*^9}, {3.759008498080146*^9, 3.759008498080914*^9}}, CellLabel->"In[49]:=", CellID->509996249], Cell[BoxData[ GraphicsBox[{LineBox[CompressedData[" 1:eJxdlltsVFUYhafttExtsdMrlxIZ1GhUMsyDF6hTzg41RShTavUBeLGKRB9M IYSHphFtYqIIxhApIfIgrWlMA4g1CjGm49mS2AzeIsYEHqiUilhtlEFEpq1O lb0+khmbtM3Xfc6//3+ttfd0ydNb27cUBgKB/v++b/z+/1eB+znmHV63YaL7 xKSnv34cP+J4xvtn9sbXZf8orPWMP8TzRY7Tvv4+FC9XPZ6f9Cfc++e9E7Ce S1NnfLhE9bw3qXcL633st4j+XlixKnouFTQp1fN3wn84vu6/CF9xPOu/y/sZ h0W2n/q/sa79U8O/8v579PcN80qPGZ6big/AC+jvbbiK/l/jfc0TMPp1LS7N g2aU/ae1v/cp/aW1v/cq/X+t/b333XqZ6vRMDQ/CC9G7H56rjexBWPoFbSf1 5jgO2ZfZL+i43D6R51/Iqkxxkn7t38xTQP3v8vTI+PtUn/fG/P357B3I44z3 FhxEnwvUC6FPJKH9lJ8Qc1ckw+rX3Em/dax35uUjaLYw7+3U34cebex/YdlD p8vbK00f/V2CbVZ5moSHyEMaPkt+rsE/Op5jR6j/u+My+w77TzmuQL8ZT3pW 2Wc1n1fqOCxdbHXjYvy4D72X4kc3+qzAj63o14wf+G9fJw8fwtJtSfJzeCKr PJ2Bd5C/X+CvNJ85Rf0fNJ9pT2h/5bPM3NGam4ew/Incm1Teqsz5tZqvhvVz LZo/ftM/9GnDv03o9wz+pdC3C/82u+drTFrzedtgLUcbd8IRzefvhsOc7wPw LvwagPvxqy6m/T7Br3BC/eg+qLR/0u91x9V2KJF7fupsWnp4Ov810j0Wa4zg 90Xp4Ufx+2fp4XfhdxS9e/H7OfwYwu/T0sN2OMEz/jhsXaDG/Ktwz7Tu16Dm sX1Z3Se1cBv+3gVvw9/7Na/di7+LGtSP8l9hQq3qV/d3pflgvebRfVotH/Y2 JHV+68zcNdJD91ONCeL/UvLx2XLpufJmfvD/EPk4hP8nycdK/PqJfIyuSuw5 4s03X0gP7yrcW6TzXNokPpvReV4MPyg9/Afges5zSx6X2afgJvJwhv2Pk4cE eTlFHl4hL6OOa+32htzPn3l2WPp5xa7/hfaex3L1ma+87DCNy8jT5Uel7yPk KbtW+m8kTyeln+0kT8XkpaRAeRogD7EC5alZ/cMB2wF3FOq+6IZ1npqSvbCZ 0X1xDO7TPwBeCv42q/yMw7NZ5WcT+YqSnwHydYz89JKvEfJTul7z6b6pNd9L H/8v6ad8bWhO3ir9zGCz9Ltb+pnxNdLXkL/l5Gsj+XupRf5sJ3+bydce8leC v6ulnzlMnrqkn2l1+tbDAfM8nJZ+3m7YrfasbhyEI9O6j0bgsPTzL8K7pJ8t TORyhY3AT5K3g+Sxm7zVNKjf/eTtjVju59U8G27VvF86XmCDD0uPS47r7RXp 7Wmc2+xHj0vPfwE1plG8 "]], {RGBColor[1, 0, 0], LineBox[NCache[{{0, 0}, { Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { 0, 3^Rational[1, 2]}, {-1, 3^Rational[1, 2]}, { Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}, {-1, 0}, { Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2] 3^Rational[1, 2]}, {1, 0}, { Rational[3, 2], Rational[1, 2] 3^Rational[1, 2]}, { 1, 3^Rational[1, 2]}, { Rational[1, 2], Rational[3, 2] 3^Rational[1, 2]}, { Rational[-1, 2], Rational[3, 2] 3^Rational[1, 2]}, { Rational[-3, 2], Rational[3, 2] 3^Rational[1, 2]}, {-2, 3^ Rational[1, 2]}, { Rational[-5, 2], Rational[1, 2] 3^Rational[1, 2]}, {-2, 0}, { Rational[-3, 2], Rational[-1, 2] 3^Rational[1, 2]}, {-1, -3^Rational[1, 2]}, {0, -3^Rational[1, 2]}, { 1, -3^Rational[1, 2]}, { Rational[3, 2], Rational[-1, 2] 3^Rational[1, 2]}, {2, 0}, { Rational[5, 2], Rational[1, 2] 3^Rational[1, 2]}, { 2, 3^Rational[1, 2]}, { Rational[3, 2], Rational[3, 2] 3^Rational[1, 2]}, { 1, 2 3^Rational[1, 2]}, { 0, 2 3^Rational[1, 2]}, {-1, 2 3^Rational[1, 2]}, {-2, 2 3^Rational[1, 2]}, { Rational[-5, 2], Rational[3, 2] 3^Rational[1, 2]}, {-3, 3^ Rational[1, 2]}, { Rational[-7, 2], Rational[1, 2] 3^Rational[1, 2]}, {-3, 0}, { Rational[-5, 2], Rational[-1, 2] 3^Rational[1, 2]}, {-2, -3^Rational[1, 2]}, { Rational[-3, 2], Rational[-3, 2] 3^Rational[1, 2]}, { Rational[-1, 2], Rational[-3, 2] 3^Rational[1, 2]}, { Rational[1, 2], Rational[-3, 2] 3^Rational[1, 2]}, { Rational[3, 2], Rational[-3, 2] 3^Rational[1, 2]}, { 2, -3^Rational[1, 2]}, { Rational[5, 2], Rational[-1, 2] 3^Rational[1, 2]}, {3, 0}, { Rational[7, 2], Rational[1, 2] 3^Rational[1, 2]}, { 3, 3^Rational[1, 2]}, { Rational[5, 2], Rational[3, 2] 3^Rational[1, 2]}, { 2, 2 3^Rational[1, 2]}, { Rational[3, 2], Rational[5, 2] 3^Rational[1, 2]}}, {{0, 0}, {0.5, 0.8660254037844386}, { 0, 1.7320508075688772`}, {-1, 1.7320508075688772`}, {-1.5, 0.8660254037844386}, {-1, 0}, {-0.5, -0.8660254037844386}, { 0.5, -0.8660254037844386}, {1, 0}, {1.5, 0.8660254037844386}, { 1, 1.7320508075688772`}, {0.5, 2.598076211353316}, {-0.5, 2.598076211353316}, {-1.5, 2.598076211353316}, {-2, 1.7320508075688772`}, {-2.5, 0.8660254037844386}, {-2, 0}, {-1.5, -0.8660254037844386}, {-1, -1.7320508075688772`}, { 0, -1.7320508075688772`}, {1, -1.7320508075688772`}, { 1.5, -0.8660254037844386}, {2, 0}, {2.5, 0.8660254037844386}, { 2, 1.7320508075688772`}, {1.5, 2.598076211353316}, { 1, 3.4641016151377544`}, { 0, 3.4641016151377544`}, {-1, 3.4641016151377544`}, {-2, 3.4641016151377544`}, {-2.5, 2.598076211353316}, {-3, 1.7320508075688772`}, {-3.5, 0.8660254037844386}, {-3, 0}, {-2.5, -0.8660254037844386}, {-2, -1.7320508075688772`}, {-1.5, \ -2.598076211353316}, {-0.5, -2.598076211353316}, {0.5, -2.598076211353316}, { 1.5, -2.598076211353316}, {2, -1.7320508075688772`}, { 2.5, -0.8660254037844386}, {3, 0}, {3.5, 0.8660254037844386}, { 3, 1.7320508075688772`}, {2.5, 2.598076211353316}, { 2, 3.4641016151377544`}, {1.5, 4.330127018922193}}]]}}]], "Output", TaggingRules->{}, CellChangeTimes->{{3.758306187279694*^9, 3.758306217224358*^9}, { 3.7583087216780043`*^9, 3.758308726111217*^9}, 3.758309568919231*^9, 3.758403558112323*^9, 3.759008518851232*^9}, CellLabel->"Out[49]=", CellID->2079180016] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[Cell["\t", "ExampleDelimiter"], $Line = 0; Null]], "ExampleDelimiter", TaggingRules->{}, CellID->1237049475], Cell[TextData[{ "The number of points in the first ", Cell[BoxData[ StyleBox["n", "TI"]], "InlineFormula", FontFamily->"Source Sans Pro"], " sides:" }], "Text", TaggingRules->{}, CellChangeTimes->{{3.758306755439684*^9, 3.75830678229506*^9}, { 3.758309576698048*^9, 3.7583096079518948`*^9}, {3.758403785831231*^9, 3.75840378620466*^9}, {3.758403875705703*^9, 3.758403895976671*^9}, { 3.854021816857231*^9, 3.854021822091618*^9}}, CellID->2013137765], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Length", "@", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "@", "#"}]}], "&"}], "/@", RowBox[{"Range", "[", RowBox[{"0", ",", "20"}], "]"}]}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.75830649989918*^9, 3.758306518662984*^9}, { 3.758306704517927*^9, 3.7583067078887453`*^9}, 3.7583095646352043`*^9, 3.758309812210556*^9, {3.758403587222048*^9, 3.7584035874193983`*^9}, 3.759008498087376*^9}, CellLabel->"In[50]:=", CellID->744314247], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "7", ",", "8", ",", "10", ",", "12", ",", "14", ",", "16", ",", "19", ",", "21", ",", "24", ",", "27", ",", "30", ",", "33", ",", "37", ",", "40", ",", "44", ",", "48"}], "}"}]], "Output", TaggingRules->{}, CellChangeTimes->{ 3.758306520814324*^9, 3.758306708756362*^9, 3.7583095821678762`*^9, { 3.7584035762686453`*^9, 3.758403604496848*^9}, 3.759008524515189*^9}, CellLabel->"Out[50]=", CellID->979495204] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Round", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "3"}], ")"}], "^", "2"}], "/", "12"}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "21"}], "}"}]}], "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.75830666712463*^9, 3.758306688329217*^9}, { 3.758306720284157*^9, 3.7583067353839273`*^9}, {3.758309616258106*^9, 3.758309636911153*^9}, {3.758403718610211*^9, 3.758403753758087*^9}, { 3.758403856383341*^9, 3.758403856594778*^9}}, CellLabel->"In[51]:=", CellID->1155380441], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "7", ",", "8", ",", "10", ",", "12", ",", "14", ",", "16", ",", "19", ",", "21", ",", "24", ",", "27", ",", "30", ",", "33", ",", "37", ",", "40", ",", "44", ",", "48"}], "}"}]], "Output", TaggingRules->{}, CellChangeTimes->{ 3.758306690098545*^9, {3.758306726023838*^9, 3.758306736226756*^9}, { 3.758309622273615*^9, 3.758309641741866*^9}, {3.758403736682766*^9, 3.758403756251361*^9}, 3.758403858747045*^9, 3.759008533834386*^9}, CellLabel->"Out[51]=", CellID->1355060892] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Neat Examples", "\[NonBreakingSpace]", Cell["(2)", "ExampleCount"], "\[NonBreakingSpace]" }], "Subsection", TaggingRules->{}, CellID->540091361], Cell["\<\ This finds the coordinate pairs that are a prime distance counting along the \ square spiral:\ \>", "Text", TaggingRules->{}, CellChangeTimes->{{3.758306295846785*^9, 3.758306331221492*^9}, { 3.758306361349299*^9, 3.758306373619851*^9}, {3.75830727691092*^9, 3.7583073676611*^9}, {3.758307457319818*^9, 3.758307467215272*^9}, { 3.758308231526018*^9, 3.758308241381768*^9}}, CellID->1039848554], Cell[BoxData[ RowBox[{ RowBox[{"UlamHexagonalSpiralPoints", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "[", "n", "]"}], "[", RowBox[{"[", "\[IndentingNewLine]", RowBox[{"Select", "[", RowBox[{ RowBox[{"Range", "[", RowBox[{"Round", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "4"}], ")"}], "^", "2"}], "/", "12"}], "]"}], "]"}], ",", "PrimeQ"}], "]"}], "\[IndentingNewLine]", "]"}], "]"}]}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.758307394004264*^9, 3.758307430080327*^9}, { 3.758309564643347*^9, 3.758309564644534*^9}, {3.75830981221485*^9, 3.7583098122156677`*^9}, 3.758404037558888*^9, {3.7584040713251743`*^9, 3.758404071490954*^9}, 3.759008498094325*^9, 3.759008648274733*^9}, CellLabel->"In[155]:=", CellID->914425330], Cell["\<\ The larger points correspond to the primes 2, 3, 5, 7, 11, 13:\ \>", "Text", TaggingRules->{}, CellChangeTimes->{{3.758307624514155*^9, 3.758307664760827*^9}, { 3.758308151929039*^9, 3.758308154000259*^9}, {3.758404101004528*^9, 3.758404102473412*^9}}, CellID->1077324486], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"n", "=", "9"}], "}"}], ",", RowBox[{"Graphics", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"LightGray", ",", RowBox[{"Line", "@", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{ LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True->GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{ LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "@", "n"}]}]}], "}"}], ",", RowBox[{"Point", "@", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{ LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{ "Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "@", "n"}]}], ",", RowBox[{"PointSize", "[", ".08", "]"}], ",", RowBox[{"Point", "@", RowBox[{"UlamHexagonalSpiralPoints", "@", "n"}]}]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", "100"}]}], "]"}]}], "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.75830772123131*^9, 3.7583077253989677`*^9}, { 3.758307783767869*^9, 3.75830781817214*^9}, {3.7583080226757507`*^9, 3.7583080873792677`*^9}, {3.7583081610428534`*^9, 3.758308203597548*^9}, { 3.75830956464948*^9, 3.758309564651239*^9}, {3.75830981221939*^9, 3.7583098122211037`*^9}, {3.758404082756373*^9, 3.758404088946466*^9}, { 3.7590084980990553`*^9, 3.759008498099826*^9}, 3.7590086358530893`*^9}, CellLabel->"In[58]:=", CellID->1012815160], Cell[BoxData[ GraphicsBox[{ {GrayLevel[0.85], LineBox[NCache[{{0, 0}, { Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { 0, 3^Rational[1, 2]}, {-1, 3^Rational[1, 2]}, { Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}, {-1, 0}, { Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2] 3^Rational[1, 2]}, {1, 0}, { Rational[3, 2], Rational[1, 2] 3^Rational[1, 2]}, { 1, 3^Rational[1, 2]}, { Rational[1, 2], Rational[3, 2] 3^Rational[1, 2]}, { Rational[-1, 2], Rational[3, 2] 3^Rational[1, 2]}, { Rational[-3, 2], Rational[3, 2] 3^Rational[1, 2]}}, {{0, 0}, {0.5, 0.8660254037844386}, { 0, 1.7320508075688772`}, {-1, 1.7320508075688772`}, {-1.5, 0.8660254037844386}, {-1, 0}, {-0.5, -0.8660254037844386}, { 0.5, -0.8660254037844386}, {1, 0}, {1.5, 0.8660254037844386}, { 1, 1.7320508075688772`}, {0.5, 2.598076211353316}, {-0.5, 2.598076211353316}, {-1.5, 2.598076211353316}}]]}, PointBox[NCache[{{0, 0}, { Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { 0, 3^Rational[1, 2]}, {-1, 3^Rational[1, 2]}, { Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}, {-1, 0}, { Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2] 3^Rational[1, 2]}, {1, 0}, { Rational[3, 2], Rational[1, 2] 3^Rational[1, 2]}, { 1, 3^Rational[1, 2]}, { Rational[1, 2], Rational[3, 2] 3^Rational[1, 2]}, { Rational[-1, 2], Rational[3, 2] 3^Rational[1, 2]}, { Rational[-3, 2], Rational[3, 2] 3^Rational[1, 2]}}, {{0, 0}, {0.5, 0.8660254037844386}, { 0, 1.7320508075688772`}, {-1, 1.7320508075688772`}, {-1.5, 0.8660254037844386}, {-1, 0}, {-0.5, -0.8660254037844386}, { 0.5, -0.8660254037844386}, {1, 0}, {1.5, 0.8660254037844386}, { 1, 1.7320508075688772`}, {0.5, 2.598076211353316}, {-0.5, 2.598076211353316}, {-1.5, 2.598076211353316}}]], {PointSize[0.08], PointBox[NCache[{{Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { 0, 3^Rational[1, 2]}, { Rational[-3, 2], Rational[1, 2] 3^Rational[1, 2]}, { Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, { 1, 3^Rational[1, 2]}, { Rational[-1, 2], Rational[3, 2] 3^Rational[1, 2]}}, {{0.5, 0.8660254037844386}, {0, 1.7320508075688772`}, {-1.5, 0.8660254037844386}, {-0.5, -0.8660254037844386}, { 1, 1.7320508075688772`}, {-0.5, 2.598076211353316}}]]}}, ImageSize->100]], "Output", TaggingRules->{}, CellChangeTimes->{ 3.7583077269025803`*^9, {3.758307794978922*^9, 3.758307819182971*^9}, { 3.758308049748094*^9, 3.7583080881187963`*^9}, {3.758308163301917*^9, 3.758308204686307*^9}, 3.758309659980236*^9, {3.758404057786284*^9, 3.7584040898496847`*^9}, 3.759008540035062*^9, 3.7590086557176447`*^9}, CellLabel->"Out[58]=", CellID->1543371489] }, Open ]], Cell["About 12% of the numbers up to 10443 are prime:", "Text", TaggingRules->{}, CellChangeTimes->{{3.7584049348365107`*^9, 3.758404949244348*^9}, { 3.758405501441153*^9, 3.758405503129024*^9}}, CellID->843341047], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"n", "=", "350"}], "}"}], ",", RowBox[{"Round", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "4"}], ")"}], "^", "2"}], "/", "12"}], "]"}]}], "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{ 3.7584048165566072`*^9, {3.7584048907708797`*^9, 3.758404901272431*^9}, { 3.758405448890163*^9, 3.758405489489677*^9}, 3.758405652409912*^9, { 3.759008544978506*^9, 3.759008561278801*^9}}, CellLabel->"In[55]:=", CellID->669943960], Cell[BoxData["10443"], "Output", TaggingRules->{}, CellChangeTimes->{ 3.758404818025424*^9, {3.75840489234787*^9, 3.758404902315119*^9}, { 3.758405451907506*^9, 3.758405490079701*^9}, 3.758405657686693*^9, { 3.7590085582364273`*^9, 3.75900856217072*^9}}, CellLabel->"Out[55]=", CellID->1996439306] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PrimePi", "@", "%"}]], "Input", TaggingRules->{}, CellLabel->"In[56]:=", CellID->991672100], Cell[BoxData["1277"], "Output", TaggingRules->{}, CellChangeTimes->{ 3.758404915618512*^9, {3.7584054727269983`*^9, 3.758405493146401*^9}, 3.75900856412171*^9}, CellLabel->"Out[56]=", CellID->1016442324] }, Open ]], Cell["Here is a plot of the first 1277 primes:", "Text", TaggingRules->{}, CellChangeTimes->{{3.758307737030126*^9, 3.758307749085785*^9}, { 3.758308123305419*^9, 3.758308138992836*^9}, {3.7584055119288187`*^9, 3.758405513641204*^9}}, CellID->1606466798], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Graphics", "@", RowBox[{"Point", "@", RowBox[{"UlamHexagonalSpiralPoints", "@", "350"}]}]}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.758307753302867*^9, 3.7583077543096848`*^9}, { 3.758307831758318*^9, 3.75830795246375*^9}, 3.758307987633847*^9, { 3.758308096308929*^9, 3.7583081098521347`*^9}, 3.75830956465534*^9, { 3.758309687159231*^9, 3.758309750219214*^9}, 3.7583098122245693`*^9, { 3.758404138333024*^9, 3.758404167935556*^9}, {3.758405400577478*^9, 3.758405408348843*^9}, {3.758405518330366*^9, 3.7584055191284246`*^9}, 3.7590086447483997`*^9}, CellLabel->"In[59]:=", CellID->2071963721], Cell[BoxData[ GraphicsBox[PointBox[CompressedData[" 1:eJxVXGdYldfSBURARKkqAipdlN47Z6hix45GFLHEHjTGbiDGgiURjSbGWNBY ozH2mEQ8Y+cmxBaN3WDHiGJBRQH9zjtr53n47p971+Wc9917Zq01s8vRLeej nsNNjIyM6kyNjLT/NjbS/lOm+6FLZvn0A490Ao0Oxm0XXKNwtX6X+rsVPq/f Aay3FPxUV6Q+X/Je+88/+lnRSQHXS0zpseD3ejyn5NBpgZV6vK9Gb4Y/EP7r ZZwTnqdfL39vTHi+KY9Xz/tvPF8Bs6n6/i31PIzPAp/Lty52V3//Sj3vjrzf nE8q7IfP83T1/TR8ntX7+SY+Tz274u8O8mcbut4Z881Q7xugnmej5vuNjM+B /hVoy1Vq/FPl81YcoMafq54f1hXvqxZsj/EXxhQjHg5kqt53VL0vQT3/d0lg te5FUtdF23WO5Czfb8Cdk4GDjDG/NHyeqUbir9up8En1vkbdMb9Xglvg/Zlp xU9NkN+FMj9ncn2L/J1UeIV8vhl/GRR5zqqnLUXLF514cxr4sU2wHY8JlL/z 02fCD50O4+MZ6v2h3TD/X9T7O2Wo8ah4W6v3+cl4TOlGn6Gpnfzb0FaZz2v9 J32BL9RpuDEfUXiYfN+Fl6XjeT7y+fc6F4m3MzdX78+NQTxy1PtvqfHIdN+3 QTx8+hXflv/hSCOjEH/E14G4M8ZnKu+35hSJhxt1l++3YgpGfLzk+278JSE+ afL95uyF/PEqY8TrM8yPV76R+el+xHz4q3cYb4ka/+N3GO95Nf5Meb4rxXYE H11qwYcX+w7lTWriQQn4PM9X41sr2JldumH8P8l43Tg3A993kHg78DjJnxtn V4I/zvg+e6r4bMXnFZ/c6Js+iLevjMeTOAn8HS3zc6GznfC+Agvwy1P46Ek3 JX72vFPhNHmeKzdIAZ8XCvbkd7HQU7lgd46MhJ5+M0I8R2C+XGECv1ircD+V 3xg1/obyeW/kd2BGsYPC88AXXZHKdys1nh3gM1UWn7qePMebMurAP/PDwL2V Hloo/IOKZzfF3y2C27I85mx48XeCPZiU/leozx9X+mz2HvkxU+N9qfQwEOPh joLd6S74yO3U+Is6Iv5X8XxqA37QQJUvV5V/C8m/NdmIHn3oqny/NS9W8+us +Hp3L/hjrOJ9DPlgO5l/S76O+fJ1CagVXcb3ub0x+Pg+AOP/Uo1vZQfMdwCE RSsTMd6DiA+59gT/4uX5zlSqxmdTBT2cP3U+2r6mHdWaQC9vT9bHTrxH4Qvg H0cFqPkKbs+mKXj+LHmfL/Ixc2JxqOB2fDYL43nyFvGPlPn6sKcl/HdMErBd Q+jfWuFQE8z3cSTyE2yM+YaEIz+B8iI/8O3GJ/F/4/00PQd8y6pBfgrxPtry FnrPF3740rBmohd9UBfgaSbID5/AfGfIg335g3bwm0uS3/Y8J03poxZ66snI z9Tb4K9LiXyfc8zB70LEj+tMwK8Gaj5NTDCftWnwH395XwDmMyo//oC8rx0t IfDtV8mfF90KQfxNpMC1pDZ/3fj69gR/Mn0q/kSV54CDGiBfF3WY/6d4Pn8e jXwF10E/ab3x/u9lPl48IhT8uv0I/nsB8eLjlyTAeh+FW0PgdLYzsLUl9OSp cEwDzM/ED897K/Pxp1ND8H7Mpx2d6gM9O2E85BKG+V2tgd4Wp2H8KyVfRhwn fh5IpfI+Rz7UC3iE8MeVa3sCD5OGxou3HkM+V8vz/XhgLOpnlMzXhx2zUQ9y 3yCfpRckfpwfJn6n23oWOMQM+TuSjvn9aor5FYchn/NMwMfx/vX9I1jymR// RXzDGuSPO2A+n72Bv1sMwHjz72q4MfmOr3qTOSmI8p21b5rymbHA08yQz7E9 8HlrU+gvPRR+uk7SEyz6y7deWvy1zDeQX4+GPzjVQI9NI5CPtLfIf3sdxr+8 EvViXqY8n33+0bA5L+wHfNkVev26N/BkM8x/cjz0+nVD+E82xsPfqfkXmtev XyESD0pfFr8D46N9HcHvsifQp5PUd0Pcpmi4UmcZBJyfAr3ODgR2tRW/4IkB wI4WyPfXoxGv9IbQb+kB+O9xCUwwr3SBnjO1x+eF8P5K9A+/S3wC2K03/Lho nfaFMl3qOHkel7eVfNBe5IOnWGD+7dLAh49MoV+TWOhpv+QjhC5ugh/1rMV8 hw0G32PegP/bO4EPa18K/+hY1bStHBhCBUIgK3Z+Bnxc9Naa71QCn5L3+/DG buDD0obQt0UQ+KATPgbxBS/oab3EP5TLXKC/g2/Bj1+yMV56Cf27ZiG/u1qI vuiqv8SXLzeGH4cpvM8S8/8H7+d0c/DfC+9nTzPE4wHezwONkf99eD9HyXjC wIc1q+I3SbxCye5nxGuW+EUwje4Bfhi/gj8ED8J8L1VBT1+EIv8DHmi4Be0+ oNXXUCLtffmhHNYc+Z4t+Q7jC579Jd+s8S8vmC0GIN9nX8HfN56FXxcNh/5n I/7saIX5nvHF/EvUfELaQO+Faj7tLvcTvsvw3qv5LVir/DyULs1D/P8WPgRR 9zjkZ2+V8sPBmF/kc+UXBzGfldLvu9O309/39XkUStkn4RfDpwKPbKq9yY3r Pga+1wh8KEpEfetmCj50iQTfzMSfQ3ijO96fYwJ+FAUpP5b5hfP6phivD+bD y0ug51yJlz9bdoP+s2+jvnv9LOPl/MmiV12zfcCTm8Af7FX8RmJ81ATj4f4N 4Bc9MB7ubox43rACX5a/QzznLMF4wuqgr39HY7zlVagvazIxX5/niKdjuPKP V+/ET65KvxdG2UniZ5wQBbzSRvydL4TXx95spvDkpvCTcxXQX0tNb/lBPPpX 1JehpvAXk13wm0QtnvmGeLmh3/5Je31eOO/JQ/9QJX7nzQdmSL64/IaGm3HA NOAi8b/X+k+nALONsfjR5snAPyF/5OGKeKFfCqNPm4OPnyB/FN4A8dtfh/i1 HwQ99XyOePUfCr6dlv7Bl8wjEC8LaXhb0t53cxoWbg6nsm9ED3rfGuBqL/jR 3GrgcDvUo92hiNcJM/DtSQc8/5DwJ5K1/zd79qbiQRgPWw3GeOG/IXyTMB+j GtSzhR3hJwUVqFf/S5DnMw2TB+r9Y4Cn2mtP9qRNYcArm4Jf9/B+LhV9RtKp NdCnLd5HCzvDL8OfwZ9zh2O8Dx8jHpm/QX/JFeDTVh3mt0Y2JDzI5JiW7wiK uajhNmR1FHhxH3khJxQDL3IAf3q8RLyKzcGfacNRr8Ibwq+StoM/S8WfIvj4 B+hX9teCP1Vj4Ve7HsGv1swCH94+1LAPPzSeqz2fzZpr7/Ogh6/kfTzUHv48 5Ufo0dcM/FkQBP3NbQC+THVDPThijHj5POkn/PlHphNFBfMRPycJfxTdbNlf /HphDfg1KRV+crMa/t16MPR4oRzxe3Uc8Xgqfu1NK+ZZ7ns9JJLydcJ33dDP gG0ctfe141bTgR+Zw6+6JSM//g3hTzkeyN8uGU+08KtoyPbi4+IXURzYRsan rxC/jeDro8A3I8HhfHc65jvstfgxm40EX0bK+Nz4DvLJZ0/D384cAV4p43On wF/qY29yPqg+7wD+fbAD8Z5jAX8bhPHzMsyH9kShf7hsgnhvNcP4mmI+FB6C +I58hXrRMRr6HDYG/tUjUctfFO0J17ALN9MBX3GCHn+LBTayhD+NLod/BYk/ RbP/qwipjxFSL2P40yOJsh+QIPXSgP0GYH+lBny0XYB+ecd97QN+3NAUfNt1 F3yML5B8KezOj+cC53vBz7oiv1wUoY3Pji7OAs6QfteLspBv7uoEvj7bj3ja mMPfcnzBVysz8HWDK/zhkvQzEWTdGfw78Rr1NnEY+IcNuRDaMhb8Qb8QSJVJ 0HPSHei9X2fEK07qmStNsizMGdwqmgpitfc15+YmwMdbabgt579bIvigC+qt SQy+n9EE/rd6H/juZ4n6OtoT+XOwAH91oRjfU6nPUXzLDuPfJPyN4as9wN8Z NfDPonnga49q+OewgYjHlVsyfp6E8SvswbXpwMPma9ia01OB27bW3udKZfHA dc7g61GMl49agp9zMF5eZYH4Z2O8PEfiHUlu3TDezFuoJ/O6qPkf1nALWvmH Vh+jKftz+GH+SeBdiRp05KpjwFhf+LA718d+fP0w8IYm4O/6p/DPTOFrHP+6 GXxdpY0nL4rDVoCf07X+Oi+Sb5/HflVVmTaeAF66EPzaJxvSrnwC+eWRxRq2 ZS8L4JXSH7SmVCPg9VaIxzdboec7ZtDrZifwr7gh/PGwNerXBPHTGLr8IlH0 fFnyGUdWn0PP2TXQd/wy1b+8Bn+vDFHr3Zfw0z4TwdfcKvC5U7xa/zyHvy5M BX+73kD8/z2HeKXLBlAr0i1pZnvqcgwVygZ5Y1q0GNj11jvx2+BFwAc7iB7Z Yz7wdQ8N+/OQfOB+tuDz60N4/oTG4O8KN4znF4lHNA9pAT/OEj7H8lsTzK9J Lfjs+Dn8zf4V/HfZaPDpjIw/kO91BR/HXZP58HvMR2E3/utP4KsHNGzPjcEv 3jdOw+a85X/AX3mK/9F88IffWsIvBkXBfxdaIH8j3KGf26bI112Ml8WOjRJI 6oluf3wfGX8snS9A/3TrDfKn7w3+X3oCf3brDv6fvKxhL5og+yuxlL1YK+iV us5dgSd1057swMYdgWNkfenNP6cAF9uC7zer4S+ztHjnR3CnLPQLdQ3A/7Kn XaS/DBd/juWuy9Af7Nb8Ic/A6yvg/6Mq+HfJM+xneGn9Tl4oO8xF/3BHW/8b 1sOejeDni7tr43Eiv7mSfz4p43OjfZ8Dv7ZFPUuJQL4uWSG+nq7wiw6WiK9P e+QX/XwC6e/1FT8rFj0kULYeevhC4hlD6ZnKv5+B/2OPo/+69Rh8L8xAfNMv wp8/bPadPmB5HBXKBrEtDbUD5j7aC6v1N5oAO9jBf7eH4ftzGsNv/wiGvsIs wd/d3vDjPVJ/daz9v9ldDhaPl/yG8JpS+OPJR+gXtl0Av+5eAH9/kvVHLBf2 gt9+2B04FxvedKwTcFRf8T/a3gE4zAHxzPoVz/O2gd9uaYd+9rCl8heMjy0t wNfJNvCfDSaI78ol6Nf2vYA/283E/DKfw18aJ0GvRZWIr3Un+EfKE/iL7WnU q63nNNyWHl/V+BJPfi2g/1+OA89oDn66GyO+fbT4GtY7A3LRrwVq/Xd+JC/7 B/1GI+k/YnjlGqyXUs3B37j52A9vI3zW8bA/0I/clGgR24yGvw8TP49j693w 99Wafxj6kxnl2H+V8RvW9z5W4G9ZBfjtWQi/Hynz8eHLjjJehT05qgVwUJbU d4qyrI9b00oL4P52yMfmH5F/56bg93kX8F3KYR6JX9ik/RY/vwr+sGMK+HRG 4htOBzoj3i8eId6be4GPc2U9EUJrL8Fftz2An3z5D+J990+pX7S/yHX8190T KEoO5Jz4wy+BXULg1/r5wHNkvBH8dDvy6dMU/LZxRv7vil8n8KW6CNFbrtQv Hbe/10/8OkXqK3FKrvCJ7DE/7t5wgOj3p1roI869v/Ctq/QjcZyQjfr3XtaT kXwjGfFZIHyL4DOnUD9P3Id+ZmB+7CAHOM78+u/62J6TLwJXyHwdSF8KrG+B enwrDvrpbA//qcR8uZcN8vN9Ed73T2PoJacN9FPZEHoZdlfmy4eNZL6UOxv+ 9Nk75HNfsMyPbMWPomhTKuLXqhz5is6CX6+7D39qewv5Svod9bifrK915Lqx Vvzfqx/wzDHw/0c9gA9L/+7OH3UD/kf632Ce0gG4iQP833kc9FXXBHr6uC/q wQRtPyU/nu8uxPpxogn0c/wc6oNjFerDztfY38oXv4/ipebTZX+rWvNXQ780 bRHqQcZ91APjNuC/61fafJpx9ErhF+d3kf5Bv205cNlY6R+oxxLgPi2gl7n7 4WenmiE/FwNRL4ybIh+/e6MeLwXfaOtA+JekJz9R9HT2d3081iOJ1Kmj8E+3 Q/qHeLLJRb9x6BH8LeUv8H11OfzN6hr05Kn193lBdMu+V9O8hobA/In68Etj 4NyPRf/8sBHwQTmAb8vB5vWxL3c0A77uAv+bZgp8T9Z/4VwSDT50bwH9OezD +/+Q9UAc/+wBP15nCf3dsUK/N7QO+vpjF/qjhIfoj471RbwOndCwL3+eI3xQ uC1vyQaeukjDbbhyIHCZvxRY/QfgGxdMEmeiUz3qY0d61B04JUbDPtSvE3CI DfLzojn0u1zyFUsT3ZGvXxtBP82NUH8Gv0N+ivpBLzekn9WR6TjUc9sn8MO1 f6Kef1QBPU3tAT9c8AD+aHYD8TI5KvOlnpM0fhNVv5b6yUfHAtfKAZ0Nnx0D /C5ew65cMao+9uPmHwIvb4F6tKoS/VS66CmaW9/FemKDHfSUPQj7p7c1enAi 7/gB9ehqJfTjOR/7/wUP0E+lO2F9sPA26s3xTUoPZdBPqYPwg/mI9CtcZQ2c /7mGW3JtU2CjN9p4rcjTUv19JuqRH/jIlKDhQOoNvvFKF+jLqRh+WO4EffWN hB9OtUP+Zq+C/5XYIH+HHZBPP1MU2AuV4v+80Rh6exYG/SUKfRLJcSzy2eQO 9LT3PvytSNZL7WnLxRX3Qu6Qwt60/QKw0WON0K91+jPAPAf16swfwNh/bcfX SoBXt4Z+vJPgd57W0EesC/TyzhT1aO6bRNG7tH/vk1j7VPW0o8XJL1GPxkwB HxdI/5PAxnno79/fgZ7WDEF8zsn6L4Q/nSz84A0yfn+ePqk+duNJE4ALZP/Y nMePBx4k4w+ggcOBA93Er2nAMIWdUH+m/QT/q2wCvexsgvrTD/Ohq/8i/t+Y IP7pW7Be/Fn65STycEU+XmO+4ocFfxyLZ+kXYuh2H+inthz62noF/jfsJvoL r6ngP/2CelQTqeXf4KsR2guf6v8IBOY01J+9vvVxAK/3AZ6hXQAz9HdrjiNf Q+xQb/6+gP6uqfY4TmLjb6EXua9AyTy+Afq3bElPMh/yzJL9pJ1aOdQn8sZz WK+UaOsXXSJvLUN/56f1q3k6vmr2l9SrEunn4nhXU9Sr0juoV7nrsR4Zqc3X 0P9lNoc+omS+7lx7XsbLrs7agExpXymwzMegn8wc5GubK/RzOQH+Fyr7IVGU 8SPyt6QF9FS0Hv3FqgbIl/s+9A/o/5IlP5fbn4g3k/zEUlgm8jPoPvKzYzD6 vYV34X8j78DvbpehfpX9C32NuIb8WcYi/jOvip/QhO8uH7rPifR0lPgH2a4A LvtL42cDerYM2KIz6tdni4DtxQ90XGGt9g9F/0kceQ/9n9yXyk/mIAv0e2bS 3yXy81tYDxtLfY3jvPPwkw/vyXx4Sxbqk+118QduA77xiH0a9uAv5Dw/kUtH oj5NCQVeeU4brzV3DgYeGSCEp0YBwKtk/J60ux3wn3LeEEQfeQO7uUJfdw7A /+JbI1/GwdD3J07IV7wXxrfGDHprXd1F/OBUA+hr8zbk75mRzJ8yFveR/q+J 6C2Zhrf+QPqNSNl/T6LnL7CfkVoDf8z6DPXNXtanCdSuF+Lr/wj9ybW/Ue/q rqG+RUyHHnNkw7otzX6j6SeJCo5IPMjpJfDa7qhfQY+Az8j6O4Rry4Efa/sb hn7wwVHocXhLI1lPjcxBfzitBfQ5+zrq2zULsXs+2wn94puGqG9xE9EvtmgA vf64Bv1ia238eQn8ix301vse6t+xNVhPnb8EvTH4qHAQF34LvPEn1Lsg8JPL ZmsGUa07uAR4oFz49KBM8JOT2kJ/NzyQ3x+awy/fLQPfZjVCvVrcVPWLFsjn BRP4qew/GfQ47JHUB3WelEimC9EfnqpAPmZ/AD1mPYB/Hr0P/S0Bn2n2aPRv KT9rOIDMs5yXr1qbpLA71fQHtrmh5asxVfYFPpSujc+L/+oOPMgTfqnvBByi 5cfQ/93fifcdbYb61vd7+LWV1NtkNpmE9ZScp+Ql8YuPsZ6ykP0R4nsLMV8r 8I0fn0V8Boq/xHB8GfwK+7GR/MsY+JnRTdS/XdGIb2+l11U1wifeehD17gn4 x0W3Ue8cXwBzjNQLXdgz4OxWml4tKOkpcKbsH7rT2MfAh52hv9ch0GOt5DOe Lir/HIH5UxsX7G+0Uvnda4P+5Jn4UQpFXYQevcQeUsVfq4OOx38tfptCuwrQ nzyX/d9EqirEettPDNlA+JXoH/ZVQp92yo9XCx/i6Kby46msYS8Kl/O+ZHqa r2F7+u4gMNUaS700PgBsk6eNxIFT9wEXyA3v9vznLuBGbaDPWZ2R/2tKX1un Yb+jhRZeXQrb/IjzvSFau6RP5qpDqIc6qYdJnPEK+xt/vUF9DDO7Kft3rTSs J7ZX/eiwStRL8yXqPPAx9Gtsj/2QclnPRfOdzWr9Jv11AL/LlPEp7M6rwGc2 mqvhZmzSR2G5QGxK53uoz0t/3Y7+7grMmv/mh9BEzJcn2yO/UZuQ7w8tkF9H 1Y/i/JOoQPWj+obw4/3m6NdeKf2OLEA+d6p8dukCf418Cj3PHIj8nXiIejpQ 9TsP7sFvxx9GPpLEv4JoU6tBY7rNTKY5sj53pRNOwDbt30v9XOMIPFPW4x7c 1aE+9uUbdsCHtfqSH8r7berjcK6zBj4k/UI0n1T9U2/heywvT0N/e9AW+nc9 AD+YY4P6vDEI8yHEg3Pbg7+bpV9M4qnYv6ArUp9SWfvU1Cm/FbeS9VcK91uH 9eMPT+EPNucQ/wFPUL8zrsAfRov/RRh8BXqdU4p6TXJ/I5kPi1+FUPpO4B2O qLdd/sXna1Q9nbYW+XzQEP7buS38uYnoNU306mpVHP8W46MTv6B+jpfz/ySy noz17wj0G3RpHuZ7VfRMZJIBvc5U68fd6NcNPi/jp/eHkN+B0m8F00s5D0+m 3G3Sb5DzjPq4BS2YDlwo+682/NMU4I2SnyBOngTsq2HDevGJyu8EG9RXb7W/ lmeN+npW7X8O1+ZPiez+IdaPc5Xel3+P/c79Qv80DvgS/fA4LR76FP5W7dcb Sb8bzw+/QT/rJPvzcRzogv2XbqLvGM6yQ397QOpzBN8DfxUO4jEtgcfKfU0f /gh85ss7NWzLU5oDL/cRPlMj8Jlv2yO/n6l8T7SFf383Av2ciTX8e3Q8/Dyk KfTtfRB8utUQfBjgAz+/Iv1VMtm6QM+jFB9yHmB/zVj8Hfyw+fNAvIPy8zf9 0U89UfxoNAl8L6mFvzc5pe5jqP2FZr2hF1/Zb42lT4ZDX1fUfp1O9Vf/SvyC ad2NoiSTASl04R/t7970wfX62J0qrwFn5L2X+r79KvCufPE//vAKsJ+rht3Y 4zJwc+FHCJv/DdzUGvpvthv8/LAB9PuvB/xrsOzXp/A+H/B/j9T/FG40F/12 ttLzQJxnc63sd+l4ujof2yfr3Xju2Rf58JT1bjQbq347SuYfwi/kPlEyF8p5 tTf9AX5zieQvlMZ8DHwf+1H0YyKed0/tD0yIVP2zWq/22gM/CVf5nRWMfiTL BPX6DM6vuZkJ6nV7W8wnXuU7sjvuj+zS/mzUgaZ2x/nPIJlvEo1U55t9sR9G 2eo8bbnkO4b8wuGnFqL/dnQ6TuNbCpX10p5nwWtjgKvlPMWdS6PrY2+ep/BM nE/x7UjgUbJ/GsO5qG+ULPVJx1sbQd9jRM/Ea+fgPDRLrXfHPsL6t60J9J4z Av31XNk/SuGCneivp0g4O3DhU+hffo9g0P+x4zif2/0O9b/xJezPHpLzu1j2 P4f16oBq6P+Tr9AvjxQ/bM+9wEeFvdlNYfpYw6/1bcFPzm+J+2MvLwGvFH17 UQ34ykflfkUsmfqDP+3NoXfHDPSPN3FeQK5/YL3l0QD8CDoC/S8yQf3OUPeL PlB6H6Duoz3UYH4H1IM1G+LnGYEPX2fhPHWEET7vk4v6sPk99P8gEnzoUId6 b54FfetqUQ9KUa/oN9yPoFzFh1XSP4dQxzV3Wh6ITKVS6T/a8qjv6mNf7qjw CFP4/bZVwBmm0G+24kupzDeOLZ/DTyYJTmDvcahHU9V5SVFnrB8isH9j6OfQ 38t5lkHvJep++mqZfyrnhmN9kot4MCdj/vh5VRqXDMb9vT/EH1NZh/vxLNu9 ecn8vj/qra9g4u7q/Ls5zldZfx75ipD4RfIdzIfHqvvf3ymc/Ze8UD9Y4d8l nxF0Jgp4jwnqwYu9qA/hWK6TGebLHmr/8GvMl1up/auTkagH11T+L/vh/uh5 pf9Vici/4PwkmheLeo+/G+L6F/ZDZH/SwMfHQdD/RnU/9HWttr5NVTiIGtQB p0t+/KjR/8Ou5Kww38N8oxU+WIv+p8f/w4E8VOGBuL/LHyn8ofhVNHdfDb7I 9o2hny+YAP3Lzwe4A2cm4v7xTKX/2tbo9xfL39N4Ygjqf4XgZD47Cv18nOyf JbI37htxF/ETYlPcz+CVUk+i+aXS70HBUTz0WxkPl4m/O3FX8JnTa1H/5oLv POE/PX2E+m4J/pB1BvL5r6rHM/rA38slP+k055X4ty7dGPmriAJfRyp/z/RG /S5QflCk7qOVCE6kfHXfoaf4iY581H7+cuUvRdj/oZviP9FU/Rbxfi9+HEyn B3o/G9UsjWq19ZVRa/ohC/iprMeqdQGDgE3L0O/sUdjkKfqhnMHAxtXol379 fziCE7Lr4ygerTDuJ8Rz+1fQ/7030NdN7GfRW6lXSeyn7j/1fAd91qr7ZX+r +j5ytLp/JvFN48vdoP95Su9b1f3JbxBvvrwC9bIG+uHlTeCn/VX/8Jk31k9h 4mc6Po/zD64WHM9/P0G/3lrqXQwPjICejdpr2IVby31ewzqit4YN9fM/LD+o sqe1tQpLvfWiQXX1sT/dVDhf+hGDz70Drn4Jf74OP+aiWvDLBfcJuLfwL5EK ES8uVf1eQTL8IhfxoTMd4HfiH0Yd6fkU8G+w8C+d9uK+JK03hr9YO4OPeQ3h HwvVeZCZFfi1AfGgMNUPrZb7uWk0Us6/HGjZYWD2k/vi+gI9cKncB2zHk44A j0B/ydnHgOX3ZTriVuo8ZJFm4PpEvo/+nTxkvzuJTRxwf/AjDesMfdkS7NeZ 18E/dszD+mGvuFM6b50O/1gj0+nIYyr6yP75Auk30vjwauwPnJP+JIXN7+D3 Cqe0+mXoT+ZOxH7dGLXfd2Um/GOn3M+I5wdq/d5T1jNxPAPrHU6Q+0ehPKq/ 8J/nyP0sTxoKvXCbSuQ3Xe4bp/KVSvRnjjrk2/w5/KWgGvxrXAc/MU4EX9cI 3zvQhgXg88/C746k/ZdNl/3F60yQz0cW0MMjqc8pdPt79P+24hfJ9HUo9Pax GerHeaxHKNIc9ajTMdSPfFv4TasK5H9QM9Sz0F+QvyOy3gujnc6jSlq9SKPZ 8oMlf2reCvhTud/jzgfcgXWaQeZF8jsv4AWih1ie1kDiQYmP4Rf+eVhPZsr9 COIVav1aIfXX0C+q8+rfxS8Met8D/r5BPHj/RqyXxgnfO/LVmTgP2t0A/UJf B6yvvKRfSeNXreHXTYT/qezshHiPkvgY+ssfUJ+vN4M/rPgI/f+XjvCHL36W eCgcwZsOArsES/3kaYeAo3IkU7wU+uFc+YGVEbeCfth1ihieLhz64btyfzqQ GkEvPFGd9/54HDhH7Zf3KUZ/8a00QClkmY7xW0t8OtC14fCDBcKfdGoWovZL jcGfdzk4v2htivq0pinWVxtMUZ+GtsR68pAj/KB4PPJTKucFceRWhXoj5/tG UeQh95E7KBxBRxTGfQBP+uRTYNwfcKTdecCFi7XxteEGc4DT5fc6vnxjLvDo 1/CDJB/4w2GND/pUHmQt++30SvyiAy+uw/qhSPYLO3Kv/+H3JBOlH+jIiT6y nmAzE/hD9TY5f+MA8QODf1zMkfO0V7L/n8xmFXL+zVcbo5/Ydgzri94u8IOl p9FPlMr+ZRBHOwm/eaXsb/rzI4WDpN7a8fDWwCNl/9Sdkj3q4/a0xbM+DqD+ 0Av/VQa/aFuC/H9xF/lvNBt8THmNejAN52Ps+Qb+kdwP9WFjDerDiy/A97u1 4MeWWTj/lZ+H5HWkdLWfvlH01En8pTz4WPGXJuDL001y3kqtG8Jv6L2cD5Gl 9B9pFIP9aJL4GfznfBH0etoK/mI8BHr+tSnqTddtWI9YuIJfFwOwfrgv92lj yW0K8r9IzgfCadzV7evOne1AzeUH6L406wbwhXHoX4bcBf5X7ru68sN7wF7X 4TeBi/G8ezfhN4EW8J9lsh+QyIEzMb6kKvQngyZh/bTnNfqRQWWYz9E36EcG 9cZ8J75FP3KoCP3KbfHvjjz9GfzJVvqPThxU11fiewrxZIsVOL8OFb2lc1UZ zre7N4b/LNuKfE6Q8/AkHhGN9WW6nHcFsPEMmY/C7rxgFnDhWqnPHA19cdlR OZ+gb2cDH5TztHZ0bh7wQLl/G0W3Fiks529x9Cfiw9evg2+X0lCvnog/p9LJ vvCbnYgH7d2E9dWTGtQrry/BL+xfptPdWuxvBLwD30Ieo551NEW/YnYZ56uP LcCnNl+ifu2V/YZUOmOP/rCh9Ccp9M9G1K871uDXFlf0l6+8UL/M/cCnLpno X35O0PwgnWjGW+lXQlOAR8o/mOHMz9KAr/6FfsW3O3BXrb/Li+et/qhfS7UN 5Twdn92G9cy05+hPDhzEfudDjT/6FPbxR/+y4BX8apKXnC/QqBr0K7tcSsSv tsl+Ric+UYP+pY2sZzrzrLM9pH/xMIM/ueagfzmj6c2wPvKZhvvJPbR+lVP5 alecZ/SxRj8zIhfnF5e0em5YH13Ox37IAX/0L5528Jcc7R8IyI9ls4uiFz4+ SItXW351HThuveiJbt+vj33oZnl97EfXHwKvPavFJ4rOPwF+KPdNE8ktCfsl E2T/NpnafoL1U3gV/Oq7cehvh74Ff3I2o163rQV/On2DetYI+qLp8nslZ8a/ H9KJdpXI+bN+USPUs/lG0Ge8FfjTaBn0O1/O95MotxX4Ei39DFHACNS3q4Hw o4qd6HeytAVqfgTdWPlk8tyd6QoHUZPVwL+ORP/Tag3wpHPi9zRuA3D5Btyn mPU9cNCv2oRe69dsAnbMkvNR/nEz8PfaBoZhvZy1pz6O5XF7gaedhn8FLIWf 2Ul/pe2ngO/lFfAzO7Pp0r8tfw7/qngJvYx+Cf86Oxp6spD4duSS9eo+ubQn XViq9MbT8SvM0E9llOK8WM6XDesrrswRvWfJ/UrDumYJ+qWuoscUzvgO/nU0 Uvt+CHO06Ikt/kE/dCgRuNpV6j8tTweeWSj1n4y6ACfIP/AUTOnd6uNI2tej Po6myz2B2z2UeFDFQtTHQMSDZqaCf7GIB0V9Ar51fgt+5e6CX3WtAb+CtoFv /RAPqYfD1pwtnmSK+ujdA/d/S2X/OpUcv0Z8L9ig/jm4YP1u2gL+9G4d6p2L E+pfdTn6p8wo8K1qPPI5NU7DMeT9Sus/DHVX/gEnV7IyLhRc9gz9U5IJsKv8 AK4Fz2sEbCO/X2rLPk3q40Bebg1ccAz18HJz4Hna/o5hPZYwAev3Q9r9KMN6 rMIX5zk+lfCzxUnov7o8g5/lxsLfZmv61XXgk7uwHpN/v0TXiSsiiuX81lP8 rTNHJkfh93Pa63Vd2GR0iuzvygVW6sQZPdF/ZVjB38oPYn2Wr/ULhvWb6Qn8 PtOnOfxt38/ox3Jbwt9KjbAek3/ZKz+KC1eIXhQO4fHf1sdBfHCVwvIvkjlx +/X/YdGnrudmhbV+LC+UKnbXx9H0+37gH56hHs54A/6ffo56aDwZ/VepCfiT /LKP9NtVsh7pTN726Kd22KDeRVtgfebpAP6sW4X1WJdmqHdHXNGPODiDT7sH w6+ayr/3EE+2z8GXyFTxc+rQO7hb9OL/cDh9kFkfB5P9B8DjF2rf96DPs4GX fSL+zlM+Av7fr+/lfuaij+vjMH4wCbjuN/iRrxP4dFrW/8Qf2KO/irgHP3qx BOM9Xy7+z5c6wX83yPl7KjctgF4sn8KfEj5DPJJeo59adwXrkTbSP3XhzOxe +L1fI/RTMw/gvpK+Gfwn7Vvsn4Q6op+61BP7Pbec0U95rcb6aYn078Qf38N+ TLncT4nna88kngpH829V9XEk+0KfXCQbui68Hfrk7Osa4e2ps2V97Ey/NQZ2 lPsxQfTIrj6Oou2OwLvk94A6+uJv9Pury+BndxzQj+F+diItWQp/eyr3WZJo aRd1f6wS/PvbuET4OAzxo0PXUE/l/NXAvx6m8K8v5PcmHWmI3M/zpW9lfZNE 1t/Cr+60Qr0MHYj8TXHVcCLNuA3/+j8siDTI "]]]], "Output", TaggingRules->{}, CellChangeTimes->{ 3.758309751887066*^9, {3.758404115366766*^9, 3.75840416894275*^9}, 3.7584054107405577`*^9, 3.758405520515115*^9, 3.759008662216303*^9}, CellLabel->"Out[59]=", CellID->2032131624] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[Cell["\t", "ExampleDelimiter"], $Line = 0; Null]], "ExampleDelimiter", TaggingRules->{}, CellID->1016030537], Cell[TextData[{ "About 11% of the numbers to 10443 are ", ButtonBox["lucky", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Lucky_number"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Lucky_number"], ":" }], "Text", TaggingRules->{}, CellChangeTimes->{{3.758380281559568*^9, 3.758380325918171*^9}, { 3.758404985811781*^9, 3.758404989811455*^9}, 3.758405604659713*^9, { 3.854021840300685*^9, 3.8540218403017282`*^9}}, CellID->1612587837], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"ResourceFunction", "[", "\"\\"", "]"}], "@", "10443"}], "//", "Length"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.7583796714853697`*^9, 3.758379735561987*^9}, { 3.758380088569213*^9, 3.758380088766447*^9}, {3.758380318513714*^9, 3.7583803193748627`*^9}, 3.758404992920609*^9, 3.758405596660761*^9, 3.759504006658078*^9}, CellLabel->"In[160]:=", CellID->981272946], Cell[BoxData["1163"], "Output", TaggingRules->{}, CellChangeTimes->{{3.758379681624024*^9, 3.7583797376296864`*^9}, 3.758380092896823*^9, 3.758380321230467*^9, 3.758404994528508*^9, 3.758405598412504*^9, 3.7595040083105917`*^9, {3.75950417198228*^9, 3.759504176683282*^9}}, CellLabel->"Out[160]=", CellID->1724653257] }, Open ]], Cell["\<\ This finds the coordinate pairs that are at lucky number distances along the \ hexagonal spiral:\ \>", "Text", TaggingRules->{}, CellChangeTimes->{{3.758307737030126*^9, 3.758307749085785*^9}, { 3.758308123305419*^9, 3.758308138992836*^9}, {3.758375357599758*^9, 3.758375360055704*^9}, {3.758380032855719*^9, 3.758380033816297*^9}, { 3.75838023056177*^9, 3.7583802470647984`*^9}, {3.758380391566204*^9, 3.75838041562714*^9}, {3.758404231461878*^9, 3.7584042338851213`*^9}}, CellID->1940924303], Cell[BoxData[ RowBox[{ RowBox[{"LuckyHexagonalSpiralPoints", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True-> GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["HexagonalSpiralPoints", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{ "\"Documentation \[RightGuillemet]\"", "https://resources.wolframcloud.com/FunctionRepository/\ resources/HexagonalSpiralPoints"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[ Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "HexagonalSpiralPoints"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["HexagonalSpiralPoints"], BoxID -> "HexagonalSpiralPoints", Selectable->False], "[", "n", "]"}], "[", RowBox[{"[", "\[IndentingNewLine]", RowBox[{ RowBox[{"ResourceFunction", "[", "\"\\"", "]"}], "[", RowBox[{"Round", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "4"}], ")"}], "^", "2"}], "/", "12"}], "]"}], "]"}], "\[IndentingNewLine]", "]"}], "]"}]}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.758374801067741*^9, 3.758374890744421*^9}, 3.758374922745802*^9, {3.7583797997804203`*^9, 3.758379800348208*^9}, { 3.758380111494788*^9, 3.758380116925457*^9}, {3.758380158183619*^9, 3.758380175827964*^9}, {3.758404242632964*^9, 3.758404254561058*^9}, 3.758405041439165*^9, {3.758405073650158*^9, 3.758405110325498*^9}, { 3.758405621758876*^9, 3.7584056242700357`*^9}, 3.7584056680556717`*^9, 3.759008498105685*^9, {3.759008693405106*^9, 3.759008698202599*^9}, 3.7595040152333927`*^9}, CellLabel->"In[157]:=", CellID->1279132408], Cell["\<\ Here are the first 1248 lucky numbers plotted along the hexagonal spiral:\ \>", "Text", TaggingRules->{}, CellChangeTimes->{{3.758307737030126*^9, 3.758307749085785*^9}, { 3.758308123305419*^9, 3.758308138992836*^9}, {3.758375357599758*^9, 3.758375360055704*^9}, {3.758380032855719*^9, 3.758380033816297*^9}, { 3.75838023056177*^9, 3.7583802470647984`*^9}, {3.758380344111113*^9, 3.758380351461541*^9}, {3.758405050553372*^9, 3.7584050517209597`*^9}, { 3.758406207249118*^9, 3.758406209777055*^9}}, CellID->838162591], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Graphics", "@", RowBox[{"Point", "@", RowBox[{"LuckyHexagonalSpiralPoints", "@", "350"}]}]}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.758374960742985*^9, 3.758374972060505*^9}, { 3.758375030188692*^9, 3.75837506018578*^9}, {3.758375182688324*^9, 3.758375182973997*^9}, {3.758380128182886*^9, 3.7583801289972897`*^9}, { 3.758405054482452*^9, 3.758405055417062*^9}, 3.758405087733027*^9, 3.758405177483634*^9, 3.758405218589514*^9, {3.758405269213189*^9, 3.758405271825583*^9}, {3.758405684197711*^9, 3.758405684578669*^9}, 3.75900870986978*^9}, CellLabel->"In[158]:=", CellID->711872390], Cell[BoxData[ GraphicsBox[PointBox[CompressedData[" 1:eJxtW2lYlWXXZQYFBRkUFfQgIIgyz+PZzIjzFGaKiJZDDvimaGpBzmYWmppp GY6ZpqKZWkpnO6Y54RhhzkMOKJgDqCAfz173e13nva6vH5/fes85z3Pfe6+1 9r4HPHIn9H3XzMTEZKGFiYn27///3964Ld0H3pu2+7XeTvB1w4+CHxrwebUe /97cf6xB+++a4aPopIC/j1nQE8ENho3q963V91cDG17hc/1v6vOm8rkFj1e/ txBsx/3xPvWeWsOX+JwxHhuSfwrtS4sxPv3dwMizdn1bUIk8vsZQrTC+51F6 RJ5ny1flc2vq2wPjMZUvOOB7Or/S3ur5g/B9mooB0HB5vzM5qPl9pbBOPnbk 25mID5shXk/xfsZ47MmmJ96X8N/3qeeX1yJ+7ZN7LNyid6VkfJ//VM8PkgHa cBow58jzqw3TFC57g/ncVDhAvW99D8zXR97nSje7It46SUCV4ai8vy2dkO+3 ZotYxD9avt+GN6ZhPgdNkY89STI+nm2OfLxWeLd835kMURhvFxmfBV0ZMDw1 0789ldggP53eAraT9znxuz3w/qWC2/K+nojHCHmeGy/JQLzGgqQ8CeNl39fg jxt+z9vwezraB9/3RCLpVhrmuxTPI10mno94OhN3w/vPyu91/Gkc5o/Hd+De g8C/f1U8jgcgn2Hm4OsdzI8fSfxb0bkYxD9djWdPL8S/XrAH+PV3dukoU+Rj iHr/MWuJp37erv0Fk5p5klud9n1zfqrwNfm9J7cagPk4yIO8WP65kBPf3RTx ckT82Vni5czjhH8eXP4a/BgQBHxUjecrPM8wT8UrLxL5O6rG54rf03wb6NdL +OVFP8rvO/I/WYiPlfzeg++p+KQLbsdhih/5ij8zMR/OakC8YtR4bqjxjED+ 2Vl+35HmIv/6KDW+P1S85qt8fqLG9yP0R1Wlv/+dPLsjbVbx8lL6WYbn82Gl D/l6cc/4SZgPP4AB0CiFz6vnl6j4dRXcgW6r+EIfXtS9G8aL+fvSi14qP3Xg Z7XM35fqoHcu+h/synkK31HxPO8Pv/pQsC+Py4b++qjPl8XD/0wVPw+lIL66 s/AP+k3mz48l/y1pPeLBe+X7PqTrC/4XCvam2lhjP/Og28gH2Yg/2FKbY+ei nV53ojoz8H/nUWChg4kfW6TgeYfl953ZIwbzDxXcicuGgF/3RR8+vKQ/4vX4 FfQeKeP35Qxn8Ot3xIMdLaFP+//BbvRTIvA8U8SfM6D/yyr+Gcl4/hDhe3sq wvOp53Pw30H8ojPtlIJjw9B/Z3KS+Xly9T7w54Y8rxPXx8GPfeoR77rd4O9q 8WsX9jou8eCpN1Fv3BAvdoiF3634HThX9N2G7BA/rjcDn8wjwbc5ZphPJ3/w bZkp5vMgBHwzk/e1pvbnryy/OdGfatuLXxgMZcBv5Pkt+YszwEHm8IeLesx/ tOQjUHhfOGR2aY83yNekbMTricSnFReK/jrz4T/lBwZfxIvtm8IPpmYCF5th fL4piP8gPI8GJINPYc8R/7qBmj8GUtkF6GH/W8DvST51XNcX2MES8S8+hfHv kvn7c1Ek4m8jOJCLPaEP4W9BEDfsUvX2Dvyy9zn5PZsEQA8zzgJTG/EfuoD4 sK4Z9GGicIgV8nEgA/M7I/MJJN0o8LtbPea3fiD8h/6V/NN345+9HDgpiEyO S/71e8YpPF7yb0hW+EMr5GNsH8x3i8y3I+ekg3/2FtBTRij89TsJVzDyZb+4 NFxwELdcBD2lvYKe/PTgz39eip45LQP6WFoFfc18W97H5TrhOy3vDzzWAfpa 2Q94N95PmZ3AvxeS3wCK8wP/rj/WnudEbYK1z4OpMEX0apgZCEwdoKdXAcBF YhAdONcf2Ff46cN2J5DfKCvoq5LhJ4dlosG8wg35HSj5DeGfq5Bf13LxIx6N ePMzJ/i9x1jgD6ww/ovBiEek6DeU5J+ey+ODEU+6no98Zoo/+NP2KPidXw38 8PvByM9q4a+ODj37cBMHhlDcQ+jPR2H+SSPINX2/f4Fzm4K/6VXAeolfEF/w xvN/Fv4GM0WCP5PeIL8VH0J/e19Bn7/kQE/Fkk9/7h2NfPg+Qz1rGirx5N73 RF88IgS4pJXUFypE/LncFn7yeRfgoibQ65oU8NvLCvn9B+PjDRKfUHLcA/++ L3wPooThGO/Wp/J+Mt2r+V8o7TqtYXtqsht4oyvq2YGdwKPMkM8wPfo9+bgw nNePzpJ+oVKbf0Eonx+LetZdnu/F255L/BRu9AfEm4NOQd93EW8uC4Gey6qB o6wxnykByL+J+F8Q+Xthfi0l3iHkMwD+NK4W+vZPhd6CbmnYna582PCW78NQ yjmK+vfuVOBRzbXneXD9B8bYmxMV3myO/NZ2wfN+k3yHcW1bvC9fiBjBG//I kviWSbzDRd/XE4pLC2V8obxiHvhQLvEP4UkYLwc9gb63Iv5MsdBzn5+A85sh v06dkW83O+jheDfo+yneJ3rgWWvil8r7wmj2F9DDhdfId5N0jP/eM+jj24Hw p2PV4jc04xeV3yro5br0I2FU+EJ7YJW+IhqYZ8Mf7BS+aQ+9W/4H8arR5lcQ xm/3Qb+orwcf/spFP3NGG09BMFceRL19Jv7Tga9Ok9/zvSvivxyAfLGr+E8b ajcJeLvkI4Q8dcj/HsSfru6U+BuiJR4R1O0R+P60DvxfPgF8P10Nv7eOgH9Z PIIfLGuYbVm0MZwcdgkfqaoOuM4R+l9XAzxO/DSUp/kinhvMwYc37vADFzPw wcER/OghfInk0FdZko/9Mr5I4UfOzA2lJRLvzuyO8fD8Svj/8QSJL+t2ih65 dSxwdUfRI30UAVzeAnyYsg18WdEcermTDn7MsIJehsbDH+wwfjqF8fLPdeDL Z5MQny+qwA+bSMQn5qL4Edkd1PqnCDrqBz/ouw94uIvwk/P2Ag/Rnl8YxjuL sb65qM2/MJKbmMH/68UvovjSX4myHpP3F4Tzs7HgR+pL8KOiHP2Um4wngC0n q/pwXsMt+H695INJCHbNcP81cFCJ9oJa/f1XwN4uiM+hEMSryhrxKE1FfDpb gU8LglBf1yF/lGYHfWJ9FyX60m36If5VDfT040jUkxGViNd7+6Gf6n+Ez7Rw TtNdNcMiycEV+nCfBmxogvrR7Ud839QafNocBT22sgKfrALQ360Sfsdw0UT4 y2HRdxQHtn9b+A4/ieRNO+AvcS/hL2/ewnzKxM/b8pkDkh+FnXgfA7MU6Gp9 QSlwULCGnSlrnzF2o+RfgVe4Ip7We4BdMR6y2Kfq70vozfNtjN9H+rdOlJip 4Sj6S/q1DlwTb4w78l9xwFdlQ8mfP56KeN2T+hfIHZ5CfyZNUX/G3EM9jhB+ xfDHB8CnNPGXCG6xAOv3jFfg18uT8JvutcqProBfJbe18XVkl3nyPs45Dv/5 dhZwobeGaww9PgE2kYJ/Xe9dCKyzBZ+Ct0B/dpK/cFqrM65XUcQm4JPYb0OM 8On6ze3xZ1/Cvyf3g5/8Iv1JMM2Lgf7iZP3hRRtti3KHukdTYbommGv6Hy2B D7trT+rMPeu/EGxjDf+Z7Iv4l5uBP8tfJYofifGYxPGmePDpJxlPrPzPA6tK ShfUgV/3JmO846R/ieAfu8Evwl+AX27joJ+/bsCvopBf3nlIw614dhqwTzvt yV60Hvnl+rbwp4O7wJ+DTaG/2V7wv7nWiN9nGD83F/+MpU/isoz2F2Mlfr3X 7ogf+xJ+/8FbiJ/XC/Bv9ATo0/ca/L6kTMt3tMKe5HUaGP2zjrcdBF7bDPxa Uw2+ZTSFn/XJQb9fZgn/8lsGf1tqDv799aa7rGcqxM9i2HkS6t+vwsdoXj8b /jZN/C2Sb54DH9u8QL389SHWKyu1fqBx/TPOao74V+Gl1+Jvc5Bv7u4h9Yie v5F88xo7xO+rTeDfdFv4u6cn4hnYBPHMD4L/fyX+H02bm6Af9Zb4xtElHeJb LnqOI7tZ4i/6vGeIZ2Y8+PidxC+Q/ugBvXK8RrDrhj8XuLT4vTyG/pYNOm92 mQUcaA6+tXwE/k3E8xv1Cf+aUgv/8hsEvWyXhiaUP8L7uJv0R8G8IxX+3Vrq dxCfOA7+jLusYQ8+f0ryx196ib9QrwPGuCM9NBjjAPIsBXZsAf1uiAQ/1zZH PH/ogvcdbIJ6kOuP+LWxQvzG20EP6A9jqbIyEet50VeC8LNY/3N8i2ro2faE qp+y3vGmibJfEktl0VoCqg1XugHf+0b7pQ1PyASe1FP8l79MB46RDd8AvpwI XN8M/Ow7Cv3XR+KHCfz5AvhhuPAvlnssAf/8X4B/m+7D/25p9bVxPebVBHxb 8xj1N2wR/C2mHPW3/zLJp8IduLII+KiMx5viZhpjf+peALzHGvGycUZ+fSwR r7LXWcK/XqaIl+HOW8IH+bqJXuIX1GVP/KKX8MuMgeivfhB/jKSiPNTLDOlP utCj1qsMAUvjFPal912Br2/WnmTH79gB75X+xZMvWhtjH+5nBews/V4obwkD vw/LejqM3bcjf2HiV9G8oyPef9sG/vpVW/htfyGAnvsXod9zxHj5DcbL4x/D P789Cf7elv2Tzuwp+62xCnuyLfjBuu9FMIbLPYDzsCFKh8APduuswS40HXzg /zQDX2e1U+tHK8S7ryX4utYM8V7xhYyPsV7QU8k16N3rJfzVbDDinfIY+m9x GvO/J/1OCDn1RXxKBmvPc+XvT2r+FU8DA2Q83Oso8PSW8NMOpojvAEfwdVAe 6nugprfG/nDJNdTzW+Kv8exWmyvryy7ipwnceTH89ILmDwVRPKEWfmn3BPU+ yGKarBdnPASfdc7wy+JUbbzmvNdB3s/+LtD31b3Q/yEn+GVSKPRf5QC/bOoL /c+xRzzHr4W/hlsgflGXoHepRwUkfHVI2xe/VdZbCVT3Hfr/1k/RH31ToPaH noC/R5Php63uwx+WvwM9H1ulYXdqtlI3fnmvBCqK1LAt/bIcOO+Y9iYL3rcY OEoOVBy5RRGwWwj812mhMfZnwzzgsS7gc9Ae1LsFVajvZ37H/Hwewn839EM8 jtyF306/JvnkT0/Bb7eUA1fK+90a/Q04PwT+m3XcGAdQ7GFgQyvE/0Yc+NrN CfGu2gI9vNMMfrHND/3F8Kbgb0tn9AMfWMAfTp5C/Js9gz8smQ49flmF+H55 HP21y2P0B5/0wHp11X3wudtfmL+H9KdB5PKu9j49zZYDq7Y0dLAxdqI1g4Bn vK+935kf9gG+Jv1rME9JB16sxbcwnI/8i35sgSP4PfEf9BPw6xj+4C30E6NE fonsbD9Y9sOipF/V88kS+HXRv+gnChvA97xK8H2YPfz62G349f21kl8uOg5/ Pr4KWPelhl04egUwdRQ+8UTwifMitPF3ouD5wLFWiO+Fy/DnxVJPiDYNhl+c EPtJpD7Z8Gu35+gXyqfA//5+DL7POIH4d3mg4XDyzYJfeN2S/oZuOPVrXmBJ lHEU/v3S0RjrKNEB+Ppd+PfZpsB7o+HfrW2AewmfIth5F/J5QvgTx3s8obdf pT4T27+IED/rKP6XyPm/oR8JEL/TczqjH3rvGfqTIZgPrxY+RXO3dPjpRlnf hfGaCvjH/iPSD/HQEZJ/hX34+xxgCVeH0viCAcApMdr/oCP7nsAd3KGHQynA s9vCj5oYoJfNLeE/l/2hx/Gilyga2QH+NMEM+Zm0D3pokP8SxY+ifA/Ex76A H+2dhP741APoo+9F+Pm6m/DzpaPAX9Mb8KOIfI2fRHXTxdm47H1j7MpXxyis +WVhLK8th3+XNIN/Z67EfvCnWvyZeH05/Fz6eUriXB/0Jw80fzcQf83ge/kz +H1mDc63KqrQv3jNw/o/6DH0UNsUfv/3A9SDYkfogQ9o4/fiSvCHTaK0BvOa wdkOmOSAvjN1twTWtZf6RIPNgTPaIf5Xk5CPqNaI/5IdyLevM/wpTYf49zdH /Fccg15mgF80fwfyEf0I/nT+DPTgeQ/xT34HenjnBvxo1hjEv7hUwwG07tKy OyG3iHKSRJ98+RhwF+FHJC+PQL2wEH7EcMIG5PNvqV/xfNcJ9aWDBfj+9BH4 31rqfSL3HPG26HmN1PMY3nIa/t8g+6Xh/O0w+HPqTfB95B3wca2Mz5vz/yP5 55Tp4o+cNwFYV6EVwCrD2HHAgbJeCaZBI4xxKA0cDjzBFXze0gn8DrSD/zia QX97pD9MIk8d/OdPDRYm0ZhbA8R/yp6A38sWwf+j/oH/jB+C+I64ivh6TwVf 6RcN+9Nr2d9KVLgTPY8wxh50Pwy4cBH83uAP7O4OPz9Ys1L6mX5aPhrXg9UT sd4Z5gj+X7oAPXwl+khgi17w+9WW0EPL6TiPnKblw5DEkSOTZb98qbafo0/i 4oPod26+gj4u1WC9eEz8P45LmoP/K+6A/yXu6HdMbqI+BHmj38n7D/z/dpnw h/m49oIaw8nTwMfS4P9Tfwceoa3fCoMo5wjwn+3Af5NfkP904VcCFTqjPrfQ Hl+QTLVPBkn/2K8O/tPzPvYTl8l6l8h+IepDjOzvhlGXD5GPGRXa5yE0cVX5 /rucSLWy/+pFwV8BV8sBZ2tqsQzY5DvN0K7pbywGtukmeuaJnwLP9hY9c5/5 xjiUE+cBl3ZAveifi/cPaAv9RG1F/XCzQb1oj/kRWUE/W+q6y/h/E39NFl/3 LTtc+kz60yRueR77JX7SEBJfLMB6571dqAfrwoU/vFLG60U7OgHf90T/028Y 9GCJ8dGROPhPfDvowzQYetwqeo+jD1vDf761gv+0q5XxMc4fE2mQrEe9+Hdz 6GfjD/CnJ6KfZOr9GfSDepEs9YI+PxrfSfYTk8jvDvrXl1LP9fRoNeIx/Q78 7Mkw6KtVBfLpEAd9BJdDbw/qNf4nUa5ckOpM37wyxp40qhbY1BN6etrQV+r5 Ihfo5Zu3sJ6Nl9en8I+/DZH+6F3RQyJvNSkVPYzT+NXYP1nbHJN6EaT1dwXx PHI5zguS/oFeXNqgPkzV/LYgmttvwPr03J/CP960UvjB67dL/8FB4BtnDBX+ setS4LxEzZDsqPhz4DHeyFdVDvJ3sz3qQ2YQ8nXOGX7WoRn6h8U2yNcFM6wH N1gjX241yF+uykezEvFnKgG/aPZMrMc6S//auD4bjvgfuAW/S3qM9dZabT+k IJTsh7ZdunJ1ksLBZJsNnLJH+74fmQ8G3p+B+nKmJ/BMT+jjxD3kZ5DwL5oT CPVprRv0kv8r3tdV6l0KO29EP4X7KCmij3tjD5XelPPAFC7bLP5gyBF/SOKv y9FvYf81mluMQLz6/416s/K1vJ83yQM78bka4N4TNdxWnR8m8UC5kNSBxj4y xgG0/AFwS6Wnu17Qm6/067GUCL3zHBX/xU2Rj0eW0EvpU+gH+2Opoo/aoMPx H6t8DP4E/dSuKtQfx4FYX8Rdhx4WGjS/TlY4iM6UAk9l1JesX4GpDvs/pruB 58uNyA68YCfwYeFTKIduM8bhPGAr8JeqHm1Rfhes9SuN64uxhVjvfWSNevNG 9WdNNL9o5O2kVejPTkn9SeEBMag/wzT9G5L52X70YwuU3pLeYL9oZK3YATez Rf0pqoTeXi/HftHHqh6Nd0E/5Szrk0ge8yfqCYn+Qnn5IOEbm8zR4uHOpwco /EKbnwWd6wNcGK9hDxrfyxj7U9MewKzxszCMNmYC5zuhPkWhH+ID5vC7XSfh f61lPYN8jpp/IP40+EpZQ6E3P6lnKXTFAP+brvq13Bj4W5LML4xc2me/33NG Ms3+Q/vcl8LdgB3kAeb0rStw9UXtTXY8uSXwjHANe3IPZ+DfND42rhfr7dXz pL7qeX0Q9I3zumQuxPqXEqTfT+GAf2Q+1BTz4WX52G/a8Qb62+mO/m5BPerV xe3Q26nXqFcb8tBfRct+RSLfmgr+b5b9ygR2OIv4OYRo2J77/yx84+pTJlIq Rv0EXPueBh3oMPjKM0I1rKMZO4yxF5WXGOMuVLQdeH5L+OVt1R/2dUJ9WxgN P5iI+kv2reGfBpVP8zrpZ3muDCeNvnNH/3FT5pdIq7Feox+foX6FXUB/vFT6 xTj6Iwf982BZDwXT84+0epFM1/vKfKnmQ+AM7K/xxUnA67F/xDs+AI7W/L2x /0tS/nlF9XdTK7Hen6zlr3G9kz8Y/d8GWf+k8rEnOC/4WaKZxgGfYz30sh71 bvs6nF/NUXrMrUD/d1f2AxJ4nRP6vd2y3xXBd9oIf7hcLsy24CngG/deK/yj AvCNl/oK/8jSyRgH0sIWwJkyn1D6wgE4pDn01FHtl+U2Qz27GYJ+/UPVb7Rc h/WDqfAxTfTlcGp3fL7S1+K16Ac/UuvRceXot5zUftj7++BnD6pQv2yvFieZ DUqhC9e0zwMo9ArwJulY3GnWZeDeBRq2pS0VwMVvNAHU6PUKlxSKn/DIv4C7 6MRP2LPcGPvztUvA2c1R3z56jPrXXuZHXIHzApqg1lPTfgKfhqr6t0udt2Up Pfo6Yb4N4CePeoP6aCH+ksrvfIB+cZLUk0SOYMRvIOLDd88i3g7YX+HfKtB/ jxU+x7LfUPSHebI+8+BYuc+SzNflwqYTL5wCXCT7C370AfjLx2R+4eQE/nKM 4CjyV/osskN9nDUKevxWzvOJ+u2EHxxR/eTLDtDjBsyfmrij3z2l/HV3Hda/ blhv0t0DWP86yPl5NI1FPSebF8h/5XToabbMN4SK5f5JCtVK/9SR50YbYz8m hWfI/mAw74oyxqE8XuHRTVAP81BPaIdWDxv7y52foz6Gq/2JpQ3Yr7ul9LhC h/uJsv1L6Rwx8W3pP10aUC8PHYYee6j12w+TsD8h96sa6+Wpj9B/7hE9J/JQ M/SfKWp/z0WdnxS+RP2M/hr7b4PkfDqGJ3+J/rJWi1djv1qq6uModV+mCvzl MumXXHihwsWyfrXltxVeaIZ89d4Kfj01Rb7uhMD/Nwo/08l6Oc7v31N+6psH /m4U/SbRP+o+iGwfNhBNUef59VI/oqjdNORv5WvUx9Hf3mq9OzKVwiTfXaiJ whh/Cxr9DTDGb05dFc6R9WitoYnCJ2U9FsidVgHLffjCRC5PxPt1gpPYIQr9 l6oHvKkz+uVBmB8HBaGey3LPJIO1/8s9Z5d2a4D+rhnA7wich7B1T9SfF1I/ 43lKFvR2qR56bDUe+hir7muuAh+55D7q5fsKy+t6z43vp7CvnM+6UqDCf0h+ wugy+MobzaC/hU/Qv7qr/Y+jkVjv2Zqi/m3B/Vq+iPmI35q4Fcafk/8nlTzT 4UfAKdRrENZvght//6+674nPE+n2cfjbenUeNw/zUziO+j2FXter8/hf0A8o HEY1cl8qlTLEH3zJvh74+r+Ih17hPNlvbcWpCu+V9YAn91F4sLo/Ofp/cDhP Vlj+HkmfzBbTsT6U5QinccZI3EeUQFAG82D8vcc3Ss/3UlFPK+X7yVw2GnoV PjXq32Ie9FhrBn84g/1mdpV+I57jT6F/HSg4llv3hR73Cm7sB74WfiocxCcU vi7+6sgjVwIXij/X6H9WOKMO9e478JuH4/4F/RwDPpwRHKPuYybzf/X37gnU ByuVb+/u0LMsdxr990wS/Dhf6XtlD9SbbYovNmtw33iUqtcDO0IvvmbgR1kJ /HqpFfhQHID+t0H8M5IyB3d8MtolTeFwyv8fHEynFbaQ9U9rvpoNbCb9pY6b DgU2Fb+L4jE5wP5yHyCaBzUg3/GvoMclE8HHfNEf8bO/sP7sKPxI4qW/ox6b qfoaNQz7gXMlXo1+kIr5yXW6gnSungp/+0rilcHly3D/wd8UfrHeA/2yB+od L45BvzNHcCJ/nQJ/CBN/0vO50fCDQXJfII4Hqv7XpAT3J9tBH2wi52velF1v jDvRaoULnys/xfy5XPwzni6EYv1aLHwhclP9QK2sr1PpFu5bwS8KM2j7bawH 5LyyMIk2RMMvZ4r/6KllF+TzU6n38RSA8VJL9D90rhbxXyk4mrLlPlgahUk/ EUC1+4GL72u4Wh9hAF5xBP1HFwY+qfqTMoV/eoL6Ni4X+d4n+//J3M5W6iP1 q0d9jTGT9SW1lXqbzlGroe+RQocMzvsM9blK9JzBo9dL/8yRZvCDMUtQny8L TmXH77F+9bSA/oNxX4JLtf6OEtncDPulTZtD//0Wov721RqAwjiervrhBO2+ SGEoj35bxs8zRHCdedUgY+zDpu8AH5TzihB6nWOMI+j3Yca4sf9CPPhL0Ucs 3ZuB/myr6D+Zhh5B/Z4l+U4j26HQt6k59P2uE/YjhptDz9fc4f/bLaHnQdug D7lP38iHc+jn6Zxab9WEol9a4Qx/X/YC+Z8p6x9vsmo3+pj700Y9Z2m4Db1Q uHqd9NuGSzrgj+X+gh9v9ATWy/lMJL/xBn63EvWzXyLe5y79QAJbq/0Rtd/I N9R+YaXov3H9cx7jHyT9RQq37IV6P0702pUrZuB84Ab4z1vPo/88Ln6Wzi0b sF70lv49jV+0w3p3qw30vEqdvy0T/Tb2vT7Q29+y3ovnZROQj8/lvCKGF8n9 xzR2k/uRvjwM+uCoXA27smUpcFGs1D86AH3wGLV/twjz52yJRwJFq/2u9tIf JtNOrFfZqR75NuRD34vE79Lp3K/wJ1vxswx6/AP8vbn0D13J9nesN/6Q/jeR 5uE8i87JfTai7ztC/4dbQ/8TgpEPnK/qyKlA289Jp7wPNOxAvQqBcT7bma/M AZbz3IbGPneeMQ7nAfONcTQvXgDcTPZjE3jNTay3QjS/MxC//w32r7pr+dU3 xnEh1stXJR2ZnPpvP+m/rbT5UVeu/SFCzkc+tkR9n70c5+OOVtD73t6o7+M1 fyPi4XOhZwvZb9Gze1fU7xnu0He3xai/J+U8MIonthK+8grZj+vE37Y1xh5s 6g6c80bDFtyqPXCx/AGfLfl7AIdp9a8gnCb4GONI8vAFPn8d9d3nGPix6Db8 vslM8O3wM9T7Gzgv4fVyfyuJFo5E/RnxCv7fvhfWY7PrwY9PJqu/95H95hQ6 Vwz9LPRCfX5+ect3Z8vSqcsSbfwtadF14JIMLcC1epebwA92a8/z5Yn/AE+W /eQQLr0PPOcK9Bz4B/w99Tr0HJgGPvV9Af1m34PfmNVAv9kjUI9ya1GvswdD z47SHzbicvQfHV6jXu9fgXp+X/SbwV3U+cZ3ltD3CLV/7iP+ncifDFXxkvto 0Xw8X/jHRas17MbDPgamadp8qw33wW/eK+cTHuQBfvNgOc8Iok/mG+MIarIQ eFs16vFzk2mi3/FPkK+GBPhHwHPoOfcB+rXxUs9TaHU2+vkAzJ92VsDft9VB z93mIn/2lqjnI27JfOmiJfq3g49xHvWD7BcmUsEt+OWEFtB31kb46Uo/+PsX k6C/jl3g7yPlvmOGwuHkHWOMg+llLPCu76U/4e1djbE/f50JXCH38UN4XHdj HMZZPYCr/oTe97+Pev+1dh6uJ779D/TfRItf43pZtxb6/1BbLzT297v3Yn3e RPqBrry4zQip/z/IejuTj7zG3xP4aP0MpfI353Hfv17D3Mivt7Ef3VmLDyWx 82s5X+JZTvCDb6xQz7P9UO9zPoP+cwNR39+UC7/5cLYWj0B+VQGskz+gqzVU 3gCemqTNtwXdvQ0ct0bDOrp5F3i19gdBBUFU8cAYR9CFR8BHLsH/d3RFPdha DX3/NAP1qLQG/j8rG/pOfQ1+XFqt7stqP2/IpJJj70j/OrEp6v+QudDPPuFH MkUGwv/by34rkWsn6LNCm29jf1+5DfodEqb9PoZKlz/On7Mtg34dpX3uT+7f GuNO5L8aeNJZ9AND1gB/Jn+f6cKl3wOnfaXhAG62FXjdCfFDji8BTjyHej+t 2RzpNyykHyY+ugD7CRWPJB4clYJ6dVL6gyTOnon1QEk1/KV/PdYDPWvgF6OG Y327yBr9gK4c/jDHBv5RfStL6v84W4kXl8yC/obI/ZXGPvwL1N8B0v+m8igX 9P/xoq9k1n2P/ttb+qUkHusGvTdrJfrjjHfh37U6bXztaan8/WYGzyjScCB5 QR+cIAvaUBrbyxhHU3lf4E734S+Vn+J5gdIvJNKMVLwv9l/4SdRkNb6n8JMV TzD+ri/Bl5I96B8GSL/QjXKGDRa+DJb+qTvN9u4v++efWaN/mHwS/lLpBP5Y ZCHeFq3QT7z5Tp1nxWmfR9HgGm3/rKvCEbSkFthE7Od0/G25v96VHOZpA/Bh 32bGuDNnNweefwj9hImjMY7mtS7AB47CT/54hvoz9w78JGEi/GXFXfQTU7dj vyhPi4chjfuHYX2xVPqNxvWEP9YXvaU/SOd7e/H3hte1+xuUwnFX5H4HF8fC HyrtUK9NtBkVhrHXCuEvmxRqE/Tm5av/i7XxdqBpW4yxHzXfBjz6EOq72zLU j3Z3kN+bvaD/pLuoH16fI98vHqJ+XPkUftBG7hMl03Drq8K/3Crkf4LJeakv p/+FX5jmo75k1yD/IWXod92kfnSjzZ1xfnS7KepL+/O5sj802x7+0Wom+oXu 0v+m0AEd/GOS3OdNpon9wIcR7qg3Z+9gvTBG+iM9PXkf9WZ/IvzFNBD1IDJV +340rRsQ3DP6s640/lPtc0+alQO8t5Pmr3b01Qhgr4ea4bdg6zHASyZr8/Xi tuOBj/8qfsrJ+cY4hl9PNcZxnDUN+N3f0J9MGYjxxDxEP3L1c8yvufQrGTzu OvbL8mU/oiufPYf11GbRUyabn8R6K1H8JJOdN6Mfkfusjf5x9TPU+7bu6Ef0 Q5HP4zrxX160Cf3ePTl/D+dV0I/CwTz9FXCxpyZQZ25rLvznoPFSf2l6C2BX Oc8PoXecjXEYnXAxxlG0xRX4xzuoL7HN0a+skPvJ6VQxBfXl72fgi9197C+e lfVTd/JbP0D2ExYJX7rSsF3in9TVEXy51tBd+jkfqS+ptGcJ/Ni/Pfwiqz3q TUI38MG2Cv5wroeG4ylwpxavrvR/hicZzw== "]]]], "Output", TaggingRules->{}, CellChangeTimes->{ 3.758374979131558*^9, {3.75837503157605*^9, 3.7583750496884737`*^9}, { 3.7583751799436617`*^9, 3.758375203914423*^9}, 3.7583801317591457`*^9, 3.758380183124422*^9, {3.75840509073319*^9, 3.7584051155074167`*^9}, 3.758405179067046*^9, 3.758405220426038*^9, 3.758405272662887*^9, 3.758405685738316*^9, 3.759008711623313*^9, 3.7595040188106937`*^9}, CellLabel->"Out[158]=", CellID->302394609] }, Open ]] }, Open ]] }, Closed]] }, ClosingSaveDialog->False, WindowSize->Automatic, WindowMargins->Automatic, WindowTitle->"HexagonalSpiralPoints | Example Notebook", Visible->False, FrontEndVersion->"13.0 for Mac OS X x86 (64-bit) (December 9, 2021)", StyleDefinitions->Notebook[{ Cell[ StyleData[ StyleDefinitions -> FrontEnd`FileName[{"Wolfram"}, "Reference.nb", CharacterEncoding -> "UTF-8"]]], Cell[ StyleData[All, "Working"], Editable -> True, DockedCells -> {}], Cell[ StyleData["Notebook"], Editable -> True, DockedCells -> {}, ScrollingOptions -> {"VerticalScrollRange" -> Automatic}], Cell[ StyleData[ "Section", StyleDefinitions -> StyleData["PrimaryExamplesSection"]], Editable -> True, ShowGroupOpener -> "Inline", WholeCellGroupOpener -> True], Cell[ StyleData["Subsection", StyleDefinitions -> StyleData["ExampleSection"]], Editable -> True, ShowGroupOpener -> "Inline", WholeCellGroupOpener -> True], Cell[ StyleData[ "Subsubsection", StyleDefinitions -> StyleData["ExampleSubsection"]], Editable -> True, ShowGroupOpener -> "Inline", WholeCellGroupOpener -> True], Cell[ StyleData[ "Subsubsubsection", StyleDefinitions -> StyleData["ExampleSubsubsection"]], Editable -> True, ShowGroupOpener -> "Inline", WholeCellGroupOpener -> True], Cell[ StyleData["Text", StyleDefinitions -> StyleData["ExampleText"]], Editable -> True]}, Visible -> False, FrontEndVersion -> "13.0 for Mac OS X x86 (64-bit) (December 9, 2021)", StyleDefinitions -> "PrivateStylesheetFormatting.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[634, 23, 168, 7, 70, "Subsection",ExpressionUUID->"707a356c-c306-4887-b83d-1b47bc49f9db", CellID->462042388], Cell[805, 32, 279, 5, 70, "Text",ExpressionUUID->"968264fd-1f7b-44f8-a91b-30015bff682f", CellID->13078159], Cell[CellGroupData[{ Cell[1109, 41, 7460, 185, 70, "Input",ExpressionUUID->"e4fa9d42-77f2-4a71-8158-0bfe775875d5", CellID->443871870], Cell[8572, 228, 874, 29, 70, "Output",ExpressionUUID->"c4e23a40-0d3f-49e7-9c5c-ff7622bc8e3e", CellID->876657973] }, Open ]], Cell[CellGroupData[{ Cell[9483, 262, 143, 4, 70, "ExampleDelimiter",ExpressionUUID->"b0ee7720-c3be-49da-a689-b034f54955a4", CellID->64435967], Cell[9629, 268, 229, 4, 70, "Text",ExpressionUUID->"3c60cbd4-392b-4e69-9257-bdb647ad66f4", CellID->697960947], Cell[CellGroupData[{ Cell[9883, 276, 8633, 210, 70, "Input",ExpressionUUID->"4c96aa73-7065-4c0c-b488-72338159ed99", CellID->1710216102], Cell[18519, 488, 1971, 37, 70, "Output",ExpressionUUID->"b01e40ce-2b1f-4147-8d33-e9d04f193b27", CellID->1560128727] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[20539, 531, 144, 4, 70, "ExampleDelimiter",ExpressionUUID->"9fe6bb98-35b2-4be8-aa04-4c8718e0e198", CellID->643539281], Cell[20686, 537, 268, 5, 70, "Text",ExpressionUUID->"b4eaa4fc-f3d7-4f08-960d-f1e90d717886", CellID->1925797651], Cell[CellGroupData[{ Cell[20979, 546, 16307, 389, 70, "Input",ExpressionUUID->"a518a695-6446-4918-ad5a-3e30d56ba655", CellID->509996249], Cell[37289, 937, 5240, 96, 70, "Output",ExpressionUUID->"da293c70-a6cf-4d54-a763-d218e2f538ef", CellID->2079180016] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[42578, 1039, 145, 4, 70, "ExampleDelimiter",ExpressionUUID->"424b3c20-5780-4276-b981-15b4f6665e22", CellID->1237049475], Cell[42726, 1045, 466, 12, 70, "Text",ExpressionUUID->"88b7e378-8c3b-433b-b967-2423a1f7af12", CellID->2013137765], Cell[CellGroupData[{ Cell[43217, 1061, 8117, 196, 70, "Input",ExpressionUUID->"2d58ec7d-ee6c-445f-9fa6-81ff4ff3d9bc", CellID->744314247], Cell[51337, 1259, 519, 12, 70, "Output",ExpressionUUID->"82e614ba-3d6a-429d-9f34-4946973479f2", CellID->979495204] }, Open ]], Cell[CellGroupData[{ Cell[51893, 1276, 595, 16, 70, "Input",ExpressionUUID->"dd536d62-dc7a-4543-806d-ba3bdbc52e17", CellID->1155380441], Cell[52491, 1294, 590, 13, 70, "Output",ExpressionUUID->"6ad40fa1-c36d-4b4e-97b3-c2f89cc5a9b5", CellID->1355060892] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[53142, 1314, 167, 7, 70, "Subsection",ExpressionUUID->"ab6bc975-c2d7-4b05-8047-5b992c84ecfd", CellID->540091361], Cell[53312, 1323, 413, 9, 70, "Text",ExpressionUUID->"e566f8a9-1d8f-4abd-ae10-90a15cc4523e", CellID->1039848554], Cell[53728, 1334, 8266, 201, 70, "Input",ExpressionUUID->"bb8b150c-e461-4072-8787-d14b65e91ad1", CellID->914425330], Cell[61997, 1537, 290, 7, 70, "Text",ExpressionUUID->"0412f9c6-40da-448f-b07a-2eff50ede2e5", CellID->1077324486], Cell[CellGroupData[{ Cell[62312, 1548, 17777, 411, 70, "Input",ExpressionUUID->"e4c6ecd8-50f3-456f-bd71-600498c2afcc", CellID->1012815160], Cell[80092, 1961, 2947, 55, 70, "Output",ExpressionUUID->"83892ec0-10b3-496e-9bab-a75dfea19258", CellID->1543371489] }, Open ]], Cell[83054, 2019, 219, 4, 70, "Text",ExpressionUUID->"c4b8192d-411a-4ec1-83b0-973419334679", CellID->843341047], Cell[CellGroupData[{ Cell[83298, 2027, 554, 17, 70, "Input",ExpressionUUID->"41ed8405-270a-4605-a3ff-77e6e312b468", CellID->669943960], Cell[83855, 2046, 309, 7, 70, "Output",ExpressionUUID->"1684a93f-a877-401d-b8a4-ae2ce6e3a5f7", CellID->1996439306] }, Open ]], Cell[CellGroupData[{ Cell[84201, 2058, 118, 4, 70, "Input",ExpressionUUID->"963544bc-64c0-4b27-b0db-583e04b4907f", CellID->991672100], Cell[84322, 2064, 213, 6, 70, "Output",ExpressionUUID->"52cbca32-0996-41ac-852e-dc030af978f6", CellID->1016442324] }, Open ]], Cell[84550, 2073, 262, 5, 70, "Text",ExpressionUUID->"17e80309-2a08-4c5a-949a-484e5c7b4029", CellID->1606466798], Cell[CellGroupData[{ Cell[84837, 2082, 659, 13, 70, "Input",ExpressionUUID->"f7fe41e8-a24b-40ba-84dd-c4c13643d14a", CellID->2071963721], Cell[85499, 2097, 14733, 245, 70, "Output",ExpressionUUID->"ae84b6bd-0eea-4baf-9561-6a7f7d6c7dc6", CellID->2032131624] }, Open ]], Cell[CellGroupData[{ Cell[100269, 2347, 145, 4, 70, "ExampleDelimiter",ExpressionUUID->"cca2409e-3b68-436b-a071-a6dae9a0f1ef", CellID->1016030537], Cell[100417, 2353, 490, 13, 70, "Text",ExpressionUUID->"ac118349-6331-4511-947c-4a2cd12851d0", CellID->1612587837], Cell[CellGroupData[{ Cell[100932, 2370, 453, 11, 70, "Input",ExpressionUUID->"ae70e0d9-d8fe-4600-8a2e-9ae4fd04692b", CellID->981272946], Cell[101388, 2383, 333, 7, 70, "Output",ExpressionUUID->"666902ec-22bf-4ebb-91cf-82b968e4eecc", CellID->1724653257] }, Open ]], Cell[101736, 2393, 516, 10, 70, "Text",ExpressionUUID->"e53eb93f-9c55-4f93-bba7-c63b4e3932b3", CellID->1940924303], Cell[102255, 2405, 8494, 203, 70, "Input",ExpressionUUID->"0aecc376-b2d3-4d0e-859e-2c7deccf99ea", CellID->1279132408], Cell[110752, 2610, 542, 10, 70, "Text",ExpressionUUID->"eabc87d7-127e-45e0-86f6-4e60eec12213", CellID->838162591], Cell[CellGroupData[{ Cell[111319, 2624, 652, 13, 70, "Input",ExpressionUUID->"d5a915b1-5edf-4166-9564-83b17b5f79ba", CellID->711872390], Cell[111974, 2639, 13673, 227, 70, "Output",ExpressionUUID->"37859f82-770d-4d32-b03f-75060374b821", CellID->302394609] }, Open ]] }, Open ]] }, Closed]] } ] *) (* End of internal cache information *)