(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 445940, 7514] NotebookOptionsPosition[ 443997, 7449] NotebookOutlinePosition[ 444414, 7467] CellTagsIndexPosition[ 444371, 7464] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Exercises (7)", "Title"], Cell["The Game of Life", "Subtitle"], Cell[TextData[{ "M. E. Mauel\nColumbia University\n", StyleBox["(with help from \nComputer Simulations with ", FontSize->12], StyleBox["Mathematica", FontSize->12, FontSlant->"Italic"], StyleBox[", \nR. Gaylord and P. Wellin, Springer-Verlag 1995)", FontSize->12] }], "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "In this exercise, you are to write a program to simulate the ", StyleBox["Game of Life.", FontSlant->"Italic"] }], "Text", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ "The ", StyleBox["Game of Life", FontSlant->"Italic"], " was invented by the mathematician John Conway and first published in ", StyleBox["Scientific American", FontSlant->"Italic"], " in 1970. The game made Conway internationally famous and launched a whole \ new field of mathematica research which includes \"cellular automation\". 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you with your exercise, we will begin with a few steps that \ illustrate the ", StyleBox["efficient use of lists.", FontSlant->"Italic"], " ", StyleBox["Mathematica", FontSlant->"Italic"], " stores data in the form of lists and lists of lists (also called \ \"arrays\" or \"matricies\".) 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The integer ", StyleBox["1", FontWeight->"Bold"], " will represent the presence of life; the integer ", StyleBox["0", FontWeight->"Bold"], " represents the absence of life." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"nGrid", " ", "=", " ", "5"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "The", " ", "dimension", " ", "of", " ", "sites", " ", "along", " ", "each", " ", RowBox[{"side", "."}]}], " ", "*)"}]}]], "Input", CellLabel->"In[2]:="], Cell[BoxData[ RowBox[{ RowBox[{"life", "=", RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"nGrid", ",", "nGrid"}], "}"}]}], "]"}]}], ";"}]], "Input", CellLabel->"In[3]:="], Cell[BoxData[ RowBox[{"life", " ", "//", " ", "MatrixForm"}]], "Input", CellLabel->"In[4]:="], Cell[BoxData[ RowBox[{"ListDensityPlot", "[", RowBox[{"life", ",", " ", RowBox[{"Mesh", " ", "\[Rule]", " ", "All"}], ",", " ", RowBox[{"InterpolationOrder", " ", "\[Rule]", " ", "0"}]}], "]"}]], "Input",\ CellChangeTimes->{3.445866776304427*^9}, CellLabel->"In[5]:="], Cell["\<\ Notice the \"white\" squares correspond to \"life\", and the {1, 1} site is \ on the lower left.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Counting Nearest Neighbors (in-class)", "Section"], Cell["\<\ How should we count the number of neighbors for each and every cell?\ \>", "Text"], Cell[TextData[{ "By using operators which act at once on entire lists, very efficient ", StyleBox["Mathematica", FontSlant->"Italic"], " programs can be written. Use the built-in command ", StyleBox["RotateRight[\[Ellipsis]]", FontWeight->"Bold"], " to define a new command, called ", StyleBox["neighbors[l_List]", FontWeight->"Bold"], ", that computes the number of nearest neighbors to every cell on the game \ board. Use ", StyleBox["doubly periodic", FontSlant->"Italic"], " boundary conditions. This means that the top boundary is adjacent to the \ bottom, and the right is adjacent to the left." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Making Life and Death Decisions", "Section"], Cell[TextData[{ "In this part, we apply the rules of ", StyleBox["Life", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "Define a command that takes two arguments. (For simplicity, make the name \ of your command be \"", StyleBox["lifeORdeath", FontWeight->"Bold"], "\".) ", "The command will be applied to every cell. 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Test your command." }], "Text", FontWeight->"Plain"] }, Closed]], Cell[CellGroupData[{ Cell["Each New Generation (in-class)", "Section"], Cell["\<\ Using your command, the game board evolves in time according to the formula \ below:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"newLife", "[", "l_List", "]"}], " ", ":=", " ", RowBox[{"l", " ", "+", " ", RowBox[{"lifeORdeath", "[", RowBox[{"l", ",", " ", RowBox[{"neighbors", "[", "l", "]"}]}], "]"}]}]}]], "Input", CellLabel->"In[22]:="], Cell[TextData[{ "Since your command has the \"", StyleBox["Listable", FontWeight->"Bold"], "\" attribute, you can apply it to entire arrays!" }], "Text"], Cell[TextData[{ "Test your function on the game board called \"", StyleBox["life", FontWeight->"Bold"], "\"." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["A Bigger Simulation", "Section"], Cell[TextData[{ "Demonstrate the ", StyleBox["Game of Life", FontSlant->"Italic"], " on larger grid (", StyleBox["nGrid = 50", FontWeight->"Bold"], " for example) and (if you would like) with various initial conditions. \ Create an animation." }], "Text"] }, Closed]] }, Open ]] }, WindowSize->{817, 710}, WindowMargins->{{Automatic, 13}, {48, Automatic}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, CellLabelAutoDelete->False, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (May 21, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 30, 0, 73, "Title"], Cell[623, 25, 36, 0, 47, "Subtitle"], Cell[662, 27, 294, 9, 87, "Subsubtitle"], Cell[CellGroupData[{ Cell[981, 40, 31, 0, 69, "Section"], Cell[1015, 42, 186, 5, 27, "Text"], Cell[1204, 49, 416, 10, 43, "Text"], Cell[1623, 61, 304553, 4996, 362, "Graphics", Evaluatable->False], Cell[306179, 5059, 708, 14, 91, "Text"], Cell[306890, 5075, 394, 8, 75, "Text"], Cell[307287, 5085, 158, 3, 27, "Text"], Cell[307448, 5090, 131297, 2155, 150, "Graphics", Evaluatable->False], Cell[438748, 7247, 202, 5, 27, "Text"], Cell[438953, 7254, 335, 10, 43, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[439325, 7269, 55, 0, 39, "Section"], Cell[439383, 7271, 382, 10, 43, "Text"], Cell[439768, 7283, 291, 9, 28, "Input"], Cell[440062, 7294, 273, 9, 28, "Input"], Cell[440338, 7305, 95, 2, 28, "Input"], Cell[440436, 7309, 283, 7, 28, "Input"], Cell[440722, 7318, 120, 3, 27, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[440879, 7326, 56, 0, 39, "Section"], Cell[440938, 7328, 92, 2, 27, "Text"], Cell[441033, 7332, 630, 16, 59, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[441700, 7353, 50, 0, 39, "Section"], Cell[441753, 7355, 115, 5, 27, "Text"], Cell[441871, 7362, 723, 23, 59, "Text"], Cell[442597, 7387, 276, 9, 27, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[442910, 7401, 49, 0, 39, "Section"], Cell[442962, 7403, 108, 3, 27, "Text"], Cell[443073, 7408, 264, 7, 28, "Input"], Cell[443340, 7417, 158, 5, 27, "Text"], Cell[443501, 7424, 126, 5, 27, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[443664, 7434, 38, 0, 39, "Section"], Cell[443705, 7436, 264, 9, 27, "Text"] }, Closed]] }, Open ]] } ] *) (* End of internal cache information *)