(* 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[ 389900, 6780] NotebookOptionsPosition[ 386832, 6682] NotebookOutlinePosition[ 387530, 6707] CellTagsIndexPosition[ 387487, 6704] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[" "], "Input", CellChangeTimes->{{3.421149856978093*^9, 3.421149862619645*^9}}], Cell[CellGroupData[{ Cell["Sierpkinski Gasket: An example of a fractal.", "Subtitle"], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{3.420489277817371*^9}], Cell[TextData[{ "The Sierpinski gasket is a simple example of a ", StyleBox["fractal", FontWeight->"Bold"], ", i.e. a figure in which the same pattern occurs at different scales (down \ to the infinitesimally small). Here we construct a simple example of fractal, \ the Sierpinski gasket, in which the repeating figure is a triangle. We will \ see that we can easily generate and plot the Sierpinski gasket for several \ levels of iteration, using the recusive functions of ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["\<\ We first construct a single triangle. This is a list of three elements (one \ for each corner) and each of these elements is a list of two numbers (the x \ and y coordinates).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"starttriangle", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.0", ",", " ", "0.0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0.5", ",", " ", RowBox[{"N", "[", RowBox[{"Sqrt", "[", RowBox[{"3", "/", "4"}], "]"}], "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"1.0", ",", " ", "0.0"}], "}"}]}], "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.5`", ",", "0.8660254037844386`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.`"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.420488960307611*^9, 3.420489209316498*^9, 3.420489284098413*^9, 3.420489389464085*^9, 3.452523041594298*^9, 3.48297353957119*^9, 3.4829735713079348`*^9, 3.5148358558088818`*^9}] }, Open ]], Cell["This is what it looks like:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"Polygon", "[", "starttriangle", "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{3.420489340991727*^9}], Cell[BoxData[ GraphicsBox[ PolygonBox[{{0., 0.}, {0.5, 0.8660254037844386}, {1., 0.}}]]], "Output", CellChangeTimes->{3.420488963046208*^9, 3.420489209337843*^9, 3.420489284118548*^9, 3.4204893421911573`*^9, 3.4204893895939283`*^9, 3.452523045165759*^9, 3.4829735396041613`*^9, 3.482973571343783*^9, 3.5148358614727373`*^9}] }, Open ]], Cell[TextData[{ "The command ", StyleBox["Polygon[list]", FontWeight->"Bold"], " contructs a graphics element comprising a filled in polygon connecting the \ points in ", StyleBox["list", FontWeight->"Bold"], ". This is then converted to a graphics object with the command ", StyleBox["Graphics[ ... ] ", FontWeight->"Bold"], "and displayed with the command ", StyleBox["Show[ ... ]", FontWeight->"Bold"], ".\n\nWe now define a function", StyleBox[" triple,", FontWeight->"Bold"], " which", StyleBox[" ", FontWeight->"Bold"], "takes a triangle and makes three triangles out of it in the same overall \ region. " }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"triple", "[", "d_", "]"}], " ", ":=", " ", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"d1", ",", " ", "d2", ",", " ", "d3"}], "}"}], ",", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"d1", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"d", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"0.5", " ", RowBox[{"(", RowBox[{ RowBox[{"d", "[", RowBox[{"[", "2", "]"}], "]"}], " ", "+", " ", RowBox[{"d", "[", RowBox[{"[", "1", "]"}], "]"}]}], ")"}]}], ",", " ", RowBox[{"0.5", " ", RowBox[{"(", RowBox[{ RowBox[{"d", "[", RowBox[{"[", "3", "]"}], "]"}], " ", "+", " ", RowBox[{"d", "[", RowBox[{"[", "1", "]"}], "]"}]}], ")"}]}]}], " ", "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"d2", " ", "=", " ", RowBox[{"d1", " ", "+", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"d1", "[", RowBox[{"[", "3", "]"}], "]"}], " ", "-", " ", RowBox[{"d1", "[", RowBox[{"[", "1", "]"}], "]"}]}], ")"}], ",", " ", RowBox[{"{", "3", "}"}]}], "]"}]}]}], " ", ";", "\[IndentingNewLine]", RowBox[{"d3", " ", "=", " ", RowBox[{"d1", " ", "+", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"d1", "[", RowBox[{"[", "2", "]"}], "]"}], " ", "-", " ", RowBox[{"d1", "[", RowBox[{"[", "1", "]"}], "]"}]}], ")"}], ",", " ", RowBox[{"{", "3", "}"}]}], "]"}]}]}], " ", ";", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"d1", ",", " ", "d2", ",", " ", "d3"}], "}"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input"], Cell[TextData[{ "The argument ", StyleBox["d ", FontWeight->"Bold"], "is a list of three elements (each of which is a list of two numbers, the x \ and y components) which give the coordinates of the original triangle. ", StyleBox["d1", FontWeight->"Bold"], " gives the coordinates of a triangle whose sides are half the size of the \ original triangle, in one corner of the original triangle. ", StyleBox["d2", FontWeight->"Bold"], " takes the coordinates of ", StyleBox["d1", FontWeight->"Bold"], " and displaces them halfway along one side of the original triangle, to get \ a triangle of half the size of the original one (same size as ", StyleBox["d1", FontWeight->"Bold"], ") ", "in another corner of the original triangle. ", StyleBox["d3", FontWeight->"Bold"], " gives the half-size triangle in the third corner of the original \ triangle.", StyleBox["\n", FontWeight->"Bold"], "\nThe function ", StyleBox["triple", FontWeight->"Bold"], " will be the building block out of which we will contruct the fractal \ iteratively. For example, if we act with ", StyleBox["triple", FontWeight->"Bold"], " on ", StyleBox["starttriangle:", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"threetriangles", " ", "=", " ", RowBox[{"triple", "[", "starttriangle", "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.25`", ",", "0.4330127018922193`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.5`", ",", "0.`"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.5`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.75`", ",", "0.4330127018922193`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.`"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.25`", ",", "0.4330127018922193`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.5`", ",", "0.8660254037844386`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.75`", ",", "0.4330127018922193`"}], "}"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4204889775291433`*^9, 3.420489209512847*^9, 3.420489284283558*^9, 3.42048938970051*^9, 3.482973539679193*^9, 3.48297357142673*^9, 3.51483588602283*^9}] }, Open ]], Cell["and plot it", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"Polygon", "/@", "threetriangles"}], "]"}], "]"}]], "Input", CellChangeTimes->{3.4204893488223467`*^9}], Cell[BoxData[ GraphicsBox[{PolygonBox[{{0., 0.}, {0.25, 0.4330127018922193}, {0.5, 0.}}], PolygonBox[{{0.5, 0.}, {0.75, 0.4330127018922193}, {1., 0.}}], PolygonBox[{{0.25, 0.4330127018922193}, {0.5, 0.8660254037844386}, {0.75, 0.4330127018922193}}]}]], "Output", CellChangeTimes->{3.420488980301602*^9, 3.4204892096736507`*^9, 3.420489284303502*^9, 3.4204893898384*^9, 3.482973539713017*^9, 3.482973571462378*^9, 3.51483589066192*^9}] }, Open ]], Cell[TextData[{ "we get three triangles occupying the same overall region (with an empty \ triangle in the middle), as desired. This, then, is the basic operation: \ start with one triangle and produce three triangles occupying the same \ overall region, as shown in the figure immediately above.\n\nThe syntax ", StyleBox["Polygon /@ threetriangles ", FontWeight->"Bold"], "is the same as ", StyleBox["Map[Polygon, threetriangles]", FontWeight->"Bold"], " and means apply the function ", StyleBox["Polygon", FontWeight->"Bold"], " to each element of the list ", StyleBox["threetriangles", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.420488735167004*^9, 3.420488885751307*^9}}], Cell[TextData[{ "Now we are going to automate the contruction of several iterations of this \ procedure. This involves acting on a ", StyleBox["list", FontSlant->"Italic"], " of triangles. We therefore make a list containing just one triangle by \ putting an extra layer of curly brackets around ", StyleBox["starttriangle", FontWeight->"Bold"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"startlist", " ", "=", " ", RowBox[{"{", "starttriangle", "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.5`", ",", "0.8660254037844386`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.`"}], "}"}]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.4204890019651413`*^9, 3.4204892096927233`*^9, 3.420489284450145*^9, 3.420489389868474*^9, 3.482973539729368*^9, 3.482973571476368*^9, 3.514835923463697*^9}] }, Open ]], Cell[TextData[{ "We M", StyleBox["ap", FontWeight->"Bold"], " (i.e. apply) the function ", StyleBox["triple ", FontWeight->"Bold"], "to each element of startlist (there's actually only one element so far, so \ M", StyleBox["ap", FontWeight->"Bold"], " is not yet necesary, but we do it anyway because it ", StyleBox["will", FontSlant->"Italic"], " be necessary in subsequent iterations and we want to use the same \ procedure for each iteration). Finally, we remove one layer of curly brackets \ with ", StyleBox["Flatten[ ... , 1]", FontWeight->"Bold"], " to again get a list of triangles on which we can act again with our \ function ", StyleBox["triple", FontWeight->"Bold"], ". 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Note that we \ could alternatively remove the outer layer of curly brackets by taking the \ first element (there is only one element so this doesn't do anything else \ apart from removing the brackets):\ \>", "Text", CellChangeTimes->{{3.420489117133686*^9, 3.420489204230523*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"list2", " ", "=", " ", RowBox[{ RowBox[{"Map", "[", RowBox[{"triple", ",", " ", "startlist"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.420489015301175*^9, 3.4204890300470657`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.25`", ",", "0.4330127018922193`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.5`", ",", "0.`"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.5`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.75`", ",", "0.4330127018922193`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.`"}], "}"}]}], "}"}], 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CellChangeTimes->{3.420489210242778*^9, 3.420489284888379*^9, 3.420489390170198*^9, 3.4829735398303633`*^9, 3.482973571579197*^9, 3.514835946448535*^9}] }, Open ]], Cell["which produces 9 triangles as shown:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"Polygon", "/@", "list3"}], "]"}], "]"}]], "Input", CellChangeTimes->{3.420489356452094*^9}], Cell[BoxData[ GraphicsBox[{ PolygonBox[{{0., 0.}, {0.125, 0.21650635094610965`}, {0.25, 0.}}], PolygonBox[{{0.25, 0.}, {0.375, 0.21650635094610965`}, {0.5, 0.}}], PolygonBox[{{0.125, 0.21650635094610965`}, {0.25, 0.4330127018922193}, { 0.375, 0.21650635094610965`}}], PolygonBox[{{0.5, 0.}, {0.625, 0.21650635094610965`}, {0.75, 0.}}], PolygonBox[{{0.75, 0.}, {0.875, 0.21650635094610965`}, {1., 0.}}], PolygonBox[{{0.625, 0.21650635094610965`}, {0.75, 0.4330127018922193}, { 0.875, 0.21650635094610965`}}], PolygonBox[{{0.25, 0.4330127018922193}, {0.375, 0.649519052838329}, {0.5, 0.4330127018922193}}], PolygonBox[{{0.5, 0.4330127018922193}, {0.625, 0.649519052838329}, {0.75, 0.4330127018922193}}], PolygonBox[{{0.375, 0.649519052838329}, {0.5, 0.8660254037844387}, {0.625, 0.649519052838329}}]}]], "Output", CellChangeTimes->{3.420489210798112*^9, 3.4204892850376387`*^9, 3.4204893903115263`*^9, 3.482973539866502*^9, 3.482973571611038*^9, 3.5148359495998907`*^9}] }, Open ]], Cell[TextData[{ "Now we define functions with which we will automate the construction of the \ Sierpinski gasket down to a specified scale. The function ", StyleBox["iterate", FontWeight->"Bold"], ", below, does one level of transformation, and this is repeated n times by \ the function ", StyleBox["newlist", FontWeight->"Bold"], ", starting from the single triangle in ", StyleBox["startlist", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"iterate", "[", "list_", "]"}], " ", ":=", " ", RowBox[{"Flatten", "[", " ", RowBox[{ RowBox[{"Map", "[", RowBox[{"triple", ",", " ", "list"}], "]"}], ",", " ", "1"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"newlist", "[", "n_", "]"}], " ", ":=", " ", RowBox[{"Nest", " ", "[", " ", RowBox[{"iterate", ",", " ", "startlist", ",", " ", "n"}], "]"}]}], ";"}]], "Input"], Cell[TextData[{ "In the ", StyleBox["Mathematica", FontSlant->"Italic"], " function N", StyleBox["est[f, expr, n]", FontWeight->"Bold"], " the function ", StyleBox["f", FontWeight->"Bold"], " is applied ", StyleBox["n", FontWeight->"Bold"], " times to ", StyleBox["expr", FontWeight->"Bold"], ".\n\n The result of ", StyleBox["newlist", FontWeight->"Bold"], " is a list of triangles. This can be converted to a graphics object using \ the ", StyleBox["Graphics", FontWeight->"Bold"], " command, and then displayed with ", StyleBox["Show", FontWeight->"Bold"], ". As an example, we show the result of 6 levels of iteration which \ produces ", Cell[BoxData[ FormBox[ SuperscriptBox["3", "6"], TraditionalForm]]], "= 729 triangles. 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This \"", StyleBox["scale invariance", FontWeight->"Bold"], "\" is a characteristic feature of a fractal.\n\nA basic way to characterize \ a fractal is by the ", StyleBox["fractal dimension", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ SubscriptBox["d", "s"], TraditionalForm]]], ", also called the Hausdorf dimension.To define it for the Sierpinski \ gasket, let the length of the side of the smallest triangle be \[Epsilon] and \ the overall length of a side of the triangular figure be L. Then, the fractal \ dimension of the shaded region is defined in terms of its area A by the \ relation" }], "Text", CellChangeTimes->{{3.420810357355835*^9, 3.420810363855421*^9}, { 3.51483605421416*^9, 3.514836056690907*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["A", SubscriptBox["A", "\[Epsilon]"]], " ", "=", " ", SuperscriptBox[ RowBox[{"(", FractionBox["L", "\[Epsilon]"], ")"}], SubscriptBox["d", "s"]]}], ","}]], "DisplayFormula", CellChangeTimes->{{3.4208103680299*^9, 3.42081037205343*^9}, { 3.514836060292964*^9, 3.514836064291957*^9}}], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ SubscriptBox["A", "\[Epsilon]"], TraditionalForm]]], " is the area of a single shaded triangle at the smallest scale (i.e. with \ side \[Epsilon]). Clearly if the whole region were shaded, we would have ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["d", "s"], " ", "=", " ", "2"}], TraditionalForm]]], ", the spatial dimension. To see this, consider, for example, the second \ figure in this notebook. If the central triangular region were also shaded, \ (so the ", StyleBox["whole", FontSlant->"Italic"], " figure would be shaded) then the large triangle of side L would be made up \ of 4 smaller triangles, each of size \[Epsilon] = L/2, and so we would have \ ", Cell[BoxData[ FormBox[ RowBox[{"A", "/", SubscriptBox["A", "\[Epsilon]"]}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox["4", TraditionalForm]]], ", and L/\[Epsilon] = 2, so 4 = ", Cell[BoxData[ FormBox[ SuperscriptBox["2", SubscriptBox["d", "s"]], TraditionalForm]]], ", which gives ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["d", "s"], " ", "=", " ", "2"}], TraditionalForm]]], ". Hence, ", StyleBox["for regular solids, the fractal dimension is equal to the space \ dimension", FontWeight->"Bold"], ". This is clearly desirable; our definition of fractal dimension should \ reproduce what we expect in simple cases of integer dimension. \n\nHowever, \ for the gasket, only three of the four regions in the figure are shaded. In \ other words, doubling L/\[Epsilon] increases the area by a factor of 3 (not \ 4), so " }], "Text", CellChangeTimes->{{3.420810381260906*^9, 3.420810429855036*^9}, { 3.5148360687543507`*^9, 3.514836109900494*^9}, {3.514837072278277*^9, 3.5148370726031322`*^9}}], Cell[BoxData[ RowBox[{"3", " ", "=", " ", SuperscriptBox["2", SubscriptBox["d", "s"]]}]], "DisplayFormula"], Cell["which gives", "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["d", "s"], "=", " ", RowBox[{ FractionBox[ RowBox[{"ln", " ", "3"}], RowBox[{"ln", " ", "2"}]], " ", "=", " ", RowBox[{"1.58496", "..."}]}]}]], "DisplayFormula"], Cell[TextData[{ "We see that the Sierpinski gasket has a fractal dimension in between that \ of the one-dimensional line and a two-dimensional triangle. \n\nIterating the \ construction of the gasket, we recover the same value for ", Cell[BoxData[ FormBox[ SubscriptBox["d", "s"], TraditionalForm]]], " since, at each iteration, the ratio of the length of a side of the figure \ to the side of the smallest triangle increases by a factor of 2, while the \ ratio of the area of the figure to the area of the smallest triangle \ increases by a factor of 3 (rather than 4 which would be the result if all \ the area in the larger triangle were included.)\n\nIf the fractal dimension \ of the object is less than the space dimension, the areal density of the \ object (i.e. the size of the shaded area per unit area of the plane) tends to \ zero as the number of iterations tends to infinity. This is because the areal \ density is given by" }], "Text", CellChangeTimes->{{3.420810446582144*^9, 3.420810512816142*^9}, { 3.420810696223563*^9, 3.4208107009587708`*^9}, {3.420810739264288*^9, 3.420810740975956*^9}, 3.420810911762364*^9, {3.4525239642191143`*^9, 3.4525239946507053`*^9}, {3.514836146283901*^9, 3.514836148156685*^9}, { 3.514836194669938*^9, 3.51483627339214*^9}}], Cell[BoxData[ RowBox[{"\[Rho]", " ", "=", " ", RowBox[{ FractionBox["A", RowBox[{ FractionBox["1", "2"], SuperscriptBox["L", "d"]}]], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["A", "\[Epsilon]"], " ", SuperscriptBox[ RowBox[{"(", FractionBox["L", "\[Epsilon]"], ")"}], SubscriptBox["d", "s"]], " ", FractionBox["1", RowBox[{ FractionBox["1", "2"], SuperscriptBox["L", "d"]}]]}], " ", "=", " ", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", FractionBox["L", "\[Epsilon]"], ")"}], RowBox[{ SubscriptBox["d", "s"], " ", "-", " ", "d"}]], " ", FractionBox[ SubscriptBox["A", "\[Epsilon]"], RowBox[{ FractionBox["1", "2"], SuperscriptBox["\[Epsilon]", "d"]}]]}], " ", "=", " ", SuperscriptBox[ RowBox[{"(", FractionBox["L", "\[Epsilon]"], ")"}], RowBox[{ SubscriptBox["d", "s"], " ", "-", " ", "d"}]]}]}]}]}]], "DisplayFormula", CellChangeTimes->{{3.420810548470702*^9, 3.420810614338724*^9}, { 3.420810660067168*^9, 3.420810684289487*^9}, {3.452524060006598*^9, 3.4525241188451138`*^9}, {3.5148361554936943`*^9, 3.5148361666616983`*^9}, { 3.5148364183353167`*^9, 3.514836458018911*^9}}], Cell[TextData[{ "since ", Cell[BoxData[ SubscriptBox["A", "\[Epsilon]"]], CellChangeTimes->{{3.420810548470702*^9, 3.420810614338724*^9}, { 3.420810660067168*^9, 3.420810684289487*^9}, {3.452524060006598*^9, 3.4525241188451138`*^9}, {3.5148361554936943`*^9, 3.5148361666616983`*^9}, {3.5148364183353167`*^9, 3.514836458018911*^9}}], " = ", Cell[BoxData[ RowBox[{ FractionBox["1", "2"], SuperscriptBox["\[Epsilon]", "d"]}]], CellChangeTimes->{{3.420810548470702*^9, 3.420810614338724*^9}, { 3.420810660067168*^9, 3.420810684289487*^9}, {3.452524060006598*^9, 3.4525241188451138`*^9}, {3.5148361554936943`*^9, 3.5148361666616983`*^9}, {3.5148364183353167`*^9, 3.514836458018911*^9}}], " . Hence the fraction of the area which is shaded equals ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"L", "/", "\[Epsilon]"}], ")"}], RowBox[{ SubscriptBox["d", "s"], "-", "d"}]], TraditionalForm]]], " . Since L/\[Epsilon] = ", Cell[BoxData[ FormBox[ SuperscriptBox["2", "k"], TraditionalForm]], FormatType->"TraditionalForm"], " where k is the number of iterations, and ", Cell[BoxData[ FormBox[ SubscriptBox["d", "s"], TraditionalForm]], FormatType->"TraditionalForm"], " is given above,", StyleBox[" the fraction of area which is shaded is just ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"3", "/", "4"}], ")"}], "k"], TraditionalForm]], FormatType->"TraditionalForm", FontWeight->"Bold"], ". ", StyleBox["This tends to zero as k\[Rule]\[Infinity].", FontWeight->"Bold"], " Qualitatively this can be seen in the figures since, the fraction of white \ increases as more iterations are performed." }], "Text", CellChangeTimes->{{3.4208107038388863`*^9, 3.4208108832066517`*^9}, { 3.420810934070261*^9, 3.420811014871364*^9}, {3.51483646601679*^9, 3.5148366564978943`*^9}, {3.51483670572816*^9, 3.51483677282546*^9}, { 3.514836871810247*^9, 3.514836986912912*^9}}] }, Open ]] }, WindowSize->{952, 986}, WindowMargins->{{154, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PrintingOptions->{"Magnification"->1, "PaperOrientation"->"Portrait", "PaperSize"->{612, 792}, "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "peter", "courses", "115", "mathematica"}, "sierpinski.nb.ps", CharacterEncoding -> "iso8859-1"]}, ShowSelection->True, Magnification->1.5, 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[568, 21, 94, 1, 40, "Input"], Cell[CellGroupData[{ Cell[687, 26, 64, 0, 69, "Subtitle"], Cell[754, 28, 115, 2, 40, "Input"], Cell[872, 32, 548, 12, 107, "Text"], Cell[1423, 46, 199, 4, 62, "Text"], Cell[CellGroupData[{ Cell[1647, 54, 413, 12, 40, "Input"], Cell[2063, 68, 464, 11, 40, "Output"] }, Open ]], Cell[2542, 82, 43, 0, 39, "Text"], Cell[CellGroupData[{ Cell[2610, 86, 181, 4, 40, "Input"], Cell[2794, 92, 337, 6, 492, "Output"] }, Open ]], Cell[3146, 101, 648, 22, 152, "Text"], Cell[3797, 125, 1941, 55, 157, "Input"], Cell[5741, 182, 1208, 37, 219, "Text"], Cell[CellGroupData[{ Cell[6974, 223, 117, 2, 40, "Input"], Cell[7094, 227, 1046, 30, 87, "Output"] }, Open ]], Cell[8155, 260, 27, 0, 39, "Text"], Cell[CellGroupData[{ Cell[8207, 264, 180, 4, 40, "Input"], Cell[8390, 270, 455, 7, 492, "Output"] }, Open ]], Cell[8860, 280, 710, 18, 152, "Text"], Cell[9573, 300, 370, 10, 84, "Text"], Cell[CellGroupData[{ Cell[9968, 314, 102, 2, 40, "Input"], Cell[10073, 318, 473, 12, 40, "Output"] }, Open ]], Cell[10561, 333, 786, 25, 107, "Text"], Cell[CellGroupData[{ Cell[11372, 362, 269, 7, 40, "Input"], Cell[11644, 371, 1080, 31, 87, "Output"] }, Open ]], Cell[12739, 405, 357, 6, 84, "Text"], Cell[CellGroupData[{ Cell[13121, 415, 257, 6, 40, "Input"], Cell[13381, 423, 1076, 30, 87, "Output"] }, Open ]], Cell[14472, 456, 131, 2, 39, "Text"], Cell[CellGroupData[{ Cell[14628, 462, 207, 6, 40, "Input"], Cell[14838, 470, 2741, 78, 225, "Output"] }, Open ]], Cell[17594, 551, 52, 0, 39, "Text"], Cell[CellGroupData[{ Cell[17671, 555, 169, 4, 40, "Input"], Cell[17843, 561, 1034, 18, 492, "Output"] }, Open ]], Cell[18892, 582, 446, 13, 84, "Text"], Cell[19341, 597, 241, 7, 42, "Input"], Cell[19585, 606, 218, 6, 42, "Input"], Cell[19806, 614, 850, 33, 133, "Text"], Cell[CellGroupData[{ Cell[20681, 651, 312, 8, 40, "Input"], Cell[20996, 661, 87219, 1431, 595, "Output"] }, Open ]], Cell[108230, 2095, 46, 0, 39, "Text"], Cell[CellGroupData[{ Cell[108301, 2099, 311, 8, 40, "Input"], Cell[108615, 2109, 270078, 4348, 599, "Output"] }, Open ]], Cell[378708, 6460, 922, 21, 174, "Text"], Cell[379633, 6483, 360, 10, 57, "DisplayFormula"], Cell[379996, 6495, 1791, 48, 246, "Text"], Cell[381790, 6545, 115, 3, 34, "DisplayFormula"], Cell[381908, 6550, 27, 0, 39, "Text"], Cell[381938, 6552, 219, 7, 52, "DisplayFormula"], Cell[382160, 6561, 1296, 21, 264, "Text"], Cell[383459, 6584, 1322, 39, 65, "DisplayFormula"], Cell[384784, 6625, 2032, 54, 122, "Text"] }, Open ]] } ] *) (* End of internal cache information *)