(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 24507, 818]*) (*NotebookOutlinePosition[ 48621, 1692]*) (* CellTagsIndexPosition[ 48577, 1688]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Mandelbrot Set", "Title", Evaluatable->False, AspectRatioFixed->True], Cell["R.G. Palmer, 9/7/99", "Subsubtitle", TextAlignment->-0.5], Cell[CellGroupData[{ Cell["Background and Copyright", "Subsection"], Cell[TextData[{ "This ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook was written by Richard G. Palmer (Physics Department, Duke \ University) for use in a course he taught. As of 1999, it has been made \ available for general non-profit use under the following copyright \ provision." }], "Text"], Cell[TextData[{ StyleBox["This Mathematica Notebook is Copyright Richard G. Palmer, 1997", FontWeight->"Bold"], ". It may be freely used by individuals, and by classes at academic \ institutions, provided:\n1. Credit is given to Richard Palmer as the original \ author; and\n2. It is not bought or sold or exchanged for profit, or \ incorporated into material that is bought or sold or exchanged for profit.\n\ Any other use requires the written permission of Richard Palmer, Dept. of \ Physics, Box 90305, Duke University, Durham, NC 27708, USA.", "See ", StyleBox["http://www.phy.duke.edu/~palmer", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for the email address." }], "Text"], Cell[TextData[{ "See ", StyleBox["http://www.phy.duke.edu/~palmer/notebooks/", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for other ", StyleBox["Mathematica", FontSlant->"Italic"], " noteboks by Richard Palmer." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Complex numbers", "Section"], Cell[TextData[{ "This is a brief introduction to complex numbers for those who haven't met \ them before. Others should only need to look at the ", StyleBox["Mathematica", FontSlant->"Italic"], " definitions -- at least evaluate the input cells and make sure you \ understand the results." }], "Text"], Cell[TextData[{ "Complex numbers are extension of the usual real number system. Real \ numbers range from -\[Infinity] to \[Infinity], with values like -234.567, ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], ", \[Pi], 42, ", Cell[BoxData[ \(TraditionalForm\`\(10\^6, \)\)]], " etc. You can think of them as lying along a line, the x-axis of an x-y \ plane. By contrast, a complex number represents a point in the plane, having \ both x and y coordinates. You can write this like" }], "Text"], Cell["(x, y)", "Text", TextAlignment->Center], Cell["or like", "Text"], Cell["x + i y", "Text", TextAlignment->Center], Cell[TextData[{ "but we'll always use the second (x + i y) notation. \"i\" is a special \ symbol used to distinguish the x (or \"real\") part from the y (or \ \"imaginary\") part. ", StyleBox["Mathematica", FontSlant->"Italic"], " use \"I\" instead of \"i\", so here is a list of 3 complex numbers in ", StyleBox["Mathematica", FontSlant->"Italic"], "'s notation (evaluate this):" }], "Text"], Cell[BoxData[ \(cnos\ = \ {2 + I\ 3, \ \(-3.5\)\ - \ 2.5 I, \ \[Pi]\ + \ \(\@2\) I}\)], "Input"], Cell[TextData[{ "It doesn't matter whether you put the number first or the ", StyleBox["I", FontWeight->"Bold"], " first -- note how ", StyleBox["Mathematica", FontSlant->"Italic"], " changed things around a bit according to its own ordering rules. But \ note that ", StyleBox["I3", FontWeight->"Bold"], " (no space) and ", StyleBox["I 3", FontWeight->"Bold"], " (with a space) are two different things; ", StyleBox["I3", FontWeight->"Bold"], " is the name of variable (a cousin of ", StyleBox["U2", FontWeight->"Bold"], "), nothing to do with complex numbers." }], "Text"], Cell["\<\ The following makes a plot of the above three complex numbers \ (don't worry about how it works). Evaluate it and identify the three points.\ \ \>", "Text"], Cell[BoxData[ \(\(ListPlot[\({Re[#], \ Im[#]}\ &\)/@\ cnos, \n\ \ \ \ AspectRatio\ -> \ Automatic, \ PlotStyle -> {PointSize[0.03]}\ ]\ \ \)\)], "Input"], Cell[TextData[{ "This is called a plot of the ", StyleBox["complex plane", FontSlant->"Italic"], ", or an ", StyleBox["Argand diagram", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "If this were all, then complex numbers would just be a way of representing \ a pair of ordinary real numbers, or equivalenly a point in the (x,y) plane. \ But there's more; ", StyleBox[ "all the usual operations you can do on real numbers are also defined for \ complex numbers", FontSlant->"Italic"], ". This includes addition, subtraction, multiplication, division, \ squaring, square roots, logarithms, and so on." }], "Text"], Cell["\<\ The definitions of addition and subtraction are equivalent to just \ treating the \"i\" as a separate algebraic unknown. Thus you just add or \ subtract the x (\"real\") and y (\"imaginary\") parts separately:\ \>", "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{ FormBox[\(\(\(x\_\(1\ \)\ + \ i\ y\_1)\)\ + \)\ \), "TraditionalForm"], \((x\_2\ + \ i\ y\_2)\)}], " ", "=", " ", \(\((x\_1 + x\_2)\)\ + \ i\ \((y\_1 + y\_2)\)\)}]}], TraditionalForm]]]], "Text", TextAlignment->Center], Cell[TextData[{ "For example, letting ", StyleBox["Mathematica", FontSlant->"Italic"], " do the work:" }], "Text"], Cell[BoxData[ \(2\ + \ 3 I\ + \ 4\ + \ 5 I\)], "Input"], Cell[BoxData[ \(2 + 3 I\ - \ \((4 + 5 I)\)\)], "Input"], Cell[TextData[{ "Multiplication works the same way, with one additional rule: whenever you \ get ", Cell[BoxData[ \(TraditionalForm\`\(i\^2\) replace\ it\ by\ - 1.\)]], " In this sense you can think of i as the square root of -1, though of \ course that doesn't really exist as a real number. In fact that's the pont; \ by introducing these complex numbers, ", StyleBox["every", FontSlant->"Italic"], " number now has a square root, ", StyleBox["every", FontSlant->"Italic"], " quadratic equation has two roots, etc -- the complex number system is \ (almost) closed, meaning that any reasonable operation on numbers within the \ system gives another number within the system. (Exceptions are things like \ 1/0, which still don't stay within the system.)" }], "Text"], Cell["In general (using * to mean multiplication):", "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{ FormBox[\(\(x\_\(1\ \)\ + \ i\ y\_1)\)*\), "TraditionalForm"], \((x\_2\ + \ i\ y\_2)\)}], " ", "=", " ", \(\((\(x\_1\) x\_2 - \(y\_1\) y\_2)\)\ + \ i\ \((\(x\_1\) y\_2 + \(x\_2\) y\_1)\)\)}]}], TraditionalForm]]]], "Text", TextAlignment->Center], Cell[TextData[{ "Check that using the ", Cell[BoxData[ \(TraditionalForm\`i\^2\ = \ 1\ \(rule.\)\)]] }], "Text"], Cell["For example (check this too)", "Text"], Cell[BoxData[ \(\((2 + 3 I)\) \((4 + 5 I)\)\)], "Input"], Cell[TextData[{ "Division is a bit more complicated, but is defined so that if ", Cell[BoxData[ \(TraditionalForm\`z\_3\ = \ \(z\_1\) z\_2\)]], ", then ", Cell[BoxData[ \(TraditionalForm \`z\_1\ = \ \(z\_3/z\_2 and\ z\_2\ = \ z\_3/z\_1\)\)]], ", as you'd expect. Here I'm using a single symbol like ", Cell[BoxData[ \(TraditionalForm\`z\_1\)]], " to represent a whole complex number. The letters z and w are often used \ for that." }], "Text"], Cell["For example:", "Text"], Cell[BoxData[ \(\((2 + 3 I)\)/\((4 + 5 I)\)\)], "Input"], Cell["Check (% means the previous result):", "Text"], Cell[BoxData[ \(%\ *\ \((4 + 5 I)\)\)], "Input"], Cell[TextData[{ "We won't worry here about other operations like square-roots and \ logarithms. But there's one more thing we need: the ", StyleBox["absolute value", FontSlant->"Italic"], " or ", StyleBox["modulus", FontSlant->"Italic"], " of a complex number (the two words mean the same thing). This is written \ as" }], "Text"], Cell["|z|", "Text", TextAlignment->Center], Cell["or as", "Text", TextAlignment->Left], Cell["abs(z)", "Text", TextAlignment->Center], Cell[TextData[{ "and means ", Cell[BoxData[ \(TraditionalForm\`\@\(x\^2 + y\^2\)\)]], ", where x and y are the usual real and imaginary parts. ", StyleBox["Mathematica", FontSlant->"Italic"], " uses ", StyleBox["Abs[]", FontWeight->"Bold"], " for the absolute value. E.g.:" }], "Text"], Cell[BoxData[ \(Abs[2\ + \ 3 I]\)], "Input"], Cell["Check that you agree!", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Definition of the Mandelbrot set", "Section"], Cell["\<\ The famous Mandelbrot set is just a region in the x-y plane, but \ with a boundary that has a very fine and interesting structure. Each point \ (x,y) is either in the Mandelbrot set, or not. To find out whether a given \ point (x,y) is in the set or not, you have to do the following:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["1. Define the complex number c by", "Text"], Cell["c = x + i y", "Text", TextAlignment->Center], Cell["2.Starting with z = 0, iterate the rule", "Text"], Cell[TextData[{ StyleBox["z ", FontSlant->"Italic"], " := ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " + c" }], "Text", TextAlignment->Center], Cell["\<\ endlessly. Here the operator := means \"set the new value of the \ thing on the left (z) to the old value of the thing on the right.\ \>", "Text"], Cell[TextData[{ "3. See whether the sequence of z values is going off to infinity (in \ absolute value, abs(z)), or not. If it does go off to infinity, then the \ original point (x,y) is ", StyleBox["not", FontSlant->"Italic"], " in the Mandelbrot set; if it never goes to infinity, then the original \ point ", StyleBox["is", FontSlant->"Italic"], " in the Mandelbrot set." }], "Text"], Cell["\<\ Here's an example, using Mathematica to do 5 iterations for the \ point (1,1), printing the absolute value of z after each iteration:\ \>", "Text"], Cell[BoxData[ \(c\ = \ 1 + I; \nz\ = \ 0; \n Do[\n\tz\ = \ z^2\ + c; \n\tPrint[Abs[z]], \n\t{5}\n]\)], "Input"], Cell[TextData[{ "Try to understand the program! ", StyleBox["Do[", FontWeight->"Bold"], StyleBox["expression", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[", {5}]", FontWeight->"Bold"], " just evaluates ", StyleBox["expression", FontWeight->"Bold", FontSlant->"Italic"], " five times. A sequence of expressions separated by semi-colons is also a \ valid expression; the sub-expressions are evaluated in order." }], "Text"], Cell["\<\ In this case, it's pretty clear that the point (1,1) is not in the \ Mandelbrot set, since abs(z) seems to be getting pretty big and heading off \ for infinity. In fact there's a theorem that says\ \>", "Text"], Cell["\<\ If abs(z) ever gets bigger than 2, then z will eventually go off to \ infinity\ \>", "Text", CellFrame->True, TextAlignment->Center, Background->GrayLevel[0.849989]], Cell[TextData[{ "so we could have stopped at ", Cell[BoxData[ \(TraditionalForm\`\@10\)]], " (which is larger than 2) and known for sure then that the original point \ (1,1) was not in the Mandelbrot set." }], "Text"], Cell[TextData[{ "Note the asymmetry of the situation: we can often know for sure that a \ point is ", StyleBox["not", FontSlant->"Italic"], " in the Mandelbrot set, because abs(z) gets bigger than 2, but there is \ usually no similar way to be quite sure that a point ", StyleBox["is", FontSlant->"Italic"], " in the set. For example, here's the same program starting from the point \ (0.24, 0.56)" }], "Text"], Cell[BoxData[ \(c\ = \ 0.24 + \ 0.56 I; \nz\ = \ 0; \n Do[\n\tz\ = \ z^2\ + c; \n\tPrint[Abs[z]], \n\t{5}\n]\)], "Input"], Cell["\<\ Is it eventually going to get bigger than 2, or not? How can you \ tell? Try 50 iterations instead of just 5 (change the 5 to 50 and \ re-evaluate). But then try changing the 0.24 to 0.241 and re-evaluate.\ \>", "Text"], Cell[TextData[{ "Actually the last example (0.241, 0.56) is ", StyleBox["not", FontSlant->"Italic"], " in the Mandelbrot set, but abs(z) only gets bigger than 2 after 86 \ iterations. And you can find points for which you have to do thousands or \ millions of iterations to see abs(z) get bigger than 2." }], "Text"], Cell[TextData[{ "In general we can only do a finite (not infinite) number of iterations, so \ we can't quite be sure about certain (x,y) points that seem to be in the \ Mandelbrot set, but might not be. We're limited by computing power. \ Nevertheless the Mandelbrot set is well defined ", StyleBox["mathematically", FontSlant->"Italic"], ", even if it's not finitely computable." }], "Text"], Cell[TextData[{ "Are there ", StyleBox["any", FontSlant->"Italic"], " points in the Mandelbrot set? Yes, certainly. The point (0,0) is a \ trivial example--z always stays at 0. And it's fairly easy to show \ mathematically that there are whole regions (e.g., close to the origin) that \ are certainly in the set." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Making a function", "Section"], Cell[TextData[{ "Here's another way to set up the question. We'll iterate for ", StyleBox["up to", FontSlant->"Italic"], " 100 cycles, and stop as soon as abs(z) gets bigger than 2, printing how \ many iterations we did:" }], "Text"], Cell[BoxData[ \(c\ = \ 0.241\ + \ 0.56\ I; \nz\ = \ 0; \nt\ = \ 0; \n While[\((Abs[z] < 2.0)\)\ && \ \((t < 100)\), \n\t\t\tt\ = \ t + 1; \n \t\t\tz\ = \ z*z\ + c; \n]; \nt\)], "Input"], Cell[TextData[{ "Try to understand what's going on here. Rather than using ", StyleBox["Do[", FontWeight->"Bold"], StyleBox["expression", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[", {n}]", FontWeight->"Bold"], ", which just evaluates ", StyleBox["expression", FontWeight->"Bold", FontSlant->"Italic"], " n times, I used ", StyleBox["While[", FontWeight->"Bold"], StyleBox["condition", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[", ", FontWeight->"Bold"], StyleBox["expression", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", FontWeight->"Bold"], ", which keeps on evaluating ", StyleBox["expression", FontWeight->"Bold", FontSlant->"Italic"], " as long as ", StyleBox["condition", FontWeight->"Bold", FontSlant->"Italic"], " is true." }], "Text"], Cell[TextData[{ "The ", StyleBox["condition", FontWeight->"Bold", FontSlant->"Italic"], " part involves two sub-conditions joined with ", StyleBox["&&", FontWeight->"Bold"], ", which is the logical AND operator. Both sub-conditions have to be true \ for the whole thing to be true. One sub-condition checks that abs(z) is less \ than 2, and the other checks that we've done less than 100 iterations so \ far." }], "Text"], Cell[TextData[{ "The new variable ", StyleBox["t", FontWeight->"Bold"], " counts how many iterations we've done. It starts at 0, and every \ iteration is incremented by 1; the expression ", StyleBox["t = t+1", FontWeight->"Bold"], " means \"add one to t\"." }], "Text"], Cell[TextData[{ "Finally, after the ", StyleBox["While[]", FontWeight->"Bold"], " loop exits, we evaluate the variable t, so that its value is printed out \ as the final answer." }], "Text"], Cell[TextData[{ "Now try changing the starting point (the value of ", StyleBox["c", FontWeight->"Bold"], ") and re-evaluating. For points that are well outside the Mandelbrot set \ you'll get a small value (the minimum is 1); for points actually in the set \ you'll get 100. Near the boundary you'll get intermediate values." }], "Text"], Cell["\<\ Now let's tidy up this program a little more, making it into a \ function of the starting point (x,y):\ \>", "Text"], Cell[BoxData[ \(MandelFun[x_, \ y_]\ := \((\n\t\tc\ = \ x\ + \ I\ y; \n\t\tz\ = \ 0; \n\t\tt\ = \ 0; \n\t\t While[\((Abs[z] < 2.0)\)\ && \ \((t < 100)\), \n\t\t\t\t t\ = \ t + 1; \n\t\t\t\tz\ = \ z*z\ + c; \n\t\t]; \n\t\tt\n) \)\)], "Input"], Cell[TextData[{ "Evaluating this won't produce any visible result; it just defines a new \ function (with two arguments) called ", StyleBox["MandleFun[]", FontWeight->"Bold"], ". The details are exactly the same as before except for the first and \ last lines, and the fact that we now have ", StyleBox["c = x + I y", FontWeight->"Bold"], " for the starting point instead of an explicit value. Don't worry for now \ about the detailed syntax of the first line." }], "Text"], Cell["We can evaluate our new function just like a built-in one:", "Text"], Cell[BoxData[ \(MandelFun[0.241, \ 0.56]\)], "Input"], Cell[BoxData[ \(MandelFun[0.1, \ 0.1]\)], "Input"], Cell[BoxData[ \(MandelFun[1, 2]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Pictures", "Section"], Cell[TextData[{ "OK, at last we're ready to make a picture. We just use the ", StyleBox["DensityPlot[]", FontWeight->"Bold"], " function to evaluate our ", StyleBox["MandelFun[]", FontWeight->"Bold"], " function at a grid of points in the (x,y) plane. Let's remind ourselves \ of the definition of ", StyleBox["DensityPlot[]", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(\(?DensityPlot\)\)], "Input"], Cell["\<\ Let's try it! I know from previous experience suitable values for \ xmin, xmax, ymin, and ymax to see the whole Mandelbrot set:\ \>", "Text"], Cell[BoxData[ \(DensityPlot[MandelFun[x, y], \ {x, \(-1.75\), \ 0.75}, \ {y, \(-1.25\), \ 1.25}]\)], "Input"], Cell[TextData[{ "Hmm, not very impressive! We need to customize ", StyleBox["DensityPlot[]", FontWeight->"Bold"], " with some options, making it use more points (", StyleBox["PlotPoints->100", FontWeight->"Bold"], "), removing the grid lines (", StyleBox["Mesh->False", FontWeight->"Bold"], "), and making it plot in color (", StyleBox["ColorFunction->Hue", FontWeight->"Bold"], "). This should be much better, though it'll take a while (be patient)." }], "Text"], Cell[BoxData[ \(DensityPlot[MandelFun[x, y], \ {x, \(-1.75\), \ 0.75}, \ {y, \(-1.25\), \ 1.25}, \ PlotPoints -> 100, \ Mesh -> False, \ ColorFunction -> Hue]\)], "Input"], Cell[TextData[{ "That should look more like it! The actual colors are somewhat arbitrary \ -- really there's a number from 1 to 100 at each point which ", StyleBox["Mathematica", FontSlant->"Italic"], " translates in some mysterious way to a color." }], "Text"], Cell[TextData[{ "Try expanding the plot -- click on it to select the graphic, and then drag \ on one of the corner handles. You'll see that the 100 x 100 grid that we \ chose (with ", StyleBox["PlotPoints->100", FontWeight->"Bold"], ") still isn't really fine enough. But there's a tradeoff between detail \ and how long it takes to evaluate." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Making it faster", "Section"], Cell["\<\ The following version is about 10-20 as fast as the above. Just \ evaluate the following (it won't produce any output) to define the function. \ Don't even try to understand it!\ \>", "Text"], Cell[BoxData[ \(\(MandelFun2\ = \ Compile[{x, \ y}, Module[{z, t = 0, c\ = \ x + I\ y}, \n\t\tz\ = \ c; \n\t\t While[\((Abs[z] < 2.0)\)\ && \ \((t < 100)\), \n\t\t\t\t t\ = \ t + 1; \n\t\t\t\tz\ = \ z*z\ + c; \n\t\t]; \n\t\tt\n]]; \)\)], "Input"], Cell[TextData[{ "Here's the same plotting command as before, using ", StyleBox["MandelFun2[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " instead of ", StyleBox["MandelFun[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ \(DensityPlot[MandelFun2[x, y], \ {x, \(-1.75\), \ 0.75}, \ {y, \(-1.25\), \ 1.25}, \ PlotPoints -> 100, \ Mesh -> False, \ ColorFunction -> Hue]\)], "Input"], Cell[TextData[{ "If you wish, you can also speed things up during experimentation by:\n\n1. \ Remove the ", StyleBox["ColorFunction->Hue", FontWeight->"Bold"], " option (gives faster rendering, but no color).\n\n2. Use fewer ", StyleBox["PlotPoints", FontWeight->"Bold"], ".\n\n3. Reduce the maximum number of iterations from 100 to e.g., 20. But \ you'll have to increase that again if you go to high magnification." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Exploration", "Section"], Cell["\<\ Now it's up to you! If you've understood everything so far you \ should be able to do the following exercise easily.\ \>", "Text"], Cell[CellGroupData[{ Cell["Exercise", "Subsubsection"], Cell["\<\ Make a picture roughly corresponding to Cohen and Stewart's Figure \ 4 panel d (page 21). Note that different spirals in the same valley look a \ little different, so it may not look exactly as in the book.\ \>", "Text"], Cell[TextData[{ "I suggest you first find the right region and ", StyleBox["then", FontSlant->"Italic"], " change the setup to get a higher quality picture. Remember how to get \ the coordinates of points in a graphic by selecting it and then moving the \ mouse around with the Command key held down -- the coordinates are then shown \ at bottom left." }], "Text"], Cell["\<\ You should use the same range for x and y to get an undistorted \ picture. Otherwise the horizontal and vertical magnifications would be \ different.\ \>", "Text"], Cell[TextData[{ "Use black and white (leaving out ", StyleBox["ColorFunction->Hue", FontWeight->"Bold"], ") for your final evaluation, unless you have a color printer. Then you'll \ probably end up with a \"negative\" of Cohen and Stewart's picture, because ", StyleBox["Mathematica", FontSlant->"Italic"], " shows larger values as lighter. You can reverse this by plotting ", StyleBox["-MandelFun", FontWeight->"Bold"], " instead of ", StyleBox["MandelFun", FontWeight->"Bold"], ", though a negative picture is OK too." }], "Text"], Cell[TextData[{ "Turn in both your picture and the command that produced it. Also your \ version of ", StyleBox["MandelFun[]", FontWeight->"Bold"], " if you changed it." }], "Text"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"4.2 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{520, 600}, WindowMargins->{{172, Automatic}, {142, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "Magnification"->1}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. Make modifications to any definition using \ commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, ScriptMinSize->12, FontSize->16], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellBracketOptions->{"Margins"->{1, 1}, "Widths"->{0, 5}}, ScriptMinSize->8, FontSize->11], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ScriptMinSize->5, FontSize->10, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], 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