(************** Content-type: application/mathematica **************
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Notebook[{

Cell[CellGroupData[{
Cell["Graphics Primitives: Creating Diagrams", "Title"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
  "Graphics primitives can be used to create 2D and 3D diagrams, including \
animated diagrams.  They are described in The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " Book, Section 2.9, Principles of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " \[Rule] The Structure of Graphics and Sound.  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s various plot commands create graphics primitives before displaying the \
graph.  For example, here is a graph of the standard parabola."
}], "Text"],

Cell[BoxData[
    \(\(squarePlot = 
        Plot[x\^2, {x, \(-2\), 2}, PlotDivision \[Rule] 2, 
          PlotPoints \[Rule] 5];\)\)], "Input"],

Cell[TextData[{
  "The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " kernel stores squarePlot in terms of graphics primitives, in this case, a \
",
  StyleBox["Line",
    FontFamily->"Courier"],
  "."
}], "Text"],

Cell[BoxData[
    \(InputForm[squarePlot]\)], "Input"],

Cell[TextData[{
  "The basic structure of 2D graphics are ",
  StyleBox["Graphics[",
    FontFamily->"Courier"],
  StyleBox["graphicsPrimatives",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", ",
    FontFamily->"Courier"],
  StyleBox["options",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["]",
    FontFamily->"Courier"],
  " and the basic structure of 3D graphics are ",
  StyleBox["Graphics3D[",
    FontFamily->"Courier"],
  StyleBox["graphicsPrimitives",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", ",
    FontFamily->"Courier"],
  StyleBox["options",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["]",
    FontFamily->"Courier"],
  "."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["2D graphics primitives", "Section"],

Cell[TextData[{
  "The 2D graphics primitives are ",
  StyleBox["Line",
    FontFamily->"Courier"],
  " (connected line segments, called a polyline), ",
  StyleBox["Point",
    FontFamily->"Courier"],
  ", ",
  StyleBox["Rectangle",
    FontFamily->"Courier"],
  " (filled rectangle), ",
  StyleBox["Polygon",
    FontFamily->"Courier"],
  " (filled polygons), ",
  StyleBox["Circle",
    FontFamily->"Courier"],
  ", ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  " (a filled circle), ",
  StyleBox["Raster",
    FontFamily->"Courier"],
  " (a rectangular arrays of gray level pixels), and ",
  StyleBox["Text",
    FontFamily->"Courier"],
  ".  Let's look at some examples."
}], "Text"],

Cell[CellGroupData[{

Cell["Line", "Subsection"],

Cell[TextData[{
  "The argument to ",
  StyleBox["Line",
    FontFamily->"Courier"],
  " is a list of coordinates for points, ",
  StyleBox["Line[{{",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "1"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "1"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, {",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "2"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "2"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, {",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "3"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "3"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, \[Ellipsis], {",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "n"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "n"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}}]",
    FontFamily->"Courier"],
  ".  ",
  StyleBox["Line",
    FontFamily->"Courier"],
  " will connect these points with line segments.  Here is a triangle.  \
Notice that the graphics primitive had to be enclosed in a ",
  StyleBox["Graphics",
    FontFamily->"Courier"],
  " command."
}], "Text"],

Cell[BoxData[
    \(triangle = Graphics[Line[{{0, 0}, {2, 0}, {1, 1}, {0, 0}}]]\)], "Input"],

Cell[TextData[{
  "You have to use ",
  StyleBox["Show",
    FontFamily->"Courier"],
  " to display the image."
}], "Text"],

Cell[BoxData[
    \(\(Show[triangle];\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Point", "Subsection"],

Cell[TextData[{
  "The argument to ",
  StyleBox["Point",
    FontFamily->"Courier"],
  " is an ordered pair of coordinates for a point.  The following example \
displays many points, so I need to create a list of objects of the form ",
  StyleBox["Point[{",
    FontFamily->"Courier"],
  StyleBox["xcoord",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", ",
    FontFamily->"Courier"],
  StyleBox["ycoord",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["}]",
    FontFamily->"Courier"],
  "."
}], "Text"],

Cell[BoxData[
    \(sinPoints = 
      Table[Point[N[{x, Sin[x]}]], {x, 0, 2  \[Pi], \[Pi]\/32}]\)], "Input"],

Cell[BoxData[
    \(\(Show[Graphics[sinPoints]];\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Rectangle", "Subsection"],

Cell[TextData[{
  "The syntax for a filled rectangle is",
  StyleBox[" Rectangle[{",
    FontFamily->"Courier"],
  StyleBox["xmin",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", ",
    FontFamily->"Courier"],
  StyleBox["ymin",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["}, {",
    FontFamily->"Courier"],
  StyleBox["xmax",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", ",
    FontFamily->"Courier"],
  StyleBox["ymax",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["}]",
    FontFamily->"Courier"],
  ".  The default fill color is black.  If you don't want your rectangle to \
be filled with color, then use ",
  StyleBox["Line",
    FontFamily->"Courier"],
  " to create it.  Here is one with perfect proportions, well at least the \
classical Greeks thought that it was perfect."
}], "Text"],

Cell[BoxData[
    \(\(Show[
        Graphics[Rectangle[{\(-GoldenRatio\), \(-1\)}, {GoldenRatio, 1}]], 
        AspectRatio \[Rule] Automatic];\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Polygon", "Subsection"],

Cell[TextData[{
  "The syntax for a filled polygon is ",
  StyleBox["Polygon[{{",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "1"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "1"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, {",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "2"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "2"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, {",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "3"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "3"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, \[Ellipsis], {",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["xcoord",
              FontSlant->"Italic"], "n"], ",", " ", 
          SubscriptBox[
            StyleBox["ycoord",
              FontSlant->"Italic"], "n"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}}]",
    FontFamily->"Courier"],
  ". "
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Circle, Disk", "Subsection"],

Cell[TextData[{
  "The syntax for a circle is ",
  StyleBox["Circle[",
    FontFamily->"Courier"],
  StyleBox["{",
    FontFamily->"Courier"],
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox["xcenter",
            FontSlant->"Italic"], ",", " ", 
          StyleBox["ycenter",
            FontSlant->"Italic"]}], TraditionalForm]],
    FontFamily->"Courier"],
  StyleBox["}, ",
    FontFamily->"Courier"],
  StyleBox["radius",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["]",
    FontFamily->"Courier"],
  ".  You can create ellipses or arcs of circles or ellipses using additional \
arguments for ",
  StyleBox["Circle",
    FontFamily->"Courier"],
  ".  See the Help Browser for details.  ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  " has the same syntax, but it creates a filled circle or ellipse.  If you \
want your circles to look like circles, you have to set ",
  StyleBox["AspectRatio\[Rule]Automatic",
    FontFamily->"Courier"],
  " in the Show command."
}], "Text"],

Cell[BoxData[{
    \(\(radius = 4;\)\), "\[IndentingNewLine]", 
    \(\(Show[Graphics[{Disk[{0, 0}, radius], Circle[{0, 0}, 1.1\ radius]}], 
        AspectRatio \[Rule] Automatic];\)\)}], "Input"],

Cell[TextData[{
  "In our example above, the argument to ",
  StyleBox["Graphics",
    FontFamily->"Courier"],
  " was a list of two graphics primitives, ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  " and ",
  StyleBox["Circle",
    FontFamily->"Courier"],
  "."
}], "Text"],

Cell["\<\
Let's draw the tangent line to a curve and place a disk at the \
point of tangency.\
\>", "Text"],

Cell[BoxData[{
    \(\(xmin = \(-5\);\)\), "\[IndentingNewLine]", 
    \(\(xmax = 5;\)\), "\[IndentingNewLine]", 
    \(\(a = 2;\)\), "\[IndentingNewLine]", 
    \(\(f[x_] = Sin[x\^2];\)\), "\[IndentingNewLine]", 
    \(\(curvePlot = 
        Plot[f[x], {x, xmin, xmax}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(tangent[x_] = \(f'\)[a]\ \((x - a)\) + 
          f[a];\)\), "\[IndentingNewLine]", 
    \(\(tangentPlot = 
        Plot[tangent[x], {x, xmin, xmax}, 
          PlotStyle \[Rule] {{RGBColor[0, 0, 1]}}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(tangentPoint = 
        Graphics[
          Disk[{a, 
              f[a]},  .01 \((xmax - xmin)\)]];\)\), "\[IndentingNewLine]", 
    \(\(Show[curvePlot, tangentPlot, tangentPoint, 
        DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"],

Cell[TextData[{
  "Notice that we used ",
  StyleBox["Show",
    FontFamily->"Courier"],
  " to combine graphics primitives with the results of ",
  StyleBox["Plot",
    FontFamily->"Courier"],
  " commands.  Also, notice that xmin, xmax, a, and f are defined at the \
beginning, so that it would be easy to change any of these to create a new \
figure.  We could package this as a function."
}], "Text"],

Cell[BoxData[{
    \(Clear[tangentPlot]\), "\[IndentingNewLine]", 
    \(tangentPlot[f_, a_, xmin_, xmax_, ymin_, ymax_] := 
      Module[\[IndentingNewLine]{curvePlot, tangentPlot, 
          tangentPoint}, \[IndentingNewLine]curvePlot = 
          Plot[f[x], {x, xmin, xmax}, PlotRange \[Rule] {ymin, ymax}, 
            DisplayFunction \[Rule] 
              Identity]; \[IndentingNewLine]tangentPlot = 
          Plot[\(f'\)[a]\ \((x - a)\) + f[a], {x, xmin, xmax}, 
            PlotRange \[Rule] {ymin, ymax}, 
            PlotStyle \[Rule] {{RGBColor[0, 0, 1]}}, 
            DisplayFunction \[Rule] 
              Identity]; \[IndentingNewLine]tangentPoint = 
          Graphics[
            Disk[{a, f[a]},  .01 \((xmax - xmin)\)]]; \[IndentingNewLine]Show[
          curvePlot, tangentPlot, tangentPoint, 
          DisplayFunction \[Rule] $DisplayFunction];\[IndentingNewLine]]\)}], \
"Input"],

Cell[BoxData[{
    \(\(f[x_] = \[ExponentialE]\^\(-x\^2\);\)\), "\[IndentingNewLine]", 
    \(tangentPlot[f, 1, \(-2\), 2, \(- .5\), 2]\)}], "Input"],

Cell[TextData[{
  StyleBox["Exercise 12.1:",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0.500008, 0.250004]],
  " Construct an animation of the tangent line to ",
  StyleBox["y",
    FontSlant->"Italic"],
  " = ",
  Cell[BoxData[
      \(\[ExponentialE]\^\(-x\^2\)\)]],
  " as the point of tangency moves along the curve from ",
  StyleBox["x",
    FontSlant->"Italic"],
  " = \[Dash]2 to ",
  StyleBox["x",
    FontSlant->"Italic"],
  " = 2."
}], "Text",
  Background->GrayLevel[0.750011]]
}, Closed]],

Cell[CellGroupData[{

Cell["Text", "Subsection"],

Cell[TextData[{
  "The syntax for inserting text in a 2D graphic object is ",
  StyleBox["Text[",
    FontFamily->"Courier"],
  StyleBox["string",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", {",
    FontFamily->"Courier"],
  StyleBox["xcoord",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox[", ",
    FontFamily->"Courier"],
  StyleBox["ycoord",
    FontFamily->"Courier",
    FontSlant->"Italic"],
  StyleBox["}]",
    FontFamily->"Courier"],
  ", where ",
  StyleBox["string",
    FontSlant->"Italic"],
  " is the text to be displayed."
}], "Text"],

Cell[BoxData[
    \(\(Show[
        Graphics[{Circle[{0, 0}, 2], 
            Text[\*"\"\<\!\(x\^2\)+ \!\(y\^2\) = 4\>\"", {1.5, 1.8}]}], 
        AspectRatio \[Rule] Automatic];\)\)], "Input"],

Cell["\<\
For a more complicated example, let's count the frequency of each \
letter within some text and then display each letter at the height of its \
frequency, similar to a bar chart.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(string = "\<Four score and seven years ago our fathers brought forth \
on this continent a new nation, conceived in liberty and dedicated to the \
proposition that all men are created equal.  Now we are engaged in a great \
civil war, testing whether that nation or any nation so conceived and so \
dedicated can long endure. We are met on a great battlefield of that war. We \
have come to dedicate a portion of that field as a final resting-place for \
those who here gave their lives that that nation might live. It is altogether \
fitting and proper that we should do this. But in a larger sense, we cannot \
dedicate, we cannot consecrate, we cannot hallow this ground. The brave men, \
living and dead who struggled here have consecrated it far above our poor \
power to add or detract. The world will little note nor long remember what we \
say here, but it can never forget what they did here.  It is for us the \
living rather to be dedicated here to the unfinished work which they who \
fought here have thus far so nobly advanced. It is rather for us to be here \
dedicated to the great task remaining before us--that from these honored dead \
we take increased devotion to that cause for which they gave the last full \
measure of devotion--that we here highly resolve that these dead shall not \
have died in vain, that this nation under God shall have a new birth of \
freedom, and that government of the people, by the people, for the people \
shall not perish from the earth.\>";\)\)], "Input"],

Cell[BoxData[
    RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(string\\)\\\" is similar to existing symbol \
\\\"\\!\\(String\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"]
}, Open  ]],

Cell[TextData[{
  "We will need a few ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " functions for dealing with text strings and lists.  ",
  StyleBox["Characters",
    FontFamily->"Courier"],
  " converts a string to a list of characters."
}], "Text"],

Cell[BoxData[
    \(stringCharacters = Characters[string]\)], "Input"],

Cell[TextData[{
  "Now we extract all of the letters from among these characters, and we use \
",
  StyleBox["ToUpperCase",
    FontFamily->"Courier"],
  " to convert them to capital letters."
}], "Text"],

Cell[BoxData[
    \(stringLetters = 
      ToUpperCase[Select[stringCharacters, LetterQ]]\)], "Input"],

Cell["We can count the number of A's in the text.", "Text"],

Cell[BoxData[
    \(Count[stringLetters, "\<A\>"]\)], "Input"],

Cell["\<\
We want to perform a count for all letters of the alphabet.  We \
first construct a list of all letters of the alphabet.\
\>", "Text"],

Cell[BoxData[
    \(alphabet = CharacterRange["\<A\>", "\<Z\>"]\)], "Input"],

Cell[BoxData[
    \(letterFreq = 
      Table[{alphabet[\([i]\)], Count[stringLetters, alphabet[\([i]\)]]}, {i, 
          26}]\)], "Input"],

Cell[TextData[{
  "Now we will place each letter at an ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-coordinate equal to its position in the alphabet and a ",
  StyleBox["y",
    FontSlant->"Italic"],
  "-coordinate equal to its frequency in this text."
}], "Text"],

Cell[BoxData[
    \(letterFreqGraph = 
      Table[Text[alphabet[\([i]\)], {i, letterFreq[\([i, 2]\)]}], {i, 
          26}]\)], "Input"],

Cell["And, finally, here is the graph.", "Text"],

Cell[BoxData[
    \(\(Show[Graphics[letterFreqGraph]];\)\)], "Input"],

Cell["What are the three most common letters in this text?", "Text"],

Cell["\<\
Perform the same analysis with the text below.  Does it have the \
same set of common letters and the same set of rare letters?  Place the \
graphs one right after the other, so that you can see them both \
simultaneously.  Do you notice any similarities?  Any differences?\
\>", \
"Text"],

Cell[BoxData[
    \(\(string = "\<When, in the course of human events, it becomes necessary \
for one people to dissolve the political bonds which have connected them with \
another, and to assume among the powers of the earth, the separate and equal \
station to which the laws of nature and of nature's God entitle them, a \
decent respect to the opinions of mankind requires that they should declare \
the causes which impel them to the separation. We hold these truths to be \
self-evident, that all men are created equal, that they are endowed by their \
Creator with certain unalienable rights, that among these are life, liberty \
and the pursuit of happiness. That to secure these rights, governments are \
instituted among men, deriving their just powers from the consent of the \
governed. That whenever any form of government becomes destructive to these \
ends, it is the right of the people to alter or to abolish it, and to \
institute new government, laying its foundation on such principles and \
organizing its powers in such form, as to them shall seem most likely to \
effect their safety and happiness. Prudence, indeed, will dictate that \
governments long established should not be changed for light and transient \
causes; and accordingly all experience hath shown that mankind are more \
disposed to suffer, while evils are sufferable, than to right themselves by \
abolishing the forms to which they are accustomed. But when a long train of \
abuses and usurpations, pursuing invariably the same object evinces a design \
to reduce them under absolute despotism, it is their right, it is their duty, \
to throw off such government, and to provide new guards for their future \
security. --Such has been the patient sufferance of these colonies; and such \
is now the necessity which constrains them to alter their former systems of \
government. The history of the present King of Great Britain is a history of \
repeated injuries and usurpations, all having in direct object the \
establishment of an absolute tyranny over these states. To prove this, let \
facts be submitted to a candid world. He has refused his assent to laws, the \
most wholesome and necessary for the public good. He has forbidden his \
governors to pass laws of immediate and pressing importance, unless suspended \
in their operation till his assent should be obtained; and when so suspended, \
he has utterly neglected to attend to them. He has refused to pass other laws \
for the accommodation of large districts of people, unless those people would \
relinquish the right of representation in the legislature, a right \
inestimable to them and formidable to tyrants only. He has called together \
legislative bodies at places unusual, uncomfortable, and distant from the \
depository of their public records, for the sole purpose of fatiguing them \
into compliance with his measures. He has dissolved representative houses \
repeatedly, for opposing with manly firmness his invasions on the rights of \
the people. He has refused for a long time, after such dissolutions, to cause \
others to be elected; whereby the legislative powers, incapable of \
annihilation, have returned to the people at large for their exercise; the \
state remaining in the meantime exposed to all the dangers of invasion from \
without, and convulsions within. He has endeavored to prevent the population \
of these states; for that purpose obstructing the laws for naturalization of \
foreigners; refusing to pass others to encourage their migration hither, and \
raising the conditions of new appropriations of lands. He has obstructed the \
administration of justice, by refusing his assent to laws for establishing \
judiciary powers. He has made judges dependent on his will alone, for the \
tenure of their offices, and the amount and payment of their salaries. He has \
erected a multitude of new offices, and sent hither swarms of officers to \
harass our people, and eat out their substance. He has kept among us, in \
times of peace, standing armies without the consent of our legislature. He \
has affected to render the military independent of and superior to civil \
power. He has combined with others to subject us to a jurisdiction foreign to \
our constitution, and unacknowledged by our laws; giving his assent to their \
acts of pretended legislation: For quartering large bodies of armed troops \
among us: For protecting them, by mock trial, from punishment for any murders \
which they should commit on the inhabitants of these states: For cutting off \
our trade with all parts of the world: For imposing taxes on us without our \
consent: For depriving us in many cases, of the benefits of trial by jury: \
For transporting us beyond seas to be tried for pretended offenses: For \
abolishing the free system of English laws in a neighboring province, \
establishing therein an arbitrary government, and enlarging its boundaries so \
as to render it at once an example and fit instrument for introducing the \
same absolute rule in these colonies: For taking away our charters, \
abolishing our most valuable laws, and altering fundamentally the forms of \
our governments: For suspending our own legislatures, and declaring \
themselves invested with power to legislate for us in all cases whatsoever. \
He has abdicated government here, by declaring us out of his protection and \
waging war against us. He has plundered our seas, ravaged our coasts, burned \
our towns, and destroyed the lives of our people. He is at this time \
transporting large armies of foreign mercenaries to complete the works of \
death, desolation and tyranny, already begun with circumstances of cruelty \
and perfidy scarcely paralleled in the most barbarous ages, and totally \
unworthy the head of a civilized nation. He has constrained our fellow \
citizens taken captive on the high seas to bear arms against their country, \
to become the executioners of their friends and brethren, or to fall \
themselves by their hands. He has excited domestic insurrections amongst us, \
and has endeavored to bring on the inhabitants of our frontiers, the \
merciless Indian savages, whose known rule of warfare, is undistinguished \
destruction of all ages, sexes and conditions. In every stage of these \
oppressions we have petitioned for redress in the most humble terms: our \
repeated petitions have been answered only by repeated injury. A prince, \
whose character is thus marked by every act which may define a tyrant, is \
unfit to be the ruler of a free people. Nor have we been wanting in attention \
to our British brethren. We have warned them from time to time of attempts by \
their legislature to extend an unwarrantable jurisdiction over us. We have \
reminded them of the circumstances of our emigration and settlement here. We \
have appealed to their native justice and magnanimity, and we have conjured \
them by the ties of our common kindred to disavow these usurpations, which, \
would inevitably interrupt our connections and correspondence. We must, \
therefore, acquiesce in the necessity, which denounces our separation, and \
hold them, as we hold the rest of mankind, enemies in war, in peace friends. \
We, therefore, the representatives of the United States of America, in \
General Congress, assembled, appealing to the Supreme Judge of the world for \
the rectitude of our intentions, do, in the name, and by the authority of the \
good people of these colonies, solemnly publish and declare, that these \
united colonies are, and of right ought to be free and independent states; \
that they are absolved from all allegiance to the British Crown, and that all \
political connection between them and the state of Great Britain, is and \
ought to be totally dissolved; and that as free and independent states, they \
have full power to levy war, conclude peace, contract alliances, establish \
commerce, and to do all other acts and things which independent states may of \
right do. And for the support of this declaration, with a firm reliance on \
the protection of Divine Providence, we mutually pledge to each other our \
lives, our fortunes and our sacred honor.\>";\)\)], "Input"],

Cell[TextData[{
  "Here is the Guatemala national anthem, if you want to try text from \
another language.  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " will count the accented characters along with the unaccented one.  So, \
for example, ",
  StyleBox["\[NTilde]",
    FontSlant->"Italic"],
  " will count as an ",
  StyleBox["n",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(string = "\<1 - \[DownExclamation]Guatemala feliz! que tus aras No \
profane jam\[AAcute]s el verdugo; Ni haya esclavos que laman el yugo Ni \
tiranos que escupan tu faz.   Si ma\[NTilde]ana tu suelo sagrado Lo amenaza \
invasi\[OAcute]n extranjera, Libre al viento tu hermosa bandera A vencer o a \
morir llamar\[AAcute].   Libre al viento tu hermosa bandera A vencer o a \
morir llamar\[AAcute]. Que tu pueblo con \[AAcute]nima fiera Antes muerto que \
esclavo ser\[AAcute].     2 - De tus viejas y duras cadenas T\[UAcute] \
forjaste con mano iracunda El arado que el suelo fecunda Y la espada que \
salva el honor.   Nuestros padres lucharon un d\[IAcute]a Encendidos en \
patrio ardimiento Y lograron sin choque sangriento Colocarte en un trono de \
amor,   Y lograron sin choque sangriento Colocarte en un trono de amor, Que \
de patria, en en\[EAcute]rgico acento, Dieron vida al ideal redentor.     3 - \
Es tu ense\[NTilde]a pedazo de cielo En que prende una nube su albura, Y ay \
de aquel que con ciega locura, Sus colores pretenda manchar!   Pues tus hijos \
valientes y altivos, Que veneran la paz cual presea, Nunca esquivan la ruda \
pelea Si defienden su tierra y su hogar,   Nunca esquivan la ruda pelea Si \
defienden su tierra y su hogar, Que es tan s\[OAcute]lo el honor su alma idea \
Y el altar de la patria su altar.     4 - Recostada en el ande soberbio, De \
dos mares al ruido sonoro, Bajo el ala de grana y de oro Te adormeces del \
bello quetzal.   Ave indiana que vive en tu escudo Paladi\[OAcute]n que \
protege tu suelo; \[DownExclamation]Ojal\[AAcute] que remonte su vuelo, M\
\[AAcute]s que el c\[OAcute]ndor y el \[AAcute]guila real   \
\[DownExclamation]Ojal\[AAcute] que remonte su vuelo, M\[AAcute]s que el c\
\[OAcute]ndor y el \[AAcute]guila real Y en sus alas levante hasta el cielo, \
Guatemala, tu nombre inmortal!\>";\)\)], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Graphics directives and options", "Section"],

Cell[TextData[{
  "There are two methods for modifying the appearance of graphics primitives. \
 You can modify the individual graphics primitives using ",
  StyleBox["graphics directives",
    FontSlant->"Italic"],
  ", and you can set ",
  StyleBox["options",
    FontSlant->"Italic"],
  " in the Show command that effect the entire graph.  We already used the ",
  StyleBox["AspectRatio",
    FontFamily->"Courier"],
  " option in the ",
  StyleBox["Show",
    FontFamily->"Courier"],
  " command to display circles properly.  Let's turn our attention to \
graphics directives.  The basic syntax for graphics directives is:  ",
  StyleBox["Graphics[{{",
    FontFamily->"Courier"],
  Cell[BoxData[
      \(TraditionalForm\`directive\_1, primitive\_\(1  a\), 
      primitive\_\(1  b\), \[Ellipsis], \ primitive\_\(1  n\)\)],
    FontFamily->"Courier"],
  StyleBox["}, {",
    FontFamily->"Courier"],
  Cell[BoxData[
      \(TraditionalForm\`directive\_2, primitive\_\(2  a\), 
      primitive\_\(2  b\), \[Ellipsis], \ primitive\_\(2  n\)\)],
    FontFamily->"Courier"],
  StyleBox["}. \[Ellipsis], {",
    FontFamily->"Courier"],
  Cell[BoxData[
      \(TraditionalForm\`directive\_m, primitive\_ma, 
      primitive\_mb, \[Ellipsis], \ primitive\_mn\)],
    FontFamily->"Courier"],
  StyleBox["}}]",
    FontFamily->"Courier"]
}], "Text"],

Cell[TextData[{
  "All of the graphics primitives allow you to specify a color as a graphics \
directive.  Colors can be specified with ",
  StyleBox["RGBColor",
    FontFamily->"Courier"],
  ", ",
  StyleBox["GrayLevel",
    FontFamily->"Courier"],
  ", or ",
  StyleBox["Hue",
    FontFamily->"Courier"],
  ".  Or you can load ",
  StyleBox["Graphics`Colors`",
    FontFamily->"Courier"],
  " and use English names for colors.  For example,"
}], "Text"],

Cell[BoxData[{
    \(\(rectangle1 = {RGBColor[1, 0, 0], 
          Rectangle[{\(-2\), \(-2\)}, {2, 2}]};\)\), "\[IndentingNewLine]", 
    \(\(rectangle2 = {RGBColor[1, 0, 1], 
          Rectangle[{\(-1\), \(-1\)}, {3, 3}]};\)\), "\[IndentingNewLine]", 
    \(\(disk = {RGBColor[1, 1, 0], 
          Disk[{2, 2}, 1]};\)\), "\[IndentingNewLine]", 
    \(\(rectanglesAndCircle = 
        Graphics[{rectangle1, rectangle2, 
            disk}];\)\), "\[IndentingNewLine]", 
    \(\(Show[rectanglesAndCircle, 
        AspectRatio \[Rule] Automatic];\)\)}], "Input"],

Cell["\<\
or here are all three graphics primitives combined into one \
statement.\
\>", "Text"],

Cell[BoxData[{
    \(\(rectanglesAndCircle = 
        Graphics[{{RGBColor[1, 0, 0], 
              Rectangle[{\(-2\), \(-2\)}, {2, 2}]}, {RGBColor[1, 0, 1], 
              Rectangle[{\(-1\), \(-1\)}, {3, 3}]}, {RGBColor[1, 1, 0], 
              Disk[{2, 2}, 1]}}];\)\), "\[IndentingNewLine]", 
    \(\(Show[rectanglesAndCircle, 
        AspectRatio \[Rule] Automatic];\)\)}], "Input"],

Cell[TextData[{
  "Other commonly used graphics directives are ",
  StyleBox["PointSize",
    FontFamily->"Courier"],
  " for the ",
  StyleBox["Point",
    FontFamily->"Courier"],
  " graphics primitive, and ",
  StyleBox["Dashing",
    FontFamily->"Courier"],
  " and ",
  StyleBox["Thickness",
    FontFamily->"Courier"],
  " for the Line graphics primitive.  These are all described in The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " Book, Section 2.9.3 Principles of Mathematica\[Rule]The Structure of \
Graphics and Sound\[Rule]Graphics Directives and Options.  Read through this \
section in the Help Browser."
}], "Text"],

Cell[TextData[{
  "In our next example, we will create an animation showing how a point on a \
rolling circle traces out a cycloid.  We will create an animation similar to \
the one show on MathWorld's ",
  ButtonBox["cycloid",
    ButtonData:>{
      URL[ "http://mathworld.wolfram.com/Cycloid.html"], None},
    ButtonStyle->"Hyperlink"],
  " page.  If ",
  StyleBox["t",
    FontSlant->"Italic"],
  " represents time, the radius of the circle is ",
  StyleBox["a",
    FontSlant->"Italic"],
  ", and the horizontal speed of the circle is ",
  StyleBox["a",
    FontSlant->"Italic"],
  ", then the center of the circle at time ",
  StyleBox["t",
    FontSlant->"Italic"],
  ", is at"
}], "Text"],

Cell[BoxData[{
    \(Clear[center]\), "\[IndentingNewLine]", 
    \(center[t_] := {a\ t, a}\)}], "Input"],

Cell["\<\
Let's set the radius and the number of steps in our \
animation.\
\>", "Text"],

Cell[BoxData[{
    \(\(a = 2;\)\), "\[IndentingNewLine]", 
    \(\(tsteps = 40;\)\), "\[IndentingNewLine]", 
    \(\(tmax = 4  \[Pi]\ a;\)\)}], "Input"],

Cell["Here are the circles moving from left to right.", "Text"],

Cell[BoxData[{
    \(Needs["\<Graphics`Colors`\>"]\), "\[IndentingNewLine]", 
    \(\(circles = 
        Table[Graphics[{Blue, Circle[center[t], a]}], {t, 0, tmax, 
            tmax\/tsteps}];\)\), "\[IndentingNewLine]", 
    \(\(circles = 
        Table[Show[circles[\([i]\)], AspectRatio \[Rule] Automatic, 
            PlotRange \[Rule] {{\(-1.1\) a, a\ tmax + a}, {\(- .2\) a, 
                  2.2  a}}], {i, 1, Length[circles]}];\)\)}], "Input"],

Cell["Now, let's add a point moving around the circle.", "Text"],

Cell[BoxData[{
    \(pointPath[t_] := 
      center[t] - {a\ Sin[t], a\ Cos[t]}\), "\[IndentingNewLine]", 
    \(\(points = 
        Table[Graphics[{PointSize[ .015], Point[pointPath[t]]}], {t, 0, 
            4  \[Pi]\ a, tmax\/tsteps}];\)\), "\[IndentingNewLine]", 
    \(\(points = 
        Table[Show[points[\([i]\)], AspectRatio \[Rule] Automatic, 
            PlotRange \[Rule] {{\(-1.1\) a, a\ tmax + a}, {\(- .2\) a, 
                  2.2  a}}], {i, Length[points]}];\)\)}], "Input"],

Cell["\<\
Now, combine the circles and points to be sure that the point is \
moving around the circle.\
\>", "Text"],

Cell[BoxData[
    \(\(Table[
        Show[circles[\([i]\)], points[\([i]\)]], {i, 
          Length[circles]}];\)\)], "Input"],

Cell[TextData[{
  "So far, so good!  Now, let's show the trail of dots as the circle moves.  \
At each step, ",
  StyleBox["i",
    FontSlant->"Italic"],
  ", we need to display the points for ",
  StyleBox["k",
    FontSlant->"Italic"],
  " = 1, 2, \[Ellipsis], ",
  StyleBox["i",
    FontSlant->"Italic"],
  ".  These are the points that have been traced out up to that time step.  \
After you evaluate the following Input cell, you may want to drag the \
bounding box of the first image in the list to make it bigger and then \
evaluate it again.  This will increase the size of all of the images in the \
animation."
}], "Text"],

Cell[BoxData[{
    \(\(trace = 
        Table[Graphics[{circles[\([i]\)], 
              Table[points[\([k]\)], {k, 1, i}]}], {i, 
            Length[circles]}];\)\), "\[IndentingNewLine]", 
    \(\(Table[
        Show[{circles[\([i]\)], Table[points[\([k]\)], {k, 1, i}]}], {i, 
          Length[trace]}];\)\)}], "Input"],

Cell["\<\
example? : animated line segment from {0,0} to {1,a}.  Can show \
previous lines as fainter gray or as dotted lines.\
\>", "Text"],

Cell[TextData[{
  StyleBox["Exercise 12.2:",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0.500008, 0.250004]],
  " Create an image of nested spiraled n-sided regular polygons.  The \
polygons should share the same center.  Each polygon should be a little \
larger than the one before and rotated by some small angle.  All of these \
polygons should be displayed together in a single image.  Suggestion: Start \
simple.  Try to display a single n-sided polygon first.  In fact, you could \
write this as an auxiliary function that you call from your main function."
}], "Text",
  Background->GrayLevel[0.750011]],

Cell[TextData[{
  "Now that we know about the ",
  StyleBox["PointSize",
    FontFamily->"Courier"],
  " directive for ",
  StyleBox["Point",
    FontFamily->"Courier"],
  ", we can improve our tangent line example.  The default size for Point is \
so small that we couldn't use it to mark the point of tangency on our graph.  \
But with a larger ",
  StyleBox["PointSize",
    FontFamily->"Courier"],
  ", we can use ",
  StyleBox["Point",
    FontFamily->"Courier"],
  " instead of ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  ".  We can plot the point of tangency using ",
  StyleBox["Point",
    FontFamily->"Courier"],
  " instead of using ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  ", since ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  " does not maintain its proper aspect ratio.  A ",
  StyleBox["Disk",
    FontFamily->"Courier"],
  " may be squashed into an ellipse if ",
  StyleBox["AspectRatio",
    FontFamily->"Courier"],
  " is not set to ",
  StyleBox["Automatic",
    FontFamily->"Courier"],
  "."
}], "Text"],

Cell[BoxData[{
    \(Clear[tangentPlot]\), "\[IndentingNewLine]", 
    \(tangentPlot[f_, a_, xmin_, xmax_, ymin_, ymax_] := 
      Module[\[IndentingNewLine]{curvePlot, tangentPlot, 
          tangentPoint}, \[IndentingNewLine]curvePlot = 
          Plot[f[x], {x, xmin, xmax}, PlotRange \[Rule] {ymin, ymax}, 
            DisplayFunction \[Rule] 
              Identity]; \[IndentingNewLine]tangentPlot = 
          Plot[\(f'\)[a]\ \((x - a)\) + f[a], {x, xmin, xmax}, 
            PlotRange \[Rule] {ymin, ymax}, 
            PlotStyle \[Rule] {{RGBColor[0, 0, 1]}}, 
            DisplayFunction \[Rule] 
              Identity]; \[IndentingNewLine]tangentPoint = 
          Graphics[{PointSize[ .03], 
              Point[{a, f[a]}]}]; \[IndentingNewLine]Show[curvePlot, 
          tangentPlot, tangentPoint, 
          DisplayFunction \[Rule] $DisplayFunction];\[IndentingNewLine]]\)}], \
"Input"],

Cell[BoxData[{
    \(\(f[x_] = \[ExponentialE]\^\(-x\^2\);\)\), "\[IndentingNewLine]", 
    \(tangentPlot[f, 1, \(-2\), 2, \(- .5\), 2]\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["3D graphics primitives", "Section"],

Cell[TextData[{
  "The 3D graphics primitives are ",
  StyleBox["Point",
    FontFamily->"Courier"],
  ", ",
  StyleBox["Line",
    FontFamily->"Courier"],
  ", ",
  StyleBox["Polygon",
    FontFamily->"Courier"],
  ", ",
  StyleBox["Text",
    FontFamily->"Courier"],
  ", and ",
  StyleBox["Cuboid",
    FontFamily->"Courier"],
  ".  The first four primitives are similar to their corresponding 2D \
primitives.   Of course, you have to use three coordinates instead of two \
coordinates to specify points in the 3D primitives.  ",
  StyleBox["Cuboid",
    FontFamily->"Courier"],
  " allows you to draw a rectangular box.  Its syntax is given in the Help \
Browser.  2D graphics pirmitives must be enclosed in a ",
  StyleBox["Graphics",
    FontFamily->"Courier"],
  " command, while 3D graphics primitives must be enclosed in ",
  StyleBox["Graphics3D",
    FontFamily->"Courier"],
  "."
}], "Text"],

Cell[TextData[{
  "Here is a bridge.  It sometimes helps to draw a diagram by hand first, so \
that you can figure out the coordinates of the polygons' vertices.  Since the \
bridge will have two similar supports, we will write it as a function and \
then call the function twice to create the two supports.  We will use the \
parameter, ",
  StyleBox["xBegin",
    FontSlant->"Italic"],
  ", to specify the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-coordinate of the left side of the support's base.  In the diagram below, \
",
  StyleBox["xBegin",
    FontSlant->"Italic"],
  " is 0 and ",
  StyleBox["l ",
    FontSlant->"Italic"],
  "\[Dash] 2.  By the way, I inserted the diagram in this ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook by opening it in a graphics program (I used Adobe Photoshop), \
selecting the entire image, copying the image using Edit\[Rule]Copy, \
returning to my ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook, and pasting the image between two cells."
}], "Text"],

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