import java.applet.Applet;
import java.awt.event.*;
import java.awt.*;
import java.awt.geom.*;


class TextLimit {

    /**This is a documentation file.  It is read by the file reader.**/

    public TextLimit() {}

    public String getText(int k) {
	if(k==0) return(text0());
	if(k==1) return(text1());
	if(k==2) return(text2());
	if(k==3) return(text3());
	if(k==4) return(text4());
	if(k==5) return(text5());
	if(k==6) return(text6());
	if(k==7) return(text7());
	if(k==8) return(text8());
	if(k==9) return(text9());
	if(k==10) return(text10());
	if(k==11) return(text11());
	if(k==12) return(text12());
	if(k==13) return(text13());
	return(null);
    }

    public static String text0() {

	String S="LIMITS:  It turns out G2(k+1) is larger than G2(k) by a factor of about s2=sqrt(3).   At the same time, G3(k+1) is larger than G3(k) by a factor of about s3=sqrt(2)+1.    When these paths are scaled by the appropriate powers of s,  they seem to converge as k tends to infinity.   This is most clear in the G3 case.  Turn on the display on the 'G3' control panel and move the 'which orbit' arrows from 1 to 7. ";

	return(S);
    }

    public static String text1() {

	String S="The situation is more subtle in the G2 case.   There seem to be three different limits\n G2(1), G2(3), G2(5), G2(7)...\nG2(2), G2(6), G2(10), G2(14),...\nG2(4),G2(8),G2(12), G2(16),...\nThe fact that the odd case (with one seeming limit) is different from the even case (with two seeming limits) does not sit well with us.  In fact, in the odd cases, the parametrizations do not converge.";

	return(S);
    }


    public static String text2() {

	String S="EXERCISE: Press the 'opposite circulation' button on the 'special configurations' control, and then use the big teal arrow keys to trace around the two displayed graphs V2(5) and V2(7).  Alternatively, nove the cursor over one of the windows and hold down the 's' key.  (The two curves V2(5) and V2(7) coincide with the arithmetic graphs G2(5) and G2(7), as the 'substitution' text will explain.)  So, really we have to 'skip 4' in both the odd and even cases to get a nice limit.";
	return(S);
    }


    public static String text3() {

	String S="THE LIMITS OF INTEREST:   Let RG2(k)=s2^(-k)G2(k) and likewise define RG3(k).   The limits of interest to us are \nRG2(1), RG2(5), RG2(9), RG2(13),...\nRG2(3), RG2(7), RG2(11), RG2(15),...\nRG3(1), RG3(5), RG3(9), RG3(13),...\nRG3(3), RG3(7), RG3(11), RG3(15),...\nWe call these the 'rescaled G2 limits' and the 'rescaled G3 limits' respectively.  We only treat the odd case. ";

	return(S);
    }

    public static String text4() {

	String S="THE IDEAL FRACTALS:    Click on the ' ideal fractal' button of the 'special configurations' window.   What you now see are the sets I2(6) and I3(6). The limit I2 is a variant of the standard Sierpinski carpet.  It is obtained by recursively removing squares from a unit square.  The rule for I3 is that each isosceles triangle is replaced by 5 smaller ones.  To see this in action, reset the program and then turn on the buttons '1', '2', '3', etc. on the 'I3' control.";

	return(S);
    }

    public static String text5() {

	String S="As we mentioned in the 'how to use' text, any object displayed in the left plot window is 'mirrored' in the right plot window in that the corresponding object is also plotted.  Thus, when you bring up the I3 objects, you simultaneously see the I2 objects.  The I2 objects are plotted in nearly-transparent colors because they consist of overlapping shapes.";

	return(S);
    }



    public static String text6() {

	String S="SUBSTITUTION RULE FOR I2:   Press the ' fractal parametrization' button on the 'special configurations' window. The sets I2(1), I2(2), I2(3), ... are also produced by a 1-to-5 substitution rule.  To see the basic shapes involved,  click repeatedly on the big arrows at the top of the middle control panel.  The set I2(0) is a cyclically ordered list of 8 trapezoids (which wind twice around). ";

	return(S);
    }


    public static String text7() {

	String S=" In passing from I2(0) to I2(1), each trapezoid is divided into 2 parallelograms and 3 trapezoids in a PTTTP pattern. In passing from I2(1) to I2(2), each parallelogram is divided into 2 trapezoids and 3 parallelograms in a TPPPT pattern.  This goes on in an alternating fashion imparts a cyclic ordering to the shapes of I2(k).  Note that these shapes overlap, and indeed multiple copies sometimes lie on top of each other.";

	return(S);
    }


    public static String text8() {
	String S="PARAMETRIZATION:  You can use this feature whenever the big arrow keys appear.  They appear when there is exactly one I2 button turned on.  Remember, you have to use the I3 controls to turn on the I2 objects.  The parametrization feature also traces out the I3 sets.  The whole thing is synchronized, and compatible with a canonical continuous surjection from I3 to I2.  For instance, the outermost points of I3 map to the edge centers of I2.";

	return(S);
    }

    public static String text9() {
	String S="The motion button that appears below the arrows turns on and off an animated tracer. Between the motion feature and the arrow keys, you should be able to get a good idea of the subdivision rule for I2.   If you can't get a good sense of it, check out OctoMap 1.  OctoMap 1 is dedicated entirely to the description of I2 and I3.";

	return(S);
    }

    public static String text10() {

	String S="MAIN THEOREM: The rescaled G2 limit exists and equals a suitably scaled copy of I2.   The rescaled G3 limit exists and equals a suitably scaled copy of I3.\n\n The main theorem gives a closed form expression for the rescaled limits of the arithmetic graphs.  In a geometric sense, this gives lots of information about the asymptotic structure of the first return map.";

	return(S);
    }

    public static String text11() {

	String S="For the expert,  the weakest (but shortest) statement we can make is that the convergence takes place in the Hausdorff topology on closed sets.  However, the substitution rules give natural parametrizations on the I2 and I3 limits, and actually the convergence takes place in the space of maps from the unit circle into the plane, equipped with the compact-open topology.  That is, we get  convergence of  parametrized curves.";

	return(S);
    }

    public static String text12() {

	String S="PLAN: The remaining text  proves the main theorem.  The 'substitution' text will explain how to view the arithmetic graphs in a more combinatorial light.   The 'ideal rule' text will explain how the combinatorially-flavored construction of the arithmetic graphs can be modified to produce a cleaner sequence of paths that have the same limit as the originals.   The 'final matchup' text will explain how to see that the 'cleaned up paths' have I2 and I3 as their limits. ";

	return(S);
    }

    public static String text13() {

	String S="If the explanation here is too informal for your taste, see my paper 'Outer Billiards, Arithmetic Graphs, and the Octagon'.  My paper covers the same ground (in 90 pages!) in a more traditional way.";

	return(S);
    }


}