(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)



(* For problems 1-5: Code implementing the chaos game and producing plots *)

h1[{x_, y_}] := N[{x/3, y/3}]
h2[{x_, y_}] := N[{x/3, y/3 + 1/3}]
h3[{x_, y_}] := N[{x/3 + 1/3, y/3}]
h4[{x_, y_}] := N[{x/3 + 1/3, y/3 + 1/3}]
h5[{x_, y_}] := N[{x/3 + 2/3, y/3}]
h6[{x_, y_}] := N[{x/3 + 2/3, y/3 + 1/3}]
h7[{x_, y_}] := N[{x/3 + 2/3, y/3 + 2/3}]
functionlist := {h1, h2, h3, h4, h5, h6, h7}

(* This defines seven contraction maps to be used in the iterated function system for 1b, and then collects them into a list *)

hch[{x_, y_}] := RandomChoice[functionlist][{x, y}]

(* This defines a function which randomly chooses one of the maps in the input list and applies it *)

(* It is also possible to code the function to make weighted choices according to a given probability list; see the documentation for RandomChoice for more information.  Weighted choices are not used on our assignments, but you may want to give them a try anyway: they're fun! *)


fracpoints := fracpoints = Drop[NestList[hch, {0.1, 0.1}, 301000], 1000] 

(* This will compute 301000 iterates of the chaos game applied to the starting point {0.1,0.1} and then delete the first 1000 terms, to clean up any slow convergence at the beginning *)

(* The double assignment causes Mathematica only to generate the list once, rather than recomputing it each time the computation is run *)


fracplot[n_] := 
 ListPlot[fracpoints[[1 ;; n]], Axes -> True, 
  PlotStyle -> Directive[Black, PointSize[0.0010]], PlotRange -> {{0, 1}, {0, 1}}, AspectRatio -> 1, 
  PlotLabel -> StringJoin["IFS plotted with ", ToString[n], " total points"]]

ifs1bplot = fracplot[100000]

(* This will plot the first n points in the list of points, using black points with size 0.0010, a range set to the square [0,1] x [0,1], and a title that displays the number of points used.  It may be advisable to change the point size if more points are plotted to create a higher-resolution picture. *)






(* For problem 1: Transcriptions of the functions used in each part *)

fah1[{x_, y_}] := N[{x/3, y/3}]
fah2[{x_, y_}] := N[{x/3, y/3 + 1/3}]
fah3[{x_, y_}] := N[{x/3 + 1/3, y/3 + 1/3}]
fah4[{x_, y_}] := N[{x/3 + 2/3, y/3}]
fah5[{x_, y_}] := N[{x/3 + 2/3, y/3 + 2/3}]
functionlist1a := {fah1, fah2, fah3, fah4, fah5}

fbh1[{x_, y_}] := N[{x/3, y/3}]
fbh2[{x_, y_}] := N[{x/3, y/3 + 1/3}]
fbh3[{x_, y_}] := N[{x/3 + 1/3, y/3}]
fbh4[{x_, y_}] := N[{x/3 + 1/3, y/3 + 1/3}]
fbh5[{x_, y_}] := N[{x/3 + 2/3, y/3}]
fbh6[{x_, y_}] := N[{x/3 + 2/3, y/3 + 1/3}]
fbh7[{x_, y_}] := N[{x/3 + 2/3, y/3 + 2/3}]
functionlist1b := {fbh1, fbh2, fbh3, fbh4, fbh5, fbh6, fbh7}

fch1[{x_, y_}] := N[{x/3, y/3}]
fch2[{x_, y_}] := N[{x/3 + 1/3, y/3}]
fch3[{x_, y_}] := N[{2x/3 + 1/3, 2y/3 + 1/3}]
functionlist1c := {fch1, fch2, fch3}

fdh1[{x_, y_}] := N[{x/2, y/2}]
fdh2[{x_, y_}] := N[{x/2 + 1/2, y/2}]
fdh3[{x_, y_}] := N[{1 - y/2, x/2 + 1/2}]
functionlist1d := {fdh1, fdh2, fdh3}

feh1[{x_, y_}] = N[{2 x/5, 2 y/5}];
feh2[{x_, y_}] = N[{2 x/5 + 3/5, 2 y/5}];
feh3[{x_, y_}] = N[{x/5 + 2/5, y/5 + 1/5}];
feh4[{x_, y_}] = N[{2/5 - y/5, x/5}];
feh5[{x_, y_}] = N[{y/5 + 3/5, 1/5 - x/5}];
functionlist1e := {feh1, feh2, feh3, feh4, feh5}

ffh1[{x_, y_}] = N[{1/2 - y/2, x/2}];
ffh2[{x_, y_}] = N[{1/2 + y/2, 1 - x/2}];
ffh3[{x_, y_}] = N[{1/2 + y/2, x/2}];
ffh4[{x_, y_}] = N[{x/3, 1/2 + y/3}];
functionlist1f := {ffh1, ffh2, ffh3, ffh4}



hch[{x_, y_}] := RandomChoice[functionlist1a][{x, y}]
fracpoints := fracpoints = Drop[NestList[hch, {0, 0.1}, 301000], 1000] 
fracplot[n_] := 
 ListPlot[fracpoints[[1 ;; n]], 
  PlotStyle -> Directive[Black, PointSize[0.0010]],  PlotRange -> {{0, 1}, {0, 1}},
  AspectRatio -> 1, 
  PlotLabel -> 
   StringJoin["IFS plotted with ", ToString[n], " total points"]]
plot1 = fracplot[10000]
plot2 = fracplot[30000]
plot3 = fracplot[100000]
plot4 = fracplot[300000]

(* This is the same code as above, but put together.  This will draw the four requested plots using the function list for part (a).  Note that the code above defining the function list needs to be run first! *)