%% Script to implement the Chaos Game, as described in Chapter 5 of
%% 'Chaos and Order in the Capital Markets,' 2nd ed, by Edgar Peters.
%% This is based on Iterated Function Systems (IFS), created by
%% Michael Barnsley of Iterated Systems.
%%
%% This script draws an equilateral triangle, with side length 2. It
%% picks a random point within that triangle, then starts iterating
%% according to the rule: at each step, randomly pick one of the
%% triangle's 3 vertices, and move the current point halfway toward
%% the chosen vertex. Repeat for the next step.
%%
%% The attracting set of points will be the Sierpinski triangle. Note
%% it might not come out so clearly from this script, since Matlab draws
%% its points kind of big. I didn't worry too much about this, since
%% I just wrote this script to play around with this system.
%%
%% Ken Gosier
%% October 2000


%% Seed random number generator with time

rand('state', sum(100*clock));


%% Random point in box (0,1) x (0,sqrt(3)). If in upper half of box,
%% reflect down to right half of triangle

pt.x = rand;
pt.y = rand*sqrt(3);

if pt.y > 2*pt.x
   pt.x = 2 - pt.x;
   pt.y = sqrt(3) - pt.y;
end


%% plot equilateral triangle bounding Sierpinski

bottom_x = [0, 2];
bottom_y = [0, 0];

sides_x = [0, 1, 2];
sides_y = [0, sqrt(3), 0];

% statements needed so later points will go on same plot as triangle

close all;
figure;
hold on;

plot(bottom_x, bottom_y, '-', sides_x, sides_y, '-');
xmin = 0;
xmax = 2;
ymin = 0;
ymax = sqrt(3);
axis([  xmin xmax ymin ymax  ]);


%% 3 vertices of trangle

pt1.x = 0;  pt1.y = 0;
pt2.x = 2;  pt2.y = 0;
pt3.x = 1;  pt3.y = sqrt(3);

vertarr = [pt1, pt2, pt3];


%% start iterating.
%%
%% threshold tells, after what iteration do we start plotting points?
%% In this system, we expect our distance to the attractor to shrink
%% by 1/3 with each iteration. So I choose threshold=5, it will be
%% 1/243-rd its original distance from the attractor. And Matlab
%% draws its points kind of big, so it will probably be close enough
%% for displaying.
%%
%% ask_iter tells, how many iterations between stopping to ask the
%% user to continue or quit? It's a kludge to say 'Press Ctrl-C to
%% stop,' but this script is just for playing around with this system;
%% I'll leave the gorgeous UI for the commercial release.

iter = 0;
threshold = 5;
ask_iter = 1000;

while 1
   
   %% point to move toward
   
   sub = fix(rand*3) + 1;
   
   pt.x = 0.5*(pt.x + vertarr(sub).x);
   pt.y = 0.5*(pt.y + vertarr(sub).y);
   
   
   %% plot if past threshold, ask to stop if
   %% right multiple
   
   iter = iter + 1;
   if iter > threshold
      plot(pt.x, pt.y, '.');
   end
   
   
   if mod(iter,ask_iter) == 0
      input('Press RETURN to go on, Ctrl-C to stop:');
   
      % user has to activate the command window to press RETURN. This
      % makes the plot window active again
   
      H = gcf;
      figure(H);
   end
end