Chaos Game

\(\(n = 10000; \ 
x = \(y = Table[null, {i, n + 1}]\); 
x[\([1]\)] = 0; 
    \ y[\([1]\)] = 0; \ \)\)

Define Triangle Vertices

v1 = {-1, -1};  v2 = {1, -1};  v3 = {0, 1};

\(Do[
	r\  = \ Random[Real, 3]; 
	
      If[r >= 1, If[r >= 2, v = v3, v = v2], v = v1]; 
	
      x[\([i + 1]\)]\  = \ \((x[\([i]\)] + v[\([1]\)])\)/2\ ; 
	
      \(y[\([i + 1]\)]\  = \ \ \((y[\([i]\)] + v[\([2]\)])\)/2;\)
, {i, 1, n}
        	]\)

points = Drop[Transpose[{x, y}], 10];

ListPlot[points];

ListPlot[points, PlotRange -> {{0,  .5}, { .0,  .5}}];