You will get fractals like this if you do anything in the correct range. Anything larger than 1/2 gets you overlapping triangles, and anything smaller than 2/3 gets you empty space. It just so happens that phi is in the right range to have both of these properties be easily observable. Values between .6 and .63 look the best to me.

I'm having trouble with imgur, but if you have java you can play around with this code, which goes quite quickly.

public class Serpenski{  
public static void main(String[] args){  
    double x=Double.parseDouble (args [0]);  
    double y=Double.parseDouble (args [1]);  
    double a=Double.parseDouble (args [2]);  
    double b=Double.parseDouble (args [3]);  
    double point1x=0;  
    double point1y=0;  
    double point2x=1;  
    double point2y=0;  
    double point3x=.5;  
    double point3y=.86603;  
    StdDraw.setXscale(0.0, 1.0);  
    StdDraw.setYscale(0.0, 1.0);  
    StdDraw.show (0);  
    for (double i=1.0; i<=100000; i++){  
        double r=Math.random();  
        if (r<(.333)){                   
            StdDraw.point(((a*point1x+b*x)/(a+b)),((a*point1y+b*y)/(a+b)));  
            x=(a*point1x+b*x)/(a+b);  
            y=(a*point1y+b*y)/(a+b);  
        }  
        else if (r<(.667)){  
            StdDraw.point(((a*point2x+b*x)/(a+b)),((a*point2y+b*y)/(a+b)));  
            x=(a*point2x+b*x)/(a+b);  
            y=(a*point2y+b*y)/(a+b);}  
        else {  
            StdDraw.point(((a*point3x+b*x)/(a+b)),((a*point3y+b*y)/(a+b)));  
            x=(a*point3x+b*x)/(a+b);  
            y=(a*point3y+b*y)/(a+b);}  
    }  
    StdDraw.show (0);  
}  

}

Change all the &lt to < . I'm not sure why reddit is switching that.