The line above one particular a shows the values of x n , for n 100. Then at 3 the diagram bifurcates, with the two visible points being those two numbers which the sequence is oscillating around. As mentioned above, at around 3.45, there is a second bifurcation, now clearly visible. Write a 2 for this second bifurcation point. And in general, let a n be the nth bifurcation point.
Lets pick out two numbers which capture the self similarity of the bifurcations.
The limit of the d n exists, and is called d . There are two points on this line. Let e 1 be the distance between the two points. There are 4 points. Let e 2 be the distance between the first two points.