Following the golden map through its period doublings, the golden ratio does not show up once. It shows up again at every stage.

The 2-cycle gives a line whose length runs to 

, plots as two conjoined similar triangles. Two pairs of their sides, call them 

, makes a bigger version of the same picture, two large triangles and two small ones. The middle triangle's side ratio 

So it is not a one-off. Each time the map doubles its period, the same ratio reappears in the geometry of the periodic points. That repetition is what makes me think it is real and not an accident of one plot.

The same golden ratio keeps coming back at each period doubling

Related to this topic

No comments yet