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

Almost everything I found in the golden map is numerical. This part you can actually solve by hand.

 and look for a fixed point, a value that maps to itself. I wanted to know where that fixed point goes as 

, since any positive number to the power 

That is the golden ratio's own defining equation turning up on its own. Solving the fixed point numerically at large 

 falls out of the map exactly, instead of as a limit of measured lengths. If there is a real reason the golden ratio runs through this whole map, I think it starts here.

The one place the golden map gives you φ exactly is its fixed point

I built a bifurcation map from the recurrence 

. It period-doubles much sooner than the logistic map, so I have been calling it the golden map.

 from 1 up to about 2.4 the map settles into a 2-cycle, two values it bounces between. I plot those two values as the points 

 and measure the length of the line between them.

. The line keeps shrinking toward it and reaches it right around 

, which is exactly where the 2-cycle gives out and the 4-cycle begins. Past that point the length turns around and grows again.

So the golden ratio is sitting on the boundary between the 2-cycle and the 4-cycle. I have the numbers but not a reason.

In the golden map, the 2-cycle line length runs straight to 1 + φ

I was looking for alternatives to the logistic map 

Its bifurcation diagram period-doubles much faster than the logistic map and splits into two distinct self-similar structures. I call it the 

, I iterate the map until the orbit settles into a repeating cycle 

 (its periodic points). Then I plot that cycle as coordinate points — 

 — join them in order, and look at the shape they trace. Measuring those shapes is where the golden ratio keeps turning up. (The full procedure, the Mathematica code, and the data tables are written up in the project.)

, that segment's length runs straight to 

 points form two "conjoined triangles." The ratios of certain side pairs drift toward 

 does the same thing again. And the fixed point 

The golden ratio has no business being here — this map is built from an absolute value and an exponent, not from 

. But it keeps surfacing in the geometry of the periodic points. I have strong numerical evidence and no proof, and I would love help understanding why.

The Sketch