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