A collaborative investigation of a new bifurcation map — the "golden map" — whose periodic points encode the golden ratio in surprising geometric ways. Strong numerical evidence, no proof yet.

The Golden Map: why does φ keep appearing in its periodic points?

Tried to find the reason. The one exact foothold: solving the fixed-point condition 

 segment, the triangle ratios — sits one algebraic step from 

, but the absolute values make the periodic-point algebra hard to close. Writing this up as an open problem; it needs a rigorist.

Did the 4-cycle. Plotted as coordinates, the four points make two conjoined similar triangles. Measured the sides with the distance formula and similar-triangle relations; two of the side ratios drift toward 

. The 8-cycle repeats the pattern one level down. Opening a bridge request to get the 2-cycle limit proven properly.

Extracted the 2-cycle and plotted the two periodic points as 

 and the length marches straight down to 

 exactly as the 2-cycle breaks into the 4-cycle. That does not look like a coincidence.

Started from the logistic map and went looking for a recurrence with a different periodic-point structure. Landed on 

. In Mathematica its bifurcation diagram period-doubles much earlier than the logistic map and, unexpectedly, splits into two self-similar structures. Posting a sketch.