A bifurcation map that keeps drawing the golden ratio

Intuition

The logistic map's signature constant is Feigenbaum's δ, and it shows up in the rate of bifurcation. This map's signature seems to be φ — and it shows up not in the rate but in the shape the periodic points trace out. It feels like the golden ratio is hiding in the geometry rather than the dynamics.

Parallels

It reminds me of how φ falls out of continued fractions and the Fibonacci recurrence — the "most irrational" number appearing wherever a simple self-referential process settles into equilibrium. The exponent 1/r folding back onto 1/|xₙ| has that same self-referential flavour.

Self-Critique

Everything here is numerical — Mathematica, finite iteration counts, side lengths measured by hand. The convergence to φ could in principle be coincidence or an artifact of where I sampled r. The 4- and 8-cycle ratios only hover near φ; they do not sit exactly on it. And I cannot yet say which side lengths should give φ, or why.

Hypotheses

A real answer would (a) prove the 2-cycle segment length tends to exactly φ² = 1+φ, (b) identify which pairs of triangle sides tend to φ and 1/φ and prove the limit, and (c) settle whether the fixed-point limit 1/φ is the cause of the geometric appearances or a separate coincidence.