Plotting Collatz (3n+1) orbits as coordinate points tiles a wedge with triangles whose side lengths encode the golden ratio: (D₂+D₃)/D₁ → φ. Strong numerical evidence, no proof yet — and a possible link to the golden-mean shift of the 3x+1 itineraries.

Collatz Triangles: why does φ appear in the triangles the orbits draw?

Plotting Collatz ($3n+1$) orbits as coordinate points $(x_i,x_{i+1})$ tiles a wedge with triangles. The $n$-th triangle has vertices $A_n=(1+2n,4+6n)$, $B_n=(2+4n,1+2n)$, $C_n=(4+6n,2+3n)$ and sides $D_1=\sqrt{20n^2+28n+10}$, $D_2=\sqrt5\,(n+1)$, $D_3=\sqrt{25n^2+36n+13}$. Numerically the side ratios tend to $\sqrt5,\ \sqrt5/2,\ 1/2$, so $(D_2+D_3)/D_1\to\varphi=(1+\sqrt5)/2$ as $n\to\infty$ (and $\to1/\varphi$ as $n\to-\infty$). Can these limits be proven cleanly? And is this geometric golden ratio the same phenomenon as the golden-mean shift of the $3x+1$ itineraries (German, 2004, entropy $