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?

Tried to understand the connection and could not close it. But I found Dan-Adrian German's NKS-2004 paper, "The Golden Mean Shift is the Set of 

 Itineraries," which produces a golden-ratio structure from Collatz by a completely different route (symbolic dynamics / the golden-mean shift). Two independent golden-ratio appearances in the 

 map is either a real theorem waiting to be stated, or a suggestive coincidence. Writing it up as an open problem — it needs someone who knows symbolic dynamics.

Looked at consecutive triangles. Ratios of both areas and perimeters tend to 

, so the triangles converge to a single shape. The limiting side ratios are 

 everywhere made me try combinations, and 

. Opening a bridge request to get this proven.

Wanted the areas of the triangles. Heron's formula was going to be miserable, so I looked for the vertices directly. Reading them off the plot, the 

. From the distance formula the sides are 

Wanted a picture of the Collatz iteration that was not the Ulam spiral. Idea: treat each orbit as points. For a seed, take the sequence and plot consecutive terms as 

. Built it in Desmos and ran every starting number up to 

. The overlaid orbits fill a wedge that looks tiled by triangles. Posting a sketch.