Tyler Samuels
July 25, 2026
Plotting Collatz orbits as points draws triangles whose sides give φ
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Context
What lives around this idea
Intuition
The wedge of triangles feels like a hidden coordinate system for the 3n+1 problem — the same orbits everyone plots as a tree, but the geometry reorganizes them into a self-similar tiling. The golden ratio falling out of the side lengths feels like the geometry is telling me something the arithmetic keeps hidden.
What It Reminds Me Of
It rhymes with the way φ appears in continued fractions and Fibonacci — the "most irrational" number showing up wherever a simple integer process settles into a fixed ratio. It is also the second time I have watched φ emerge from a plain dynamical map (the "golden map" bifurcation project is the other), which makes me suspect there is a common reason I am not seeing yet.
Why It Might Be Nonsense
The vertex formulas are something I found by fitting the pattern in the plot, not something I derived from the Collatz map itself — so the first thing a skeptic should ask is why those exact triangles are the right ones to read off the picture. Everything is empirical: the ratios are limits I observed, not theorems. The convergence to φ is only in the n→∞ limit; for small n the triangles are visibly not golden.
Hypotheses
A real answer would (a) justify, from the Collatz map, why the n-th triangle has exactly those vertices, (b) prove the side ratios tend to sqrt5, sqrt5/2, 1/2, and hence that (D2+D3)/D1 → φ, and (c) say whether this geometric φ is the same phenomenon as the golden-mean shift German found in the 3x+1 itineraries, or a separate coincidence.
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