Inspired by Henri Poincaré (qualitative dynamics). We study the irrational rotation map T(x) = (x + α) mod 1 with α = (3−√5)/2 and ask whether the orbit behaves as theory predicts. Following P3, the page delivers an Answer, a Reason Why, and an independent Check suite.
We rotate the unit circle by a fixed irrational angle α = (3−√5)/2 — the golden angle — which is about 0.381966 of a full turn, i.e., ≈ 137.5°. The orbit of 0 never repeats, returns arbitrarily close to its start, and spreads evenly around the circle. In finite data (N = 5,000 steps) this appears as near‑uniform bin counts, at most three distinct gap lengths, and tiny low‑order Fourier modes. The checks below confirm this behaviour.
Rotation by an irrational angle preserves length and never closes up. Because α is irrational, the points {nα mod 1} do not repeat; they become evenly spread (equidistributed). A length‑preserving map on a bounded space also has the Poincaré recurrence property: some iterates come back arbitrarily close to where they started. For any finite N, the circular gaps between the sorted points take at most three sizes (the three‑gap theorem). These facts explain why the orbit looks uniform and why our specific tests should pass.
Unit circle, marked with the first 60 orbit points starting at 0. Each step advances by a fixed angle α; the blue arrow shows one such step. The cloud of points hints at equidistribution while the arc lengths suggest the three‑gap phenomenon.
Each diagnostic targets a different facet: validity, bijection, nonperiodicity, coverage, distribution, recurrence, structure, and spectral flatness. Expand any row to see details.