Walk around a circle in equal strides and drop a mark at every step. Wherever you stop — three steps in, three hundred — the gaps between your marks come in at most three lengths. Never a fourth.
Pick any number — call it α. Start at the top of a circle and turn by α of a full turn. Mark the spot. Turn by α again, mark again. Keep going. You are dropping the points {α}, {2α}, {3α}, … — the fractional parts of the multiples of α — around the rim.
Those marks chop the circle into arcs. You would expect the arcs to be a mess: a long one here, a sliver there, every width different. They are not. However many points you have placed, and whatever α you chose, the arcs come in no more than three distinct lengths — and when there are exactly three, the longest is precisely the sum of the other two. That is the whole theorem. Hugo Steinhaus noticed it; it took three separate proofs in the late 1950s to nail it down.
Below, the circle draws itself. Add points and watch. The arcs are coloured by length: short, medium, long. A fourth colour never appears.
Two things are worth slowing down for. First, the new point always falls into one of the widest gaps and cuts it in two — it never lands in a sliver. So the biggest gap only ever shrinks; the construction tidies itself. That is why no arc is ever left absurdly large, and why a fourth length can't accumulate: each step replaces a long arc with two shorter ones drawn from the lengths already present.
Second, the smallest gap is not arbitrary — it is the best fraction you could use to fake α. The closest two of your marks ever come is exactly ‖qα‖, the error of α's best rational approximation with denominator below N. The gaps, in other words, are α's continued-fraction approximations made visible. (The readout names the fraction.) This is the hinge that turns a fact about arcs into a fact about which numbers are hard to approximate — and it ties this layer to The Most Irrational Number.
If the gaps are best approximations, then the number whose marks stay evenest for the longest is the number that is hardest to approximate by fractions — the one whose denominators have to grow fastest before they catch it. That number is the golden ratio, φ = (1+√5)/2, whose continued fraction is all ones, the slowest-converging there is. Its turn is the golden angle, ≈ 137.5°.
The payoff is exact and a little startling. For the golden angle the ratio of the largest gap to the smallest stays bounded by φ² ≈ 2.618 forever — the marks are within a factor of three of perfectly even at every count, not just in some limit. Nudge α a hair off a simple fraction — to 0.2500001, say — and the marks rush toward four spokes before crawling onward: the gap ratio runs past 10⁵. Even π misbehaves where it is well-approximated — its famous near-miss 355/113 shows up as a spike to ≈ 290×. Watch both at once — the running gap ratio as you add points, and the same points grown into a seed head by Vogel's rule (angle n·α, radius √n).
Every value above is recomputed in your browser from α and N alone — nothing is stored. But a browser uses floating-point, so the proof that the count is never four lives in an offline verifier that works in exact integer arithmetic (take α = p/q and every gap becomes a whole number of 1/q units — no rounding, no tolerance). It checks:
→ research/three-gap/verify.mjs · 9/9 · check log in research/three-gap/facts.md
The leap to all real numbers — the page proves the theorem for fractions, exhaustively; that it holds for every real α (including the irrationals the sunflower actually uses) is the cited theorem of Sós, Surányi, and Świerczkowski, not something a finite computation can finish. Said plainly, not smoothed.
The exact count of each length — exactly how many short, medium, and long arcs there are at a given N, and precisely when there are two versus three, is given by α's continued fraction (the Steinhaus–Slater structure, N. B. Slater, 1967). We cite that formula rather than re-derive it; the things we display — the best-approximation identity and the golden two-lengths-at-Fibonacci fact — are the ones we verified.
"Golden packs best" — shown here as: its gap ratio is bounded for all N while a near-rational angle's runs away. That is a sharp, true comparison. It is not a claim that φ is uniquely optimal by every measure a botanist might use — the phyllotaxis literature is richer than one ratio, and real seed heads are shaped by more than geometry.