Only Three Gaps

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, a new distinct point always falls into one of the widest gaps and cuts it in two. The largest gap cannot grow. When there are three lengths, a long gap splits into the two shorter lengths already present. At a transition, a split can introduce a new length: for α = 1/4, the third mark splits a 3/4-turn gap into 1/4 and 1/2, and that half-turn length is new.

Second, the smallest gap records how closely a fraction can imitate α. With N ≥ 2 distinct marks, it is exactly min_1≤q<N ‖qα‖, where ‖qα‖ = |qα − p| and p is the nearest integer to qα. This is the scaled error used for a best approximation of the second kind; the fraction itself is p/q. The gaps make α's continued-fraction approximation errors 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.

The number that refuses to clump

If a fraction with a small denominator gets very close to α, some marks nearly coincide. The golden ratio, φ = (1+√5)/2, is badly approximable: q‖qφ‖ stays above a positive constant. Its continued fraction is all ones, and it makes Hurwitz's approximation bound sharp. 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).

Show the check

Every value above is recomputed in your browser from α and N alone; nothing is stored. The browser uses floating-point arithmetic. An offline verifier checks the finite rational cases below with exact integer gaps: take α = p/q and every gap becomes a whole number of 1/q units, with no rounding or tolerance in those gap checks. The universal theorem rests on mathematical proof. The verifier checks:

• 2,160,746 exact cases (every reduced α=p/q with 0<p<q, 2≤q≤220 and 1≤N≤q): the gap count is never more than three, and with three the longest = short + medium; zero violations.
• 4,000 random α with denominators to ~9 million: same, never broken.
• The smallest gap equals the minimum scaled approximation error min_1≤q<N ‖qα‖ in all 826,129 tested cases with N≥2 distinct marks.
• The new point always lands in a current largest gap, which never grows — 2,166/2,166.
• The rational golden proxy F₄₁/F₄₂: gap ratio ≤ φ² + 10⁻⁹ at 714 sampled counts N=5,12,…,4996. The near-quarter turn exceeds 600,000× in those samples. Gap ratios use floating-point division.
• The same proxy has exactly two lengths at the 27 tested Fibonacci counts F₄=3 through F₃₀=832040. For the true golden angle, two lengths occur at Fibonacci N≥2; N=1 has one full-turn gap.

→ research/three-gap/verify.mjs · 9/9 · check log in research/three-gap/facts.md

What this layer does and does not claim

The leap to all real numbers: the verifier checks every reduced fraction α=p/q with 0<p<q, 2≤q≤220 and 1≤N≤q, plus 4,000 random rational cases. That the theorem holds for every real α is the cited result of Sós, Surányi, and Świerczkowski; a finite computation does not prove the universal claim.

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": the golden angle's largest-to-smallest gap ratio is bounded by φ² for all N, while the displayed near-quarter turn reaches 625,000.25× at N=5. This is a finite spike: that rational turn eventually gives equally spaced distinct marks. The golden bound follows from the three-distance gap structure; it does not say the golden angle wins at each fixed count. Nor is φ 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.