Three gaps and no more

Every system in this field so far has been complicated. Here is the tamest one there is, and it has a theorem in it that nothing so far has prepared for.

Take a circle. Turn it by a fixed fraction $\alpha$ of a full turn, mark the landing point, and repeat. If $\alpha$ is rational the points close up into a finite cycle and stop. If it is irrational they never repeat, and the orbit fills the circle.

Now count the gaps between neighbouring points. There are twenty-one of them, and they have two distinct lengths. Take another step and there are twenty-two gaps with three lengths. Keep going, at any number of steps, with any angle: never more than three.

The three-distance theorem #

That is the statement, and it is worth being precise about how strong it is.

For any $\alpha$ and any $n$, the $n$ points $\{0, \alpha, 2\alpha, \ldots, (n-1)\alpha\}$ taken modulo one divide the circle into $n$ arcs, and those arcs have at most three distinct lengths. When there are three, the largest is the sum of the other two.

No hypothesis on $\alpha$ — it holds for rationals too, where the repeated points contribute gaps of zero. No exceptions for special $n$, and no error term. The theorem was conjectured by Steinhaus and proved several times over in the 1950s, and the number of independent proofs is a sign of how many directions it can be approached from.

What makes it surprising is that nothing about the construction suggests a small number. Twenty-one points dropped anywhere on a circle would give twenty-one gaps of twenty-one different lengths. These are not dropped anywhere; they are the orbit of a rotation, and the rotation constrains them far more than it appears to.

Why three and not more #

The mechanism is a shift argument, and it fits in a paragraph.

Each point $k\alpha$ has a neighbour on its clockwise side. Which point is that neighbour? It is $(k+j)\alpha$ for some $j$, and the gap has length equal to the distance $j\alpha$ travels — which depends on $j$ only, not on $k$. So all gaps produced by the same $j$ have the same length, and the number of distinct gap lengths is the number of distinct $j$ values in use.

That is where the counting happens. The value of $j$ can only change at the two ends of the sequence — for points near the start and near the end of the orbit, the neighbour that would have been used does not exist yet. Everywhere else it is one fixed $j$. That gives one common length and two exceptional ones, and the sum relation follows because the largest gap is the one that has not yet been split.

The whole argument is about which points are missing, not about the geometry of the circle, which is why it holds for every $\alpha$ without a case analysis.

The angle that spreads them best #

All irrationals give three gaps. They do not all give good gaps, and the difference is the whole reason $\varphi$ appears in this essay.

The gaps are most even when the largest and smallest are closest in size. For most angles this fails badly at some stages: the orbit produces many tiny gaps and a few enormous ones, then evens out, then goes lopsided again. The stages where it is worst are exactly the stages just after a good rational approximation to $\alpha$.

That figure is what a good rational approximation does to a rotation. $\pi \approx 22/7$ is accurate to a part in two and a half thousand, so the orbit nearly repeats after seven steps and lands almost on top of itself. Compare the hero figure, where the two lengths are $0.0344$ and $0.0557$ — a ratio of $1.62$, which is $\varphi$, and is the closest any angle gets.

The golden ratio is the number for which this never happens badly, because it is the hardest number to approximate by fractions. Its continued fraction is all ones, the convergents advance as slowly as any can, and no rational ever gets a bargain. So the rotation never nearly-closes, and the points stay as evenly spread as the three-gap constraint allows, at every number of steps and not merely at good ones.

Two gaps at the Fibonacci counts #

The hero figure was drawn at twenty-one points and not twenty, and the reason is worth making into a pair of figures rather than a remark.

Reading down the three golden figures gives $0.0902, 0.0557, 0.0344, 0.0213$ — each the previous one divided by $\varphi$, and each pair of consecutive values the two gap lengths at some stage. The gap lengths are a geometric sequence in $\varphi$, which is the arithmetic version of the statement that this rotation looks the same at every scale.

Nothing similar happens for a general angle. For $\sqrt2 - 1$ the analogous counts are the denominators $2, 5, 12, 29, 70$, and the gap ratio there is $1 + \sqrt2 \approx 2.414$ rather than $1.618$: still bounded, still eventually even, and never as even.

The connection to fractions #

The two exceptional gap lengths are not arbitrary numbers; they are quantities this collection has already computed.

At any stage, the three lengths correspond to the shifts $j$ that are denominators of continued fraction convergents to $\alpha$, or intermediate steps between them. So the sequence of gap patterns as $n$ increases is the continued fraction expansion of $\alpha$, read out geometrically.

For $\varphi - 1$ the convergent denominators are the Fibonacci numbers — $1, 2, 3, 5, 8, 13, 21, 34$ — and those are exactly the step counts at which the picture is at its most even, with only two distinct gap lengths rather than three. The hero figure has twenty-one points for that reason.

That is the tie this essay exists to make. The continued fraction of an angle, which looks like a fact about arithmetic, is the same object as the pattern of gaps its rotation leaves, which looks like a fact about a circle.

The gaps split in a fixed order #

Watching the pattern change as points are added makes the theorem feel less like a coincidence, because each new point does exactly one thing.

Add a point. It lands inside one of the existing gaps and splits it in two. Which gap? Always one of the largest ones, and always at the same relative position within it, so the two pieces are always the same two sizes. So the process is: keep splitting largest gaps into a fixed pair of smaller ones until all the largest are used up, then start on the new largest.

That immediately gives the bound. At any moment there are gaps of the current largest size, gaps of the two sizes it splits into, and nothing else — and once every gap of the largest size has been split, one of the three lengths disappears and the count drops to two. The stages with two lengths are exactly the moments a round of splitting finishes, which is why they are the Fibonacci counts for $\varphi$ and the convergent denominators in general.

The picture of a length repeatedly split into two smaller ones in a fixed ratio is Euclid’s algorithm run as squares, and it is the same process: subtract the smaller from the larger, repeat, and the sequence of counts is the continued fraction. The circle version and the rectangle version are the same algorithm drawn on different objects, which is why the same integers come out of both.

The parallel goes one step further. The two gap sizes at any stage are neighbours in the Farey sense — their numerators and denominators satisfy the same unimodular relation that makes the mediant land between them — and the point that splits a gap is exactly the mediant of the two convergents bounding it.

Where it is visible #

Rotation by $\varphi$ is not a curiosity; it is the arrangement plants use.

Successive leaves or seeds placed at $137.5°$ — which is $\varphi - 1$ of a turn measured the other way round, since $0.618$ of a turn clockwise is $0.382$ anticlockwise — end up spread as evenly as possible around the stem for every count of leaves, which is what a plant needs, since it does not know in advance how many it will have. Any rational angle would eventually stack leaves directly above one another; any less irrational angle would go through stages of clumping like the $\pi$ figure above.

The three-distance theorem is why this works at every stage rather than only at the end, and the golden ratio is why the three distances stay close in size. Neither fact is about botany, and both are consequences of a rotation with a badly approximable angle.

Every arc gets its share #

There is a second theorem here, older and in a different direction, and the two together say almost everything about the rotation.

Weyl’s equidistribution theorem, from 1916: for irrational $\alpha$, the fraction of the first $n$ points that land in any given arc converges to the length of that arc. Not merely “the orbit visits everywhere” but “it visits everywhere in the right proportion”, which is a much stronger statement and the reason irrational rotations are used as sampling schemes.

The two theorems answer different questions and neither implies the other. Equidistribution is asymptotic — it says what happens in the limit and says nothing about any particular $n$. The three-distance theorem is exact at every $n$ and says nothing about the limit. In practice the second is the useful one, because a sampling scheme is used at whatever count it was stopped at, not in the limit.

The elementary half of this is a counting argument the collection has already met. Divide the circle into $m$ equal arcs and take $m+1$ points of the orbit; two of them must share an arc, by the pigeonhole principle, and their difference is a multiple of $\alpha$ landing within $1/m$ of zero. Iterating that multiple steps round the circle in fine increments, so the orbit is dense. Dirichlet’s approximation theorem is the same argument stated about fractions instead of arcs, and the multiple it produces is a convergent denominator — which is the third time in this essay that the same integers appear from a different direction.

What the rotation is good for #

Because the points are even at every count and never repeat, the sequence is a practical thing rather than only a pretty one.

Sampling without a grid. To estimate an average over an interval, a grid of $n$ points commits to $n$ in advance and starting over with $2n$ throws the old work away. The golden rotation does not: the first $n$ points are near-evenly spread for every $n$, so a computation can be stopped, inspected, and continued. That property is the whole selling point, and low-discrepancy sequences built on the same idea are standard in numerical integration.

Deterministic in place of random. A random sample of $n$ points has gaps of wildly varying size — clumps and voids are what randomness looks like, and they cost accuracy. The rotation has a bounded worst gap, guaranteed rather than probable, which converts an error that shrinks like $1/\sqrt n$ into one that shrinks nearly like $1/n$.

Choosing a step for a lattice. Anything that must step through a cyclic structure without falling into a pattern — a hash probe sequence, a colour assignment, an angular offset for repeated marks — wants a step whose multiples do not nearly close. That is arithmetic modulo a wrap with the multiplier chosen for how badly it approximates a fraction, and the answer is again the golden ratio scaled to the modulus.

None of the three needs a theorem about chaos. They need a rotation that stays even at every stage, which is what this whole essay establishes.

The dynamics are as simple as it gets #

Set against the rest of this field, the rotation is a control case, and what it lacks is as informative as what it has.

No sensitivity. Two starting points a distance $d$ apart stay exactly $d$ apart forever, because rotation is an isometry. The Lyapunov exponent is zero — not negative, not positive. Every digit of the starting position is worth exactly as much after a million steps as it was at the start, which is the property every chaotic map on this site lacks.

No mixing. The orbit visits everywhere, so it is ergodic, but it does not mix: a small arc rotated a thousand times is still a small arc of exactly the same size, just somewhere else. Nothing is ever stretched or folded, so two points that start close never explore independently — the orbit of one is the orbit of the other, shifted.

Complete predictability. Position after $n$ steps is $n\alpha$ modulo one, in closed form. There is no transient, no attractor, and nothing to compute numerically — the millionth point is as cheap to name as the second, which is the exact opposite of the irreducibility the last two essays were about.

It is the system a random walk is the opposite of, too. The walk has an unpredictable rule and a completely regular long-run distribution; the rotation has a completely predictable rule and a distribution that is regular for a different reason — not because fluctuations average out, but because there are no fluctuations at all. Two routes to the same evenness, one statistical and one arithmetic.

And yet the orbit never repeats, fills the circle densely, and produces a non-obvious counting theorem. So irregularity and unpredictability come apart here: this system is irregular forever and predictable forever, which is the case chaos is usually contrasted against and is worth having drawn.

That last assertion is the one that makes the pair of figures an experiment rather than an illustration. The same code, the same number of steps, and a different angle, with the difference between a closed cycle and a dense orbit turning entirely on whether one number is a ratio of integers.