Two lines that cross once

Inside every rung there is a rise at which the two contact steps change places. That is a measurement — sweep the rise, read the hop ranking, watch it swap — and measured across eight rungs it always lands in the coarse half. A measurement that comes out the same way six times invites the question of what would have made it come out otherwise, and here that question has an answer that needs no stems at all.

The curve where two families are equal #

Fix a rise. The step joining organs k places apart runs some fraction of the way around the cylinder and k times the rise up it, so its length depends on the divergence and the rise and on nothing else. Ask for the divergence at which the p-step and the q-step are exactly equal, and there is one, found by bisection in a second.

Do that at every rise of a rung and the answers trace a curve. Across the golden 5/8 rung it runs from 135.80° at the coarse end to 138.11° at the fine one — a climb of 2.281°. Nothing has been grown to produce it. It is a statement about lattices, in the sense that if the arrangement of points is a lattice with those two counts, this is the divergence that would make its two nearest-neighbour families equidistant.

That figure is worth a moment, because the curve is not new to this subject even if this collection has not used it this way before. The van Iterson diagram is built from precisely this condition; each of its branches is a locus of equal contact distances, and the tree structure comes from the places where a third family joins the tie.

The curve the rule actually sits on #

Now the second line. Grow a stem at each rise under the placement rule — put each new organ where the sum of inverse powers of distance to the existing ones is least — and read off the divergence it settles at. That is a measurement, one stem per rise, and it is the quantity every table here quotes.

Across the same 5/8 rung it climbs from 136.64° to 137.87° — 1.262°, a little over half the climb of the balanced curve.

Two smooth lines, one about twice as steep as the other, over a range where the shallower one starts above the steeper. They cross once. That crossing is where the two step lengths are equal, which is to say it is the handover.

Why the balanced curve is worth computing at all #

There is a version of this subject in which the balanced curve is the whole story. It is the version the classical diagram encodes: lattices on a cylinder, parameterised by rise and divergence, with the touching-circle condition drawing a tree through the parameter space. Every rung of the ladder is one branch of that tree, every transition is a node, and the numbers that come out are the Fibonacci ones.

What that version does not contain is a rule. It says which lattices are possible and which are equidistant; it says nothing about where a growing thing ends up. This collection’s whole apparatus is on the other side of that distinction — a placement rule with no reference to spirals in it, and a counter that never sees the divergence — so the two objects have been kept apart on purpose, and comparing them has needed a reason.

The handover is that reason. It is a property of the rule’s own stems, measured by growing them, and it turns out to be located by a condition from the classical diagram. That makes the comparison a result rather than a juxtaposition.

What that explains #

Three things, and none of them needed the sweep that found them.

Why there is exactly one handover per rung. Two smooth curves cross once when one is monotonically steeper than the other across the whole interval, which is what these are. There is no possibility of a rung with two handovers unless the rule’s line changes its slope relative to the balanced line, and nothing in the range measured comes close.

Why it is in the coarse half. The balanced line moves about twice as far across a rung as the rule’s line does. A shallower line starting above a steeper one, with both spanning the same interval, crosses nearer the start when the ratio of slopes is large. At a ratio near two the crossing is around a third of the way in, which is where these are: six, twelve, fifteen, thirty-three, thirty-five and forty per cent.

Why the rule is never on the balanced curve. This is the part that took the longest to see and is the most interesting. The rule does not sit where the two families are equidistant. It crosses that condition and leaves it, once per rung, and spends the rest of the rung on one side or the other.

That is worth stating plainly because there is an appealing story in which the rule seeks the balanced condition — a lattice with equal nearest neighbours is the most even one, and evenness is the intuition every popular account of this subject reaches for. The measurement says otherwise. The rule’s line and the balanced line agree at one rise per rung and disagree everywhere else, by up to 0.9° on the 5/8 rung, which is several times the scatter of the rule’s own settled value.

What the rule is doing instead #

The honest answer is that this collection does not know, and the shape of the disagreement narrows it usefully.

The rule’s line is shallower. It moves less than the balanced condition demands as the rise falls, so it starts a rung above the balanced value and ends below it. A rule tracking the balanced condition would have the same slope; a rule ignoring it entirely would have no particular relation to it. What the measurement shows is a rule that follows the same trend at about half the rate.

One reading is that the rule is responding to more than two families. The balanced curve is a condition on two of them; the placement rule sums over the whole neighbourhood, with a falloff that reaches tens of organs rather than two. A condition involving the third and fourth families would sit somewhere else, and the rule’s compromise between them would be shallower than any single pairwise condition. That is testable — compute the divergence balancing the second and third families, and see whether the rule’s line sits between the two conditions — and it has not been done here.

The other reading is that there is nothing to explain: the rule settles where it settles, the balanced curve is a different function, and their crossing is the coincidence of two lines in a plane rather than a fact about anything. That reading is harder to refute than it sounds, and the argument against it is the one this essay opened with — the crossing predicts the handover, which is an independently measured quantity on six rungs.

The size of the disagreement #

A gap of 0.9° sounds small for an angle that is itself around 137°, so it is worth putting beside the quantities this collection measures things against.

The rule’s settled divergence has a scatter — the run-to-run spread of the last sixty angles — of about 0.1° to 0.2° on these rungs. So the gap between the rule’s line and the balanced line is four to nine times that scatter at the ends of a rung: not a near miss, and not something a longer run or a finer grid would close.

It is also large compared with the slide the rule’s line makes across a whole rung, which is 1.262°. In other words, the distance from the balanced condition at the coarse end of a rung is most of the distance the rule travels across the entire rung. The two lines are not nearly the same line with a crossing in it; they are two lines that happen to intersect.

Against the grid, finally: the azimuth grid is 1,536 steps, or 0.234° each, so 0.9° is about four grid steps. That is comfortably resolved, and it is the reason the flat band needed re-measuring on a finer grid while this comparison did not.

Two routes to one number #

That prediction is the check worth stating carefully, because it is the only evidence here that the arithmetic is about the same object as the sweep.

The handover was measured by growing stems: sweep the rise at one per cent, read each stem’s settled divergence, compute the hop ranking, and note where the first two entries swap. The crossing was computed by bisection on a formula, with no stem anywhere in it. They agree on all six rungs to within one sweep step, which is one per cent in the rise.

Agreement between a measurement and an arithmetic prediction is the strongest thing a figure here can do, and it is this collection’s habit rather than an achievement: a cylinder’s ladder predicting a disc’s counts and a branching tree predicting a dynamical model’s attractors are the same shape of argument. What it buys is not certainty about the mechanism but certainty that two unrelated failure modes would have had to fail the same way.

The thing this makes cheap #

Once the handover is computable, an experiment that needs to straddle one is a morning’s work rather than an exploration. Compute the crossing, grow a band of rises around it, check that the counted pair holds and the divergence does not move, and cut.

Both bands in the experiment that followed were located this way. Neither needed a search: the arithmetic said where to look, the sweep confirmed it, and the expensive part — a hundred and fifty cut stems — went entirely on the question rather than on finding the place to ask it.

The refutation this could have been #

It is worth naming what would have falsified the account, because an agreement between two curves is the kind of thing that can be arranged after the fact.

The balanced curve was computed for a stated pair, and which pair to use is fixed by the rung rather than chosen: on the 5/8 rung the condition is on the five-step and the eight-step, and nothing else was tried. Had the crossing landed at the wrong rise, there would have been no second pair to fall back on that also gives 5/8.

The curve was computed before the handovers were re-read at one per cent, not after. The earlier sweep, at a coarser step, put the golden 5/8 handover at 0.0155 and the arithmetic said 0.0154; the finer sweep moved the measurement to 0.0156 and the arithmetic did not move. That is the right order of events for a prediction, and it is the reason the agreement is reported as one.

And the account makes a claim it could still fail on: any rung anywhere, on any branch, should have its crossing where the bisection puts it. The two rungs without a measured handover are the test cases nobody can run at present, because their crossings sit above the coarsest rise this collection’s stems settle at — but a coarser sweep would reach them, and if either disagreed the account here would be in trouble rather than merely incomplete.

There is one more thing the account does not do, and the collection has a habit of saying so. It explains where the ordering changes hands. It does not explain why the rule’s line has the slope it has, which is the quantity doing all the work. That number — a little over half the balanced line’s slope, on both branches — is measured on two rungs and unexplained on both.

What is left #

The obvious next computation is the one named above: the balanced condition for other pairs of families, and whether the rule’s line sits inside the envelope they make. It costs nothing — the same bisection on different lags — and it would say whether “a compromise across the neighbourhood” is a description or a phrase.

A second one is whether the ratio of slopes is itself computable. On both rungs measured it is close to two — 2.281 against 1.262 on the golden 5/8, and 3.537 against 2.602 on the Lucas 4/7, which are 1.81 and 1.36 — so “about twice” is already an overstatement of a quantity measured twice. Whether it varies with the pair, with the branch or with neither is four bisections and four sweeps away, and it is the number that would turn the account here from a description into a prediction of where a handover sits rather than a confirmation of where it was found.

The less obvious one is what happens at a transition. A transition is where a third family joins the two, which on the van Iterson diagram is a node. If the rule’s line crosses the balanced curve once per rung, then near a transition it is some distance from it, and how it hands over from one rung’s condition to the next’s is not something any figure here draws. The forks have a closed form and the rule’s approach to them does not.