The angle the ladder returns to

A stem grown at a fixed rise settles on one divergence angle, and the counter is never shown that angle: it reads the arrangement’s spirals from the points and returns a pair. Sweep the rise from coarse to fine and the pair changes at computable places, which is the ladder this collection has been walking for several rounds.

What the ladder has always been drawn as is a staircase of pairs. This essay draws the other quantity — the angle itself — down the whole of both branches, and it does not do what a staircase does.

What a slide would look like #

The convenient picture is a slide. The golden angle is 137.5078°, the ladder is the sequence of Fibonacci pairs, and a stem at a coarse rise sits some way off that angle and creeps towards it as the rise falls. On that picture the divergence is a monotone function of the rise: every value is taken once, the approach is from one side, and asking which rise settles on 137.3° has exactly one answer.

Nothing in the rule promises that. The rule places each organ where a sum over its neighbours is least, and what a stem settles on is an output of that placement rather than a parameter of it. A monotone approach would be a fact about the output, and facts about outputs are measured.

The measurement #

The ladder sweep already has the number in it. Each branch is swept from a rise of 0.0700 down to 0.0040 at a constant ratio of one per cent — a geometric step, because the ladder’s own transitions are geometric and a constant absolute step would resolve the coarse rungs finely and the fine ones hardly at all. At each rise a stem is grown, a counter is run over its points, and the settled divergence is read from the last stretch of the run.

That gives 288 rises a branch and, on the golden one, four rungs: 2/3 from 0.0700 to 0.04702, 3/5 from 0.04655 to 0.01809, 5/8 from 0.01791 to 0.00689 and 8/13 from 0.00682 to 0.00482. The divergences at the ends of those four are 137.930° and 140.703°, then 140.703° and 136.641°, then 136.641° and 137.902°, then 137.902° and 137.785°.

Read that list again as a path rather than as four intervals. It goes up, then down, then up, then down.

Two things about that list are worth pinning before anything is built on it. The first is that the rung bounds are measurements and not choices: a rung is a maximal run of consecutive rises returning one pair, so its ends are the rises at which the counter’s answer changes, found by the counter and not by a person deciding where a rung should be. The second is that the divergence at a rung’s end and the divergence at the next rung’s start are not the same number — 0.04702 gives 140.703° and 0.04655 gives 140.703° as well, but 0.01791 gives 136.641° where 0.01809 gives 136.641°. The two agree at the transitions to the resolution the grid allows, which is what says the curve is continuous through them.

Three turns in four rungs #

The divergence climbs across the 2/3 rung, falls across the 3/5, climbs across the 5/8 and falls across the 8/13. Three changes of direction in four rungs, and the turns happen at the transitions rather than inside the rungs — each rung is traversed in one direction and the direction reverses when the counted pair does.

That is a zig-zag about the limit angle, and it is not a shape anybody drew here before. Every previous figure of this quantity has been a figure of one rung — the slide across the 5/8 that a sweep of a single rung measured, or the flat middle a band is cut out of. Inside a rung there is nothing to see but a slide. The turn is between rungs and needs the whole ladder in one picture.

The excursions, which do not simply shrink #

The obvious reading of a zig-zag about a limit is a spiral closing in: each excursion smaller than the last, the pattern converging on the golden angle as the rise falls. Measured, the widest excursion from 137.5078° on each rung, coarse rung first, is 3.195°, 3.352°, 0.961° and 0.422°.

The second is larger than the first. The 3/5 rung swings further from the limit than the 2/3 rung does, and only after that does the collapse begin — and when it begins it is abrupt, a factor of three and a half between the second rung and the third and another factor of two between the third and the fourth.

A phrase like “converging on the golden angle” would have hidden that. It is true of the last three rungs and false of the first two, and the one number in the set that does not fit is the one worth printing.

There is a second reason not to summarise the four numbers as a trend. An excursion is a maximum over a rung, and the rungs are not the same width: the 2/3 rung carries 41 rises, the 3/5 carries 96, the 5/8 carries 97 and the 8/13 carries 36. A wider rung has more chances to reach a large value, so a sequence of maxima taken over runs of unequal length is not a sequence of comparable quantities. Here that argues in the direction of the finding rather than against it — the two rungs whose excursions collapse are one wide and one narrow — but a comparison that is only valid in one direction should say which.

The collapse has a rate, and it is the ladder’s own #

Refusing to call four numbers a trend is right, and the last three of them do have a rate that can be checked against arithmetic rather than fitted.

The angles a stem’s divergence can be bounded by are the convergents of the limit — 144°, 135°, 138.4615°, 137.1429°, 137.6471° — and their distances from 137.5078° are 6.492°, 2.508°, 0.954°, 0.365° and 0.139°. Each is 0.382 of the one before, which is one over φ squared, and it is exact rather than approximate because that ratio is what a continued fraction of all ones does.

Set the measured excursions beside those. The 5/8 rung’s widest excursion is 0.961° against the 0.954° that bounds it, which is agreement to under one per cent; the 8/13 rung’s is 0.422° against 0.365°, which overshoots by a sixth. The two coarse rungs sit well inside their bounds — 3.195° where 6.492° was available and 3.352° where 2.508° was — so the coarse half of the ladder is not pressing against anything.

That splits the four numbers cleanly and explains why they refuse to be one trend. On the fine rungs the excursion is set by the convergent, so it falls at the ladder’s own factor of φ squared; the measured ratios of 0.287 and 0.439 bracket 0.382 with two points, which is as much as two points can do. On the coarse rungs the divergence never reaches its bound, so its excursion is set by whatever the rule does rather than by the arithmetic, and there is no reason for consecutive values to stand in any ratio at all — which is why the second is larger than the first.

So the honest summary of the column is two regimes rather than one trend: a coarse half where the excursion is unconstrained and does not shrink, and a fine half where it is pinned to a convergent and falls by a factor this collection already measures elsewhere on the same ladder.

The Lucas branch does something else #

The same sweep on the Lucas branch gives four rungs — 1/3, 3/4, 4/7 and 7/11 — and a different shape. The divergences run 104.766° to 102.617°, then 102.422° to 101.777°, then 101.777° to 99.141°, then 99.141° to 99.402°. Down, down, down, and then a single small climb on the finest rung.

One turn in four rungs, against three. The excursions from the Lucas limit of 99.5016° are 5.264°, 2.920°, 2.295° and 0.427°, and those do shrink monotonically. So the Lucas branch is the picture everyone expects and the golden branch is not.

Why the difference matters more than it looks #

A curve that turns takes some of its values twice. A curve that slides takes each value once. That is the whole of it, and it is the reason this measurement is worth a figure rather than a footnote.

On the golden branch, 137.266° is the settled divergence somewhere on the 3/5 rung and again somewhere on the 5/8. 137.844° is reached on the 5/8 and again on the 8/13. 139.297° is reached on the 2/3 and again on the 3/5. Those are pairs of rises with different counted pairs and the same settled angle, and they exist because the curve came back.

On the Lucas branch only one such pair exists, between the 4/7 and the 7/11 rungs, and it exists because of the one turn at the fine end. Every other pair of Lucas rungs has divergence ranges that do not overlap, so no search however fine will produce a match between them.

The asymmetry is worth stating as a negative rather than as a shortfall. It is not that the Lucas branch was searched less carefully; it is that the search has nothing to find there, and a report saying “one match on the Lucas branch” is reporting a property of that branch’s curve. A file that had asserted a floor of two matches per branch would have turned that property into an error, and the measurement would have been thrown away as a bug in the search.

What that buys, stated now and used later #

Sweeping the rise along a rung moves three things at once: the pair a counter returns, the divergence the rule settles on, and which of the two contact steps is the shorter. A result that changes along a rung cannot be attributed to any one of them, which is exactly the position a sweep of one rung was left in.

A band holds the first two and moves the third. What the zig-zag makes available is the design that holds the second and moves the first: two rises, one angle, two different counted pairs. That is the last of the three quantities and nothing here could separate it before.

The turn is not an artefact of the sweep #

Two cheap ways for a turn to be manufactured were checked before anything was built on it.

The first is the sweep’s resolution. At one per cent in the rise the narrowest rung here carries thirty-six rises and the widest a hundred, so a turn is not two points joined by a line; it is a direction held across dozens of consecutive rises and reversed at a transition.

The second is the azimuth grid. The divergence is read from a stem placed on a grid of 1,536 candidate azimuths, which quantises it at 0.234°. The excursions being compared are 0.422° to 3.352°, so the smallest of them is under two grid steps and would be worth doubting on its own. It is not on its own: the sign of the excursion is what the turn is about, and the sign is stable across a whole rung.

What a counter would see #

Nothing. This is a fact about the angle and a counter is never shown the angle; it reads the arrangement’s spirals from the points and returns a pair. Two rises that share a divergence return different pairs, so a counter separates them perfectly and says nothing about the quantity they share.

That cuts both ways, and the second way is the useful one. A real specimen is counted, not protractored — the divergence is the harder measurement and the count is the easy one — so an observation that a plant sits at 137.3° is an observation nobody makes, while an observation that it carries 5 and 8 is made routinely. A curve that takes the same angle at two counted pairs is a curve on which the count carries information the angle does not.

Where the turns sit #

The direction reverses at a transition, which is where the counted pair changes. That is worth stating separately because it is the one place where the two quantities — the pair and the angle — are locked together.

Inside a rung the two are independent: the pair is constant and the angle slides. At a transition both change at once, and the angle’s change is a change of direction rather than a jump. The curve is continuous through the transition on every one of the six transitions swept here; what is discontinuous is its slope.

Two routes to the same pair #

A matched pair can be found two ways and both were run. The first is the sweep: take every rise on both rungs, look for two whose settled divergences agree. The second is arithmetic on the sweep’s own output: take the interval of divergences each rung covers, intersect the two intervals, and if the intersection is non-empty search inside each rung for the rise nearest its midpoint.

The second is what the file does, because it separates the question is there a match from the question where is it. A pair of rungs whose intervals do not overlap cannot produce a match at any resolution, and saying so costs two comparisons rather than a sweep. Of the six pairs of golden rungs, four overlap and four match; of the six Lucas pairs, one overlaps and one matches. So the overlap test and the search agree everywhere, which is the check that the search is finding what the arithmetic says is there and not something else.

What is not answered #

Why the golden branch turns three times and the Lucas branch once is not answered here. Both branches are the same rule at the same rises; the difference is which lattice the stem was started into, and it produces two qualitatively different curves.

Nor is the size of the excursions accounted for. Three of them shrink by factors of three and a half and two; the first two do not shrink at all. A curve converging on a limit at a geometric rate would give a constant ratio, and this one gives 0.94, 3.49 and 2.28 — so whatever sets the excursion is not one parameter with one exponent in it.

And the number of rungs is a property of where the sweep stopped rather than of the ladder. It runs from 0.0700 to 0.0040 because below about 0.004 a stem does not settle onto a lattice at any run length and above 0.0700 the coarse rung has no interior worth sampling. Four rungs a branch is what that range holds. Whether the alternation continues below the wall is not a question this sweep can be extended to answer, because there is nothing to measure down there: the runs that reach a lattice at all settle on 106.1° and 151.3°, which is neither branch.

What is answered #

That the divergence curve returns to its own values, on one branch and barely on the other, and that the return is a property of the rule rather than of the sweep.

Everything the next few essays do rests on that one sentence. A matched pair is two rises the curve reaches the same angle at; without the turn there is no pair, and the third of the three quantities that move along a rung stays where it has been for four rounds — moving with the others, and untestable.

The one line #

Down the golden branch the settled divergence climbs, falls, climbs and falls again, turning three times in four rungs and swinging 3.195°, 3.352°, 0.961° and 0.422° from the limit angle. Down the Lucas branch it turns once and shrinks steadily. The golden branch therefore offers four rises where two different counted pairs settle on the same angle, and the Lucas branch offers one.