Phyllotaxis
the spirals are counted, not admired
Concept

Exchange — where it appears

The pair of adjacent chains that change places in a wrecked stem's profile, one displaced forwards and one back by the same amount. Its direction is the same on every row measured, its size is one divergence step less about a fifth of the surviving hop, and rows without a balanced pair have none.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Three chains, each moved a third of a turn, on g008 cut at offset 6. The eight chains of a stem that kept a lag of 8, set round the circle, with the three that are displaced marked and their displacements written on. Each is about a third of a turn the same way round, and they sum to -359.6 degrees — one whole turn to within half a degree, on an azimuth grid whose step is a quarter of a degree. A rearrangement that closes by going once round sums to a turn, not to nothing, which is why the test it was set had the wrong number in it.

A cycle sums to a whole turn

The test proposed for whether three displaced chains are a three-cycle was that their displacements sum to zero. Three chains rotating into one another's places each move about a third of a turn the same way round, and a third of a turn three times is a whole turn — which the unfolded test calls the worst row in the census.

mechanism · Damage shape How far each rearrangement is from closing, against the lag it kept. One bar per excluded cut, its length the distance from the displacements summing to nothing, ordered smallest first, and each labelled with the lag the cut left standing. The seven shortest bars are the cuts that kept a lag of 7 or 8 and the six longest are the cuts that kept 4 or 5, with a gap of 6.3 degrees between the two groups and no row in it. The rule sorts every one of them.

The lag decides whether it closes

Seven of the thirteen excluded rows have displacements that cancel and six do not. Every row that closes kept a lag of seven or eight and every row that does not kept four or five, thirteen times out of thirteen — and then a lattice nobody had cut broke it.

mechanism · Damage shape The 13 cuts the exchange sets aside, chain by chain. One row per wrecked cut with no balanced pair, one cell per chain of the lag it kept, and the chains displaced away from the common level filled. The lag is printed at the left and the sum of the displacements at the right, folded into half a turn either way. seven of the 13 sum to within 12 degrees of nothing and six do not, and every row that closes kept a lag of 7 or 8 while every row that does not kept 4 or 5. The rows are stacked by that lag.

The rows nobody added up

Seventeen of the census's thirty wrecked cuts come back as one balanced pair of displaced chains, and every claim about the exchange is quantified over those seventeen. The other thirteen were set aside as having three or more exceptions and never looked at again. They are one addition each.

mechanism · Damage shape No majority and no balanced pair, over the whole census. One row per wrecked cut. The bar is the share of that row's residue classes sitting at the level they agree on, and the vertical rule is a half — a bar reaching past it has a majority and a median is safe there. The mark at the right says whether the exchange keeps that row. 15 rows are on both lists of 16 and 16, and the two part on g008/6, which is excluded and has a majority of 5 of 8, and l013/4, whose lag is 4 so that a balanced pair leaves two classes each way and a majority is arithmetically unavailable.

No majority and no pair

Whether a wrecked stem's chains agree on a level and whether its exceptions form a balanced pair are computed from different things. Sixteen rows have neither, sixteen have neither, and fifteen rows are in both — and the two they part on are the two most interesting rows in the census.

mechanism · Damage shape The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement.

A fifth cluster

A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having.

cylinder · Ablation g005 at offset 6, read against two levels. One wrecked cut's four residue classes, drawn twice. On the left the level is the median of the four class means, which falls on the class at -5.5 degrees sitting on its own; three classes are then exceptions and the row is set aside as having no balanced pair, summing to -173.6 degrees. On the right the level is the pair of classes that agree, at -138.6 degrees; two classes are then exceptions, they are adjacent and equal and opposite at 133.1 and 134.2 degrees, and the row is an exchange. The exchange table goes from 20 rows to 21.

The twenty-first row

Recomputing the level moves one row out of the set the exchange sets aside and into the exchange itself. Its hop is four times larger than any the correction was fitted over, and the correction fails on it in the one way it had never failed.

mechanism · Damage shape

Named alongside it

The objects these essays reach for when they reach for this one.

Claim testingHonest limitsExceptional chainDamage profileFalsifiabilityResidue classNegative resultOut of sampleAblationCensus designContact familyHop length

All concepts