Phyllotaxis
the spirals are counted, not admired
Concept

Exceptional chain — where it appears

A residue class of a damage profile sitting ten degrees or more from the level the rest sit at. The classes at the common level are within 5.55 degrees of it and the exceptions are 12.4 to 160.4 away, so the line is a gap rather than a threshold anybody tuned.

Named by 3 essays across one field — 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

Named alongside it

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

Claim testingDamage profileExchangeHonest limitsResidue classFalsifiabilityMeasurement errorAblationArtefactAzimuth gridBalanced pairCensus design

All concepts