Phyllotaxis
the spirals are counted, not admired
Concept

Replication — where it appears

Running the same instrument at the same settings on a second case, to find out whether a result belongs to the design or to the thing it was measured on. Here two bands cut whole disagreed, which put a boundary round a striking finding rather than extending it.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.

A band with nothing inside it

Five offsets wreck on the Lucas 7/11 band and every one of them keeps the same family at every rise it wrecks at. There are no islands, no transition region and no period to look for, which is what makes the picture from the other band a picture of that band.

cylinder · rung interior 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 Twenty rises at the fine end of both branches, cut at every offset. One row per rise searched, coarse at the top of each block. The bar names the counted pair the stem shows and the numbers on the right are the lags its wrecking cuts leave standing. A pale row is a rise whose settled divergence has left the branch it was started from by more than 20 degrees, which is what happens below the ladder's finest rung — the pairs there are 2 and 4, 8 and 16, 11 and 22, which are not two consecutive terms of any additive sequence. One rise on the Lucas branch keeps a lag of 11 while still on it.

One rise below the census

Ten lattices were cut at every offset and their surviving lags came back as four numbers. One rise further down a rung the census already sweeps, three cuts keep a lag of eleven — which is a fifth number, on a lattice nothing about was unusual except that nobody had cut it.

cylinder · ablation The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.

The second band, cut whole

One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it.

cylinder · rung interior Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.

The wrecking set moves again

Which offsets wreck a stem was assumed to be a property of the lattice. On one band it turned out to be a property of the lattice and the rise, changing on nearly a fifth of that band's steps. On the second band it changes more, and one offset's wrecking is broken into five separate stretches.

cylinder · rung interior

Named alongside it

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

AblationClaim testingContact familyHonest limitsLattice offsetRiseRungSamplingHandoverCensus designControlHop length

All concepts