Nearest neighbour — where it appears
Named by 39 essays across 7 fields — each of them below, with the objects they name alongside it.
Counting the spirals
Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by a count that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.
Packing, measured four ways
The claim is that the golden angle packs best, and it is measurable. Read on the interior of a head, the two criteria about distance put the golden angle first among the angles near it and the two about cells are won by rational angles — which makes the claim half right, and makes the right half a statement about a class of angles. An earlier reading of the same four criteria, divided by the cells at the head's edge, said the opposite.
What a spire buys
Translation along the coiling axis enters the contact boundary as its square, so a shell that has only just begun to walk along its axis has not moved the boundary at all. It is also strictly one-way, and it has a threshold above which no distance from the axis whatever puts the whorls in touch.
Counting without an index
A person counting spirals on a cone puts a finger on one scale, follows a family round, and counts how many distinct chains there are. That needs no order of arrival — and building it turns out to be strictly more general than the counter that reads the order of arrival, and to find a bug in the counting of a bijugate stem that nothing had caught.
Errors that pass between organs
An organ's neighbours are the ones eight and thirteen places back — that is what a parastichy pair is. So a disturbance transmitted by contact is correlated at exactly the two lags the readout examines, and it does not have to be told them. Driven into a lattice with no rule in it, it returns the counted pair on eight stems out of eight.
What a forgery has to know
A lattice with transported errors reproduces the comb and the pair, so one quantity is left: the two combs' relative strength. Weighted by distance the forgery puts more in the second comb than the first; the rule does the opposite. It matches only if the coupling is turned three to one towards the further neighbour, which no falloff supplies.
The organ that was taken away
Every observable this site has is read off an arrangement that was finished before the reading began, and earlier work here showed what that costs. So remove one primordium from a settled stem and place the next one against what is left. The rule has to answer. The rival account cannot, because in it no organ's position was ever computed from its neighbours.
A period that is not a count
Eighteen wrecked stems settle into a block whose period is one of their own spiral counts, and one settles into a block of four on a lattice counted 8 and 13. The odd one is not noise. It is the case that shows what the rule is actually conserving, and it is the reason this thread is about lattice steps rather than about spirals.
The exponent that barely matters
An unchecked claim, repeated since the first essays, held that the repulsion's falloff exponent hardly changes the answer. It could not be tested, because the function that would have taken it never passed one down. Tested at last, it is true on a disc — by three and a half degrees across a sixteenfold range — and on a stem it decides whether there is a pattern at all.
Fractions with the same neighbours
Every instrument this collection has for the width of a disorder dip has a free parameter set by how close the next rational sits — which makes a hypothesis about the neighbourhood untestable with any of them. The repair is not a better instrument. It is a set of fractions whose neighbourhoods are identical and whose denominators are not, and the arithmetic supplies twenty-four of them.
The second comb
The autocorrelation of a divergence sequence has peaks at the smaller parastichy number and at every multiple of it. It also has a second set of peaks, at the same spacing, offset by the difference of the pair — so a list of angles with no coordinate in it returns both numbers rather than one.
A harmonic is a step taken twice
The spectrum contains the larger parastichy number, their sum, and echoes of the smaller one, and no ranking of peak heights separates them. What separates them is arithmetic: a harmonic is a multiple of the spacing and a family is not, and the two kinds sit in different residue classes.
Two shapes, one threshold
Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.
The boundary belongs to the pattern
Three kinds of noise, in three incommensurable units, destroy a lattice at the same place — about a degree and a half of divergence scatter. The earlier work measured that of two kinds and called it a scale rather than a constant. With a third it looks less like a coincidence and more like a property of what a lattice is.
The nearest organ is not the nearest neighbour
Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit.
The ratio was the floor of a curve
One number was left standing between a placement rule and a transported disturbance, measured at one rise, with the explanation that the geometry there happens to favour the larger parastichy number. Swept across two rungs the number is a U — a floor of about 0.79 two thirds of the way up a rung, climbing past 2.8 as a transition approaches — and the geometry is flat exactly where the curve is steepest.
A counter that cannot be slid
The counter that needs no order of arrival follows each family into chains and counts them, and on an ideal lattice it agrees with the counter that does. On a stem whose rise falls it works only inside a band of widths, and outside the band it returns a pair rather than refusing: the rung below when the band is too narrow for the larger count, a pair on no rung when the band spans more than about a third of a rung of rise. The upper edge moves with the rate, so a width that is right on one stem is wrong on another, and on the fastest bijugate stem measured no width works at all.
Removing a neighbour costs least
Take away an organ that is a direct chain-neighbour of the growing tip and the next organ moves by under thirty-one degrees. Take away anything else inside the front and it moves by at least sixty-three. Thirty cuts, two groups, a factor of two between them and nothing in the gap.
Packing, measured against the interior
An earlier reading of these heads reported that no packing criterion singles out the golden angle and that three criteria give three winners. Every one of those readings was divided by a mean cell area that, on a head of 150 organs, was 38.8 where the interior's is π. Divided by the interior's own, the criteria about distance put the golden angle first of 72 angles and the criteria about cells go to rational ones.
The residual was the window
After the depth and the q over n squared scale are taken out of a disorder dip, something looked left over and looked ordered by how crowded the fraction's neighbourhood is. Measured on fractions whose neighbourhoods are identical by construction, seven widths across a factor of two and a half in denominator agree to one per cent. There is no residual; there was a comparison made at different effective windows.
One level and two exceptions
Inside a wrecked stem's period most residue classes sit at one level and a couple do not. On seventeen of the thirty cuts the exceptions are exactly two, equal and opposite to within five per cent — and on all seventeen they are neighbouring residues, which was not looked for.
A disturbance the organs share
This collection has put three kinds of noise into the placement rule and found the lattice fails at about the same recorded scatter whichever kind it was. None of them asked what happens when the displacements are correlated between organs. At equal displacement per organ, a lattice survives three times as much of a disturbance the organs share — and what a protractor records is the part they do not.
The disturbance that travels
If a lattice survives three times the displacement when the organs share it, then a disturbance passed between the organs that actually touch should be the gentlest of all — it is correlated at exactly the offsets the rule places against. It is the harshest. Half the displacement destroys what independent noise leaves standing, and the reason separates two things that had been one.
Both walls of the slot
The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°.
The first three hundred organs
Over the counts a head shows, forty-five angles resist approximation within a per cent as well as the golden angle, and what separates them is at the centre. Grown as heads and measured there, the golden angle has the widest closest pair of all forty-six — by organs 1 and 4, the count its arithmetic names — and keeps first place only while the centre is in the reading. Its rivals stay a per cent apart from it out to a radius that tracks where their spiral counts start to add up, and every one of them is within a per cent by the 289th organ. By the largest hole it is never the best.
A removal that changes nothing
On one of the six lattices, taking away both walls of the slot moves the next organ 11.953125° — which is exactly, to the last digit, what taking away the larger wall alone moves it. The smaller wall's removal contributes nothing at all when the larger one is already gone.
The organ that guards the second slot
An organ twelve places back is the furthest of any from where the next one goes, and removing it moves the next one by a whole divergence. The reason is that the rule's profile has two low points rather than one, the second is the slot after next, and that organ is holding it up. The comparison between what it holds up and how far behind it is decides the whole thing.
The rung that two organs wreck
On the coarse 3/5 stem both walls of the slot heal when removed alone and wreck when removed together — and the wreck keeps no rigid hop at all. Two of the six pairs in the design end at a destination single removals almost never reach.
Where a handover sits
Inside every rung there is a rise at which the two contact steps change places, so that the shorter hop belongs to the other family below it. Six of the eight rungs on this ladder have one, each has exactly one, and every one of them sits in the coarse half.
Two lines that cross once
The divergence at which a lattice's two contact steps are exactly equal is a curve across each rung, computable from the geometry with nothing grown. The rule's own settled divergence is a second, shallower curve, and where they cross is where the step ordering changes hands.
A band that holds the angle still
Around every handover the settled divergence has a shallow floor, so a run of rises either side of it share a divergence to a twentieth of a degree while their two contact steps change places. That is a matched pair with one quantity varying, and it is the design the ablation thread had no way to state.
The hop that survived
A stem that never repairs after an organ is removed settles into an exactly repeating block of angles, and the period of that block is a spiral count of the lattice it was cut from. Nobody could say why. Read the wrecked stem by lags rather than by neighbours and the answer is one line: one family of the original lattice is still standing, organ by organ, and the block is its period.
Six lattices were not enough
The interaction between the two walls of a slot came back at −25.8° to +132.9° on six lattices, three above zero and three below, with no ordering by rise, by counted pair or by branch. A quantity that looks free on six rows is usually a quantity that has been sampled at six rows.
When the second wall is free
On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.
The rung decides the sign
Twenty-four lattices where both walls of the slot are really there. Thirteen give a strongly positive interaction, at 85° to 135°; eleven give a negative or null one, at −25° to −0.5°. Nothing lies between. Every rung's lattices fall on the same side as each other.
A survivor has to be a neighbour
A stem that never repairs after a removal keeps exactly one lattice hop rigid, and nothing predicted which one. Sweep every offset at twelve lattices and the answer narrows sharply: at twenty-nine of thirty the surviving hop is one of the two families a counter returns, and the one exception is a step six times too long to be one.
Not the shorter of the two
If a damaged stem keeps one contact family standing, the obvious guess is that it keeps the nearer one. Across thirty wrecked offsets that is true twelve times and false seventeen, and on one lattice the two steps differ by a quarter of a per cent — where the words shorter and longer are doing no work at all.
The ordering was not the actor
Cut an organ out of every rise of a band where the counted pair and the settled divergence are held and the two contact steps change places, and the family left standing does not change. Fifty-five wrecked cuts on two branches, and the ordering reverses underneath every one of them.
The organ that moved furthest
A reading that works on nine of thirty rows needs a reference organ, and the obvious repair is to measure one rather than to choose it. Measured, the disturbance turns out to have no far edge at all, so there is no organ that moved furthest in any sense the reading can use — and the generalisation that does work needs no reference organ.
Named alongside it
The objects these essays reach for when they reach for this one.
MeasurementParastichy pairHonest limitsThe placement ruleLattice offsetDivergence angleRiseAblationLatticeClaim testingNegative resultRung