Phyllotaxis
the spirals are counted, not admired
Concept

Matched design — where it appears

A set of comparisons built so that everything but the quantity under test is held fixed by construction rather than by argument. Two stems on different lattices at one rise is one; seven such pairs, across which the ordering reverses, is what makes it a design rather than an anecdote.

Named by 22 essays across 3 fields — each of them below, with the objects they name alongside it.

The same rule, the same rise, two lattices, two fronts. How many organs back a single removal is still felt, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice, at seven rises. Everything but the seed is identical at each rise — the rule, the spacing, the heights, the azimuth grid — and the two fronts differ at every one of them. Which branch has the wider front changes hands four times going down the range, so no function of the rise gives the column. The golden branch carries 5/8 at four of these rises, across a factor of two in the rise, and its front is eight at all four.

Seven rises and two seeds

One organ removed from a stem is felt out to the larger of its two spiral counts. Every test of that has confounded the count with the rise, because on one branch the two move together. Grow a second branch beside the first at the same rise and they come apart — and doing it at seven rises turns a matched pair into a design whose last column changes hands four times.

cylinder · Branch choice The newest member of the front is the weakest. For every cell of the design whose rung boundary is inside the range, how far the next organ moves when the organ exactly as many places back as the larger parastichy number is removed — the offset that arrived when the stem entered this rung — against how far below that boundary the stem sits. Each line is one lattice on one branch. The horizontal line is the threshold that decides whether an offset counts as felt, and the three cells below it are the three whose front reads one offset short. Nothing is a different kind of thing: the boundary is a step everywhere, and near the top of a rung its last stair is shallow.

The front that reads one short

Eleven cells of a fourteen-cell design put the boundary exactly at the larger spiral count. Three put it one offset earlier, and the tempting move is to lower the threshold until all fourteen agree. Measured instead of tuned, the three turn out to be the three cells nearest below their own rung's boundary — and the last offset of a front is weak because it has only just arrived.

cylinder · Branch choice Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls.

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°.

mechanism · Both walls The offset past the front that is felt anyway. The four cells of the design whose response has a hole in it: a run of felt offsets, a stretch of quiet, and then one isolated offset well outside the front at which a removal moves the next organ by tens of degrees. The open circle on each row is the count coming in at the next rung of that branch's ladder, and the filled point is the isolated offset. It sits one inside the incoming count on every row, including on the Lucas branch, where the incoming counts are 7 and 11 rather than the Fibonacci numbers the rule was found on.

The hole on the other branch

Near a transition, the run of offsets a stem notices stops being a run: there is quiet past the front and then one isolated offset, felt as hard as anything inside it. Where that offset sits was pinned down on Fibonacci lattices, where the numbers to check it against are 5, 8 and 13. On the Lucas branch they are 4, 7 and 11 — and the rule holds there too.

cylinder · Branch choice A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.

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.

cylinder · Rung interior What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.

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.

mechanism · Both walls Where taking the second wall as well changes nothing. Each row is one lattice, with three marks: how far the next organ moves when the smaller wall alone is removed, when the larger alone is removed, and when both are. On these lattices the third mark sits on the second, to within two steps of the azimuth grid. The smaller wall is free — taking it away as well changes nothing — and on a row like that the interaction is minus the smaller wall's own cost by construction, which is arithmetic and not a measurement. Three of them are the whole of one rung and the others are the fine ends of two more.

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.

mechanism · Both walls The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°.

The angle the ladder returns to

Down the golden branch the settled divergence climbs across one rung and falls across the next, turning three times in four rungs. That is why the same angle is reached at two different rises — and why the Lucas branch, which turns once, almost never offers the same thing.

cylinder · Same angle Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter.

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.

mechanism · Both walls The interaction across each rung, coarse end to fine end. One line per rung, drawn against where in the rung each lattice sits — nought at the coarse end, one at the fine end, measured in the logarithm of the rise. The lines are flat. Inside a rung the interaction moves by 3.3 to 13.6 degrees, against a spread of 240 degrees across the ladder, and it falls from the coarse end to the fine one on 6 of the 7 rungs. Position inside a rung was the candidate this design was built to test and it is not what decides the sign.

The exception was already labelled

The larger counted number sorts twenty-two of twenty-four lattices by the sign of their slot interaction. Both misses are on the Lucas 3/4 rung — the one rung a different measurement had already singled out, for reasons with nothing to do with this one.

mechanism · Both walls Which rungs of the golden branch share a divergence. One row per pair of rungs. A pair whose divergence ranges overlap has a rise on each rung where the rule settles on the same angle; a pair whose ranges do not overlap has none, whatever the search. On this branch four of six pairs match, and three of those match to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09, so the design holds one angle while changing everything the rise controls.

Two rungs, one angle

Five pairs of rises settle on the same divergence while a counter returns different pairs at them, and four of the five agree to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09.

cylinder · Same angle A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.

The angle is not the actor

Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.

cylinder · Same angle Shared counted numbers against shared survivors. One row per matched pair, over both branches. The third column is the counted numbers the two rungs have in common and the fourth is the families both stems leave standing; on every row the two are the same set. The row whose rungs share no counted number is the one whose stems share no survivor, which is what makes this a claim about an intersection rather than a restatement that a survivor is usually a contact family. four rows, and the empty case is one of them.

Where the survivors meet

At a matched pair the two stems keep exactly the counted numbers their two pairs have in common — the 5 where 3/5 meets 5/8, the 8 where 5/8 meets 8/13, the 7 where 4/7 meets 7/11, and nothing at all where 3/5 meets 8/13. Four rows, including the empty one.

cylinder · Same angle Which rungs of the golden branch share a divergence. One row per pair of rungs. A pair whose divergence ranges overlap has a rise on each rung where the rule settles on the same angle; a pair whose ranges do not overlap has none, whatever the search. On this branch four of six pairs match, and three of those match to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09, so the design holds one angle while changing everything the rise controls.

The last of three quantities

The pair, the divergence and the step ordering move together when the rise is swept, and for a long time no result could be attributed to any of them. Two designs later, two are ruled out as sufficient and the third has never been held still — because holding it is what a rung already does.

cylinder · Same angle Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.

Four crossings nobody visited

Six rungs of the ladder carry a handover and two of them had a band built on them. The other four are here: 70, 126, 16 and 124 rises wide, found by sweeping at a ratio rather than at a fixed step in the rise, which is why the fine ones had been stepped over.

cylinder · Rung interior What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.

A band that moves nothing

One of the six bands holds the counted pair, holds the divergence, and does not move the ordering: its two contact steps stay within four parts in a thousand of each other across the whole of it, so the ordering changes hands three times and neither end has one worth the name.

cylinder · Rung interior The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism.

The ordering on six bands

A hundred and fourteen wrecked cuts across four bands, and at every offset of every one of them the family left standing is the same immediately above the handover and immediately below it. Where the answer does change — on the widest band, at three offsets — it changes somewhere else.

cylinder · Rung interior Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it.

Every rise of a band

A band is cut at nine rises because the quantity it was built to test is a constant, and a constant is checked at the ends and at the crossing. On the widest band that quantity turned out not to be constant, which makes nine the wrong number. This is all hundred and twenty-six.

cylinder · Rung interior Offset 7 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 126 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark one islands — runs of 2 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.

Three offsets, three crossings

The claim the band design rests on is that the survivor does not change where the two contact steps change places. It holds at full resolution: nineteen changes and not one at the handover. Where they are is three different rises, eight, nineteen and twenty-nine below it.

cylinder · Rung interior The ordering changes and the survivor does not. Every offset that wrecks, at every rise of the band, with the family left standing written in the cell. The counted pair is 5 and 8 at all 18 rises and the settled divergence is held to a twentieth of a degree, so the one quantity moving across the columns is which of the two contact steps is the shorter — and it changes hands at the marked rise. The cells do not: the 5 family survives at all 24 wrecked cuts, on both sides. Scored on this band, the reading that a wrecked stem keeps its shortest hop is right 14 times out of 24, for an answer that never changed.

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.

emergence · 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 What nine rises find on each band, against what all of them find. Two bars per band: the changes of surviving family a nine-rise design finds, and the changes the full sweep finds. On the Lucas band the two agree exactly, at none and none. On the golden band they agree that something changes and disagree about how much — 5 against 19 — because one step of that design is 16 rises and the band carries features one to three rises wide. A sample was never wrong about whether; it was wrong about how many.

When nine rises are enough

A coarse design was shown to be misleading on one band and it has been criticised on that ground ever since. On the second band it is exactly right, and the difference between the two cases is a property of the band rather than of the design — which is the awkward part.

cylinder · Rung interior

Named alongside it

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

Parastichy pairRungAblationControlRiseNegative resultMeasurementClaim testingLattice offsetHandoverHonest limitsResolution

All concepts