Phyllotaxis
the spirals are counted, not admired
Concept

Geometric ladder — where it appears

A ladder whose steps sit at a constant ratio rather than a constant spacing, so its rungs are equal in the logarithm of the rise. Consecutive transitions here sit at about the square of the golden ratio apart, which is why every position inside a rung is measured logarithmically.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

The rise falls as one over the distance from the apex. A straight line of slope −1 on log axes. The dashed horizontals are the transition rises of the cylinder's ladder, computed with no cone anywhere in them; where they cross, the count changes. Consecutive crossings are 2.62, 2.62, 2.62, 2.62 apart — φ² is 2.618.

Transitions a factor of φ² apart

The ladder's rungs are a factor of 1/φ² apart in rise. A disc's rise falls as one over radius squared and a cone's as one over distance, so the same rungs land a factor of φ apart on a seed head and a factor of φ² apart on a cone — measured, on both, by a counter that has never heard of either.

cylinder · Cone transitions 264 exceptions in a 2400-organ head, and eight circles. A golden, 137.508° head of 2400 organs, tessellated inside 86% of its radius. Every interior cell is drawn, and the 135 five-sided cells and 129 seven-sided ones are marked apart from the 1367 hexagons, and the pale circles are radii computed from the divergence angle through the lattice's third-shortest vector — nothing is fitted. Every defect sits within 0.64 of a cell spacing of one of those eight circles, and between them there is not one exception in hundreds of cells.

The defects lie on rings

The cells in a seed head that are not hexagons are not scattered through it. Every one of 264 sits within a cell of a radius computed from the divergence angle alone, the radii are a factor of φ apart, and between two of them lie 422 consecutive cells without a single exception.

tissue · Topological charge The defect rings are not where the counts change — they are √φ further out. A logarithmic radius axis for a golden, 137.508° head. The lower marks are the radii at which the counted parastichy pair changes, where the two shortest lattice vectors change places; the upper marks are the radii at which a cell's neighbours change, where the third-shortest does. They interleave, and the ratio of each ring to the transition inside it is 1.2715, 1.2723, 1.2723, 1.2719, 1.2719, 1.2723 against √φ = 1.27202. Consecutive rungs are a factor of φ apart in radius and √φ is their geometric midpoint, so a defect ring sits exactly halfway between two parastichy transitions. Anyone looking for the defect line at the radius where the counts change will not find it there.

The rings are not the transitions

A seed head has two ladders on it — the radii where the counted parastichy pair changes, and the radii where the exceptional cells sit — and the obvious guess is that they are the same ladder. They are not: the second sits a factor of the square root of phi outside the first at every rung of two different divergence ladders, which is exactly halfway between two consecutive transitions.

tissue · Topological charge A stem is a thread at six rises of the sweep, a ribbon at 23 and a surface at 111. The 140 rises of the sweep, split by how many index families the tessellation carries. On 6 rises above 0.5010 a node has 2 walls and the strip is a thread; on 23 between 0.2187 and 0.4833 it has 4 and is a ribbon; on 111 below 0.2109 it has 6 and is a surface. The comparison means something different in each: two families leave 33.333 per cent in dispute on the surface and 0.00 on the ribbon, and three families leave 0.000 and 33.333.

The third family

On a seed head no threshold makes the counted contacts and the shared cell walls the same relation. On a stem they are the same relation exactly, at every one of a hundred and eleven rises and to three decimal places of nothing, provided the contact cut keeps three families where a count keeps two.

tissue · Neighbour definition Which offsets give short hops, at a rise of 0.013. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.

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.

cylinder · Rung interior Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose.

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.

cylinder · Rung interior 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 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 Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.

How wide a band should be

A band ends where the divergence has slid a twentieth of a degree, so its width should follow from how fast the divergence slides. Predicted from the rung's slope that is wrong by factors of 0.20 to 5.92; predicted from a stationary point it is 0.41 to 1.08, and the outlier is the rung that has no stationary point.

cylinder · Rung interior The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences plants are observed to follow. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.

Ten sequences, two of them the ladder's

Every parastichy pair the settling table produces is two consecutive terms of a sequence in which each term is the sum of the two before it. Ten such sequences account for all fifteen destinations — nine, once one of the ten turns out to be a reading rather than a ladder — and the two the collection is built on are neither the largest nor the smallest.

emergence · Off-ladder The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.

The hops cross once

Walking a whole rung at the grid its rises are named on costs a few hundred stems and no cuts at all, and it answers a question nobody had asked: whether a rung has one handover or several. It has one, and the ladder had recorded it in the wrong place.

cylinder · Ablation Six rungs walked at the grid, 10 crossings between them. Each row is one rung, drawn from its coarse end on the left to its fine end on the right and scaled to its own width so positions inside different rungs can be compared. The shaded stretch is the band the ladder grows around the rise it recorded as that rung's handover. five of the six rungs carry exactly one crossing, and on each of those the whole stretch a second one could sit in has been walked at the grid and closed at both ends. The 3/4 rung of the Lucas branch carries five, spaced 22, 12, 15 and 20 grid steps apart, and its band holds three of them. The rule on each row is a located crossing.

Five rungs walked

Six rungs of the ladder carry a handover and only one of them had ever been walked at the resolution its rises are named on. Walking the other five costs 1,224 grown stems and no cuts at all, and it returns a crossing count per rung — five ones and a five.

cylinder · Handover grid Six handovers relocated, in steps of the sweep that recorded them. The ladder finds a handover by stepping at a ratio of one per cent, which never lands on the grid the rises are named on, so a recorded handover is the nearest rise the sweep visited to a crossing nobody had located. This is the difference, in units of the sweep's own step at that rise: 0.082 to 0.835, mean 0.374. All six are positive and all six are inside a single step of the sweep, and both of those are predictions rather than summaries: the ladder reports the first rise it visits at which the ordering has already changed, so the rise it records must sit on the fine side of a crossing and within one of its own steps. The dashed rule is one step of the sweep, which is the bound the sampling predicts.

The handovers corrected

Six recorded handovers, relocated to the grid against where a one-per-cent sweep put them: all six sit on the fine side of a crossing and all six inside a single sweep step. Nothing about the rung explains the size of the discrepancy, which is what a sampling artefact is supposed to look like.

cylinder · Handover grid

Named alongside it

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

RiseMeasurementRungHandoverLadderParastichy pairClaim testingDivergence angleHonest limitsResolutionTransitionsDiscretisation

All concepts