Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

The column that cost no stems

Every table in this collection records the rise a stem was grown at. None records where inside its own rung that rise sat, and the fraction turns out to be computable from numbers already written down — which makes it the cheapest column anybody here has ever added and the one that changes the most about how the tables read.
13 min read 21 figures State the claim, state the testCounted, not admired

Worth reading first: The survey this site cannot do · Counting the spirals · A head is a set of points.

Every experimental table on this site has a rise in it. It has to: a stem is grown at a stated rise, the counted pair follows from that rise, and a row that did not say which rise it came from could not be reproduced. So the number is there, in the census of wrecked stems, in the front-depth tables, in the jugacy tables, in every sweep that has ever been published here.

What is not there is where that rise sat inside its own rung — and a rung is a range, not a point. The 5/8 rung on the golden branch runs from a rise of 0.0179 down to 0.0069, which is a factor of two and a half. A stem grown at 0.016 and a stem grown at 0.008 are both counted at five and eight spirals, and they sit at opposite ends of that range. Nothing in any table said so.

The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.
Fig. 1 The column, computed for the ten lattices the ablation census ever wrecks at. The bar is the rung; the mark is where the stem was grown.

This essay adds that column. It is the cheapest thing this collection has done in some time — no stem is grown for it that has not been grown twice already — and it is worth an essay because of what it does to readings that were scored without it.

What a fraction of a rung means

A rung is bounded by the rises at which a counter’s answer changes. Above 0.0179 a stem on the golden branch is counted at three and five spirals; below 0.0069 it is counted at eight and thirteen. In between it is counted at five and eight, and that constancy is the whole reason the word rung exists: it names the range over which the instrument returns one answer.

So the bounds are a measurement rather than a definition, and they have to be taken the same way every other measurement here is taken. The ladder is swept at one per cent in the rise — a constant relative step, which matters because the rungs are geometric, with consecutive transitions at a ratio near the square of the golden ratio. A constant absolute step would resolve the coarse rungs finely and the fine ones hardly at all, which is precisely backwards for a quantity whose answer is a fraction.

The golden ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 12 and 35 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; seven of them sit past the mark.
Fig. 2 The golden branch, cut into rungs by where the counter’s answer changes, with the census’s own lattices dropped onto the bars.
The Lucas ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 6 and 40 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; three of them sit past the mark.
Fig. 3 And the Lucas branch, which carries four rungs of its own over the same range of rises.

Swept that way, the two branches give eight rungs between 0.0040 and 0.0700. The position of a rise inside its rung is then the obvious thing: how far it is from the coarse transition to the fine one, taken in the logarithm of the rise because a geometric ladder has no other natural coordinate. Zero is the coarse end and one is the fine end.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 4 The ladder the rungs are cut out of, over the whole range this collection can reach.

Two housekeeping points, both of which turn out to matter.

The first is that a sweep below the finest rung still returns pairs. Below about 0.0048 on the golden branch and 0.0057 on the Lucas one, the counter answers 2/4, 8/16, 3/11, 11/13 — at settled divergences of 129° to 226°, which neither branch goes anywhere near. Those runs of rises are not rungs and are excluded here on the divergence rather than on whether a pair came back, because a pair always comes back. That is the same region where, measured separately, a stem does not settle onto a lattice at any run length.

The second is that the fraction is only defined for a rise that is inside a rung at all. A rise below the finest one is refused a position rather than given one by extrapolation, and the gate this site runs checks that refusal.

How precise a fraction is

A position is a ratio of two logarithms, and both of them are measured, so the question of how much to trust the third decimal place is real. Three things bound it.

The sweep step is one per cent in the rise, so a rung boundary is located to within one per cent and a rung spanning a factor of two and a half carries about ninety sampled rises. On such a rung a position is good to about one part in a hundred, which is far finer than any use made of it here. The narrowest rung in the table is the golden 8/13, which spans 0.0048 to 0.0068 and carries thirty-six rises; a position on it is good to about three per cent. That is the worst case and it is still an order of magnitude better than the distinctions being drawn.

The second bound is the counter itself. A rung boundary is where the counter’s answer changes, and near a boundary the two candidate pairs have contact steps of nearly equal length, so a boundary is not a knife edge in the geometry even where it is one in the answer. What that means in practice is that the first and last one per cent of a rung are places where the pair a counter returns is a decision about a near-tie. No census row here sits within five per cent of a boundary, which is checked rather than hoped.

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.
Fig. 5 The same boundary problem on a disc, where a count is a statement about an annulus and the transitions are places rather than lines.

The third is the one worth stating loudest, because it is the one that would invalidate the column rather than blur it. The bounds are measured on the ideal lattice — the hop-length ranking of the settled divergence at each rise — and the census’s stems are grown by the placement rule. Those agree here, and they have to be checked to agree rather than assumed: a rise where the rule’s own stem is counted at a different pair from the ideal lattice at the same rise would be a rise whose position is meaningless. None was found in the range swept.

The angles against the positions, rise by rise. three rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.
Fig. 6 The two routes to a counted pair — from the angle and from the positions — which have to agree before a rung boundary means anything.

The column

Here is the whole of it, for the twelve lattices the ablation census is built from. Ten of them ever wreck; the other two are in the census precisely because they do not, since a table assembled only from the lattices that produce an answer is a table of its own selection.

The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.
Fig. 7 Every lattice in the census, including the two that never wreck, placed inside its own rung.

The positions run from twelve per cent to eighty-nine. That looks like a good spread, and reading it as one is the mistake this essay is about. What the positions are a spread of is not obvious until the rung carries a second mark.

The mark that changes what the spread means

Inside every rung there is a rise at which the two contact steps change places: above it the shorter of the two hops belongs to one family, below it to the other. Where that sits is fixed by the arithmetic of the lattice and not by anything a stem does, and it is measured here along with the bounds.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 8 Where that rise sits, over every rung on both branches that has one.

Six of the eight rungs have such a rise inside the range swept, each has exactly one, and every one of the six sits in the coarse half — at twelve, fifteen and thirty-five per cent on the golden branch, and forty, six and thirty-three on the Lucas one. Never past the middle.

Now put the census’s own lattices back on the same bars. Eight of the ten that wreck sit past the mark. One sits before it. One sits so close to it that the two steps differ by two and a half parts in a thousand, which is a rise where the words shorter and longer are doing no work at all.

The readings, and where in their rungs they fail. Each bar is one candidate account of which family a wrecked stem keeps, scored across every wrecked cut in the census. Under each bar are the positions inside their own rungs of the cuts it gets wrong, as percentages from the coarse end. The best of them is right 25 times of 30, and the positions of its failures are the point: two of them are the single lattice grown at the far fine end of its rung, which is also the only census row past three quarters of the way down. Nothing here rescues a reading. What it shows is that the table these readings were scored on varies a quantity nobody chose, over a range nobody stated.
Fig. 9 The candidate accounts of which family survives, scored across the census, with the positions of the cuts each one gets wrong.

Counted by cuts rather than by lattices it is starker: twenty-five of the thirty wrecked cuts were made past their rung’s handover.

That is what the spread is a spread of. It is not a sample of the ordering; it is very nearly a hold on it. The census varied the counted pair across its rows, varied the offset within each row, and — without anybody choosing to — held the step ordering nearly fixed while appearing to leave it free.

What it does to the readings

The reading that a wrecked stem keeps its shortest hop scores twelve of thirty on this census. That number has been quoted here as evidence the reading is wrong, and it is still evidence the reading is wrong, but the column changes what kind of evidence it is. On a census sitting almost entirely on one side of the handover, “the shortest hop” is very nearly a synonym for “the larger family” — and the larger family survives eleven times of thirty. The two readings score within one of each other because on these rows they are close to the same reading.

The two contact steps change places inside the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 10 The ordering inside one rung, and how little of its range a census sampling one rise per rung ever visits.
The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 11 And the other quantity moving over the same range, which no counter can see either.

The reading that does best — the smaller count while the cut lands no further back than it, the larger beyond — is right at twenty-five of thirty. Its five failures sit at eighty-nine, eighty-nine, nineteen, sixty-six and thirty-three per cent. That is not a fine-end artefact and it is not a coarse-end one. Two of the five are the single lattice grown at eighty-nine per cent of the 8/13 rung, and the pair of runs that closed the class outright sit at nineteen and sixty-six per cent of the 4/7 rung.

So the column rescues nothing. It was never going to: a rule that is wrong on five rows is wrong on five rows whatever coordinate the rows are plotted in. What the column does is tell the reader what those five rows have in common, which is that they were drawn from a range nobody stated.

Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.
Fig. 12 The census the readings are scored on, with the winning rule marked, and with the position column now available to every row of it.

The two things this makes possible

The first is a matched comparison. If the fraction of a rung is a coordinate, then two stems at the same fraction of two different rungs are comparable in a way two stems at the same rise are not — and, more usefully, two stems at different fractions of the same rung differ in a quantity that can now be named. That is what makes a band where the divergence is flat and the ordering is not a design rather than a coincidence.

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.
Fig. 13 The sweep that first showed a survivor changing along one rung, which is the measurement the column was needed for.

The second is a caveat that can be checked instead of repeated. Every reading scored on the census can now be asked whether its failures cluster in the coordinate the census did not control. The answer here is that they do not cluster — the five failures are spread across the range — and that is a real answer rather than a shrug.

What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.
Fig. 14 The neighbouring version of the same discipline: a quantity that the instrument fixes rather than measures.

Why nobody computed it

Because it needs the rung bounds, and the rung bounds were never wanted. Every experiment here starts by choosing a rise that produces a pair; once it does, the question is about the pair, and where the rise sits inside the range that would have produced the same pair is not part of the question. The bounds were computable at any point — one stem per rise, six hundred stems for the whole ladder, about ten minutes — and they were never computed because no result needed them.

That is the same shape as the finding that a census taking one rise per rung is a sample rather than a plateau: a quantity settled early, quoted thereafter, and never varied. The difference is that this one is arithmetic rather than an experiment, so the correction costs nothing at all. The whole column is six hundred stems that were already grown for other purposes, a logarithm and a subtraction.

The plane of stems: divergence across, rise up. Each shade is one parastichy pair. The marked points are the lattices where three families are equally short — the forks — and the Fibonacci ones run up the middle towards 137.51°.
Fig. 15 The diagram the rung bounds fall out of, drawn from the geometry rather than from any run.

What it does not do

It does not make the census a two-parameter sweep. Ten lattices at ten positions is still ten rows; putting a coordinate on them does not add a row. A census that genuinely varied the position would grow several stems inside one rung, which is what the rung sweep did and what every table published before it did not.

It does not make the position a cause. The position is a coordinate. Something varies along a rung and something decides which family survives, and the second of those has now been narrowed by an experiment rather than by a column: on a band where the divergence is held and the ordering flips, the answer does not move.

And it does not reach the readings whose subject is the pair rather than the geometry. Whether a pair’s two numbers share a factor is a property of the pair, and a census that fixes one rise per pair fixes nothing relevant to it. Sorting readings into those the column can touch and those it cannot is, as ever, most of the work of applying any of this.

The block is one of the two numbers, and not always the smaller. Every lattice a single organ was removed from, one row each: golden, rise 0.020 carrying 3/5; golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; golden, rise 0.005 carrying 8/13; Lucas, rise 0.020 carrying 4/7; Lucas, rise 0.013 carrying 4/7. The two open ticks on each line are that lattice's own parastichy numbers; the filled dots are the blocks the stems that never recovered settled into, one per offset that failed. The claim this table was built to test is that the block is the smaller of the two, which held at the two rises it was first measured at. It does not hold here: 3/5 gives 5, 5/8 gives 5 and 8, 4/7 gives 4 and 7. What survives is weaker and still worth something — every filled dot but 1 sits on one of that row's own ticks, so the orbit carries a count of the lattice it was cut from, and which of the two it carries is decided by the offset rather than by the pattern.
Fig. 16 One of the tables the column does not reach, because its reading is about the arithmetic of the counted pair.
The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block; At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 and 16 rows.
Fig. 17 And the destinations that table feeds, each measured at the rise its own pair required.

The other tables it reaches

The ablation census is not the only table here built one rise per rung, and the column belongs on all of them.

The front-depth table — how many offsets never repair, at each rise — is already a statement about position in disguise, because the front deepens as the rise falls and the deepening is smooth within a rung. Reading its rows as a function of the pair rather than of the position is what made the growth look like a step when it is a slope with steps drawn on it.

How many offsets wreck, along the 5/8 rung. The count of offsets that never repair, at each rise on one rung. It runs from 1 at the coarse end to 5 at the fine end, while a counter returns 5 and 8 spirals at every one of them. The front — the run of recent organs at which a removal is felt at all — deepens as the rise falls, so there are simply more places a cut can land and fail to heal. That is the mechanism under the grid: the offsets that appear at the fine end are the ones beyond the smaller contact number, and those are the ones that keep the larger family.
Fig. 18 The front deepening inside one rung, which is the quantity the front-depth tables were reading across rungs.
The band that never heals is what two fixed edges leave over. Each row is one rung. A stem is grown to 400 organs, one organ is removed, and the stem is followed for 300 more. A filled mark is an offset the stem recovers from, with the number of organs it took printed above it; an open ring is an offset it never recovers from. At the 3/5 rung, a front of 5, every offset heals. At the 5/8 rung, a front of 8, the offsets 4, 5 never heal. At the 8/13 rung, a front of 13, the offsets 4, 6, 7, 8, 9 never heal. What is the same at every rung is the two edges: the first three offsets heal, and so do the last few of the front. What is left in the middle is the band, and it widens because the front does.
Fig. 19 The same quantity as the earlier tables reported it, one rise per rung, with three positions that were never stated.

The noise tables are a different case and a more interesting one. Those grow stems at a fixed rise and vary the disturbance, so the position is constant down each table and cannot confound anything within it. What the column says there is about comparisons between tables: the three rises the depth thread works at sit at very different fractions of their rungs, so a difference between them carries the position difference along with everything else.

On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.032 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 14 places back at the right. A cell is filled where removing that organ moves the next organ by more than 2.5°, and empty where it does not. On a rung the filled cells are a run from one to the larger parastichy number — 5, 8, 13 at the pairs shown on the left. Every rise shown here is in the middle of its rung, so every row is a run and nothing past it is felt — what happens between the rungs is a separate matter.
Fig. 20 Three rises the noise work compares, which are three positions as well as three pairs.

And the jugacy tables are the case where the column is genuinely irrelevant, which is worth having an example of. Their reading is about the arithmetic of the counted pair, and the pair is exactly what a rung holds constant.

Where this leaves the collection

With a coordinate that every future table should carry, and with a small piece of evidence that the tables already published are not badly compromised. The readings that were right are still right; the reading that was wrong is wrong for a reason the column makes precise rather than for the reason it was quoted with.

The uncomfortable part is the same one as last time. Nothing about this needed a new idea, a new library or a new measurement. It needed somebody to notice that a number appearing in every table has a denominator, and to go and measure the denominator. That took a morning, and it had been available since the first census was written.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 21 The whole ladder again, with every rung of it now carrying two marks a table could have quoted and never did.

Where this goes next

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it. One rise per rung is a sample Every census on this site takes one rise from each rung, because the question was always which pair. Any rule later scored on those rows inherits a variable that was never varied — and two of this collection's results turn out to be about the sampling as much as about the rule. The readings, and where in their rungs they fail. Each bar is one candidate account of which family a wrecked stem keeps, scored across every wrecked cut in the census. Under each bar are the positions inside their own rungs of the cuts it gets wrong, as percentages from the coarse end. The best of them is right 25 times of 30, and the positions of its failures are the point: two of them are the single lattice grown at the far fine end of its rung, which is also the only census row past three quarters of the way down. Nothing here rescues a reading. What it shows is that the table these readings were scored on varies a quantity nobody chose, over a range nobody stated. The side the census sat on Eight of the ten lattices the ablation census wrecks at were grown past their rung's handover, one before it, and one so close that the ordering it quotes differs by parts in a thousand. A reading scored over the step ordering was therefore scored against a quantity the census was nearly holding fixed. The golden ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 12 and 35 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; seven of them sit past the mark. 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.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • What a count cannot decide — both name counting blind, control, honest limits, lattice, measurement, parastichy pair, rise, rung, summary statistic, underdetermination
  • A stem too fine to settle — both name counting blind, control, honest limits, lattice, measurement, parastichy pair, rise, rung, underdetermination
  • Two accounts of one number — both name counting blind, honest limits, ladder, lattice, measurement, parastichy pair, rise, rung, underdetermination
  • A stem coarse enough to cut — both name counting blind, control, honest limits, lattice, measurement, parastichy pair, rise, rung
  • One rung, two answers — both name control, honest limits, lattice, measurement, parastichy pair, rise, rung, underdetermination
  • Seven rises and two seeds — both name control, honest limits, ladder, lattice, measurement, parastichy pair, rise, rung

Named objects

A flat tag is an object no other essay names yet.

Counting blindCensusControlHandoverHonest limitsLadderLatticeMeasurementParastichy pairRiseRungSamplingSelection effectSummary statisticUnderdetermination