Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

A fifth of the hop

The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent.
12 min read 20 figures State the claim, state the testTwo routes, one number

Worth reading first: The damage has a period · The organ that was taken away · Counting the spirals.

One step of the control’s own divergence is the best of four candidate units for the size of the exchanged pair in a wrecked stem, and it is wrong by up to 11.8 per cent.

The miss is not scatter. Every row whose stem kept a lag of 5 or 7 overshoots one step; every row whose stem kept a 4 or an 8 falls short. No lag appears on both sides, on seventeen rows across two branches and eight lattices.

That was already in the record as an unexplained pattern — the essay that measured the pair reports it and says nothing about it. This essay is the attempt to say something, and most of the work is deciding how much the attempt is worth.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 1 What the exchange misses one divergence step by, against the surviving hop’s own angle, with no line fitted to it.

The quantity the residual sorts by

Fold the surviving lag into the divergence: the angle from an organ to the one k places above it is k times the divergence, taken round the circle and folded into half a turn either way. That is the surviving hop’s own angle, and it is what a counter would read if it counted that family.

It is negative for lags of 5 and 7 and positive for lags of 4 and 8, at about −34°, −24°, +39° and +22°. The residual is positive where the hop’s angle is negative and negative where it is positive, on all seventeen rows.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 2 The lag spectrum that names the surviving hop. Its angle, not its length, is the quantity this essay is about.

The correction

Take one step of the control’s divergence and subtract a fifth of the surviving hop’s angle. The worst row goes from 11.8 per cent to 4.0, the average from 4.3 to 1.0, and every one of the seventeen rows falls inside five per cent.

That is a factor of three on the worst row and a factor of four on the average, bought with one coefficient.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 3 The same residuals with the line the correction draws: a straight line through the origin with a slope of about a fifth and the opposite sign.

A fifth is stated, not fitted

The least-squares coefficient over the seventeen rows is 0.188. A sixth, a fifth and a quarter all take the worst row under six per cent, and the fifth is the roundest number in that range.

Quoting 0.188 would be quoting three digits of a coefficient fitted to a handful of points, which is a precision the data does not carry. Quoting a fifth says the same thing and does not pretend.

It is the same discipline the tolerances in this thread are chosen by: a threshold that sits in a gap is stated once and never tuned, because a number that could have been anything in a range and was chosen at one end is a number carrying a decision nobody declared.

Four accounts of one angle, scored on the same 17 rows. Each bar is how far an account of the exchanged pair's size sits from the measurement, as a share of the control's divergence, on its worst row. A step of the wrecked stem's own divergence is the obvious candidate and the worst of the four. The surviving family's own step is not the unit at all: that angle is a fifth of the exchange. A step of the control's divergence is close, and correcting it by a fifth of the surviving hop takes the worst row from 11.8 per cent to 4.0.
Fig. 4 The four candidate units on their worst rows. The correction is the difference between the third bar and the fourth.

The sign is the result, not the size

A coefficient fitted to seventeen rows is worth about seventeen rows. What is not a fit is the direction: the surviving lags in this census give hops on both sides of zero, and the residual follows every one of them.

If the correction were an artefact of fitting, it would have no reason to change sign where the predictor does. Four lags, two signs, seventeen rows, and no exceptions.

Every stem that never repaired, and the lag it kept. The 19 offsets across six lattices at which a single removal leaves a stem that never returns to its divergence. For each one: which organ was removed, the period of the block of angles the stem settles into, the lag whose hop survived the cut unchanged, and how many whole turns the stem gains over one period of that lag. The block and the surviving lag are the same number in every row. The marked row is the one whose survivor is not a parastichy number of the lattice that was cut — a hop 6.8 times the length of a contact hop, which no census would report and which the rule held rigid all the same.
Fig. 5 The census with its surviving lags. Four distinct lags, two of them giving a positive hop angle and two a negative one.

The honest denominator is four

This is the part the essay exists to say. The hop’s angle is k times the divergence, and the divergence barely moves across the census — 99° to 138°. So the seventeen rows carry eleven distinct hop angles which fall into four tight clusters, one per surviving lag.

The correction is therefore a line fitted to four positions on its own axis, scored on seventeen rows. A reader told “seventeen rows, no exceptions” and not told “four clusters” has been told something misleading by a true sentence.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 6 The residuals again. What looks like seventeen points spread along an axis is four groups of points at four places on it.

And inside a cluster it does not work

Within the rows sharing a lag of 5 the hop angle runs from −37.3° to −30.2° and the residual runs from 3.9° to 10.4°. If the relationship held inside a lag, the largest residual would sit at the most negative hop. It does not: the largest residual sits at −36.1° and the smallest at −31.2°, and the ordering is scrambled.

So the correction accounts for the differences between surviving lags and nothing at all for the differences within one. The coefficient implied row by row runs from 0.121 to 0.301, a factor of two and a half, and the file records that rather than averaging it away.

The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here.
Fig. 7 The exchanged pairs by surviving lag. Inside a lag they still differ, and this essay’s rule says nothing about that.

Why the hop’s angle and not its length

The surviving family is identified by a length: it is the lag whose step across the surface the cut stem holds unchanged, and the census reports that length for every row. The length is not the quantity the residual follows.

The angle is. A lag’s step has a height, prescribed by the rise, and an azimuth, which is that many divergences round the circle. Two lattices can have the same step length with very different azimuths, because the height makes up the difference — and it is the azimuths this residual sorts by.

That is worth stating because a reader who has followed the survivor thread has been reading lengths for several essays. The correction changes the unit halfway through, and the change is the reason it works.

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.
Fig. 8 Every lag’s step at one rise. The ranking is by length and the quantity this essay uses is the azimuth underneath it.

What a fifth might be

An honest guess, offered as one. The two exchanged chains are one place apart in the sequence and therefore one divergence step apart in azimuth, and they are also some number of hops apart in the family the stem has kept rigid. If the exchange relaxes slightly towards the geometry of that family, the relaxation would carry a share of the family’s own angle.

A share of a fifth would then be a property of how stiff the surviving family is against the rest of the arrangement, which is a quantity nobody here has defined, let alone measured. So the guess names a thing that would have to exist rather than explaining anything.

The hops of a 8/13 lattice, shortest first — golden, rise 0.005. Every lattice hop at this rise, ordered by how long its step is across the surface of the stem, with the two that a wrecked stem here leaves standing picked out. The two shortest are the contact families the counter returns, 8 and 13, and they differ in length by a factor of 1.088. The lags left standing after a removal are 4 and 8, sitting at rank 37 and 2 in this order, so the family the rule holds is a short step but not always the shortest one.
Fig. 9 The lags ordered by step length at one lattice. The surviving family is one of these and the arrangement’s stiffness against it is not a quantity in this thread.

What would test it

A wrecked cut whose surviving lag is not 4, 5, 7 or 8. The correction predicts a residual of about a fifth of that lag’s hop angle, with the opposite sign, and it predicts it before the row is measured.

Lags outside those four do occur: the census’s own length ranking places a survivor as deep as thirty-seventh, and lags of 11 and 13 are perfectly ordinary steps on a fine lattice. No wrecked cut in this census keeps one, so finding a lattice that does is the test — a search rather than a sweep, and cheap.

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. 10 Which offsets wreck at each lattice and what each keeps. A cut keeping a lag outside the four would be a fifth cluster on the correction’s axis.

The prediction is specific

At a lag of 11 on a golden lattice the hop angle is about +32°, so the correction predicts an exchange about 6° short of one divergence step. At a lag of 13 it is about −20°, so about 4° long.

Two numbers, either of which could be wrong by more than their own size. That is what makes it a test rather than an extension — and it is the same shape as the test the slot interaction’s rule needs and cannot get from inside the ladder: a value of the predictor that the existing design does not reach.

Which offsets give short hops, at a rise of 0.005. The two lowest points are at 8 and 13, 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.
Fig. 11 The steps at every lag on a fine lattice, where a survivor of 11 or 13 would be an ordinary contact family rather than a curiosity.

The worst row is the row a longer run doubts

The correction’s largest error is 4.0 per cent, on the Lucas 0.013 stem’s cut four organs back. The next worst is 2.3, and the mean is 1.0, so that row is nearly twice as far out as anything else.

It is also the only row in the census whose surviving lag is 4, so it is a cluster of one and the correction has no other row to be checked against there. Both of those were known, and a cluster of one is the shape a rule with a single exception has when the exception is also the only test of one of its terms.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 12 The census by stem. The Lucas 0.013 row is the one this essay’s rule fits worst.

And a separate measurement says the same row is unreliable

Running every cut twice as far asks whether a profile’s levels are steady over six hundred organs rather than three hundred. Twenty-six of thirty rows are; one that was periodic over the shorter run is not over the longer, at a spread of 171.2° against 6.1°.

That row is this one. So the exchange it reports is a reading taken over a window still inside a transient, and the row that fits this essay’s rule worst is the row a different measurement, made for a different reason, says should not have been read.

The 3 cuts the two run lengths disagree about. Each block is one wrecked cut, with its widest within-class spread drawn at both run lengths and the 10 degrees that separates periodic from not marked by the rule. Two of these become periodic when the run is doubled, at spreads falling from about seventy degrees to about eight. One goes the other way, from six degrees to a hundred and seventy — and that one is the row an entirely separate reading of the same census independently reports as its worst fit.
Fig. 13 The rows the two run lengths disagree about. One of them stops being periodic when there is room to look, and it is the worst fit here.

Which is a check, not a rescue

Dropping the row would take the worst error from 4.0 per cent to 2.3 and the cluster count from four to three. That is a better-looking table and a weaker one, and the essay does not drop it.

What the coincidence is worth is different: two readings of one census, taken for unrelated reasons, agreeing about which row is least trustworthy. Neither knew about the other, and that is the kind of agreement worth more than either measurement.

How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.
Fig. 14 How steady each class is at both run lengths. The row this essay fits worst is one of the three that cross between quadrants.

The alternative nobody can rule out

Four clusters and one coefficient is exactly the amount of data that cannot distinguish a linear relationship from any other monotone one. A rule of the form minus a fifth of the hop and a rule of the form minus a constant per lag agree on every row here, because the hop is nearly constant inside a lag.

The second is a much weaker claim — four numbers fitted to four groups, which is no compression at all — and nothing in this census separates it from the first. What separates them is a fifth cluster, which is the test above and the reason it is the test.

The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here.
Fig. 15 The exchanged pairs by size. Four groups of rows and a rule with one parameter is a fit that four separate parameters would match exactly.

What is not claimed

That the exchange is exactly a step less a fifth of a hop. It is a rule that fits seventeen rows to within four per cent, rests on four clusters, and has no account behind it.

Nor that the residual is understood. Something makes the coefficient run from 0.121 to 0.301 inside a single lag, and nothing here says what. The rule accounts for the between-lag structure and leaves the within-lag structure exactly where it was.

Four accounts of one angle, scored on the same 17 rows. Each bar is how far an account of the exchanged pair's size sits from the measurement, as a share of the control's divergence, averaged over the census. A step of the wrecked stem's own divergence is the obvious candidate and the worst of the four. The surviving family's own step is not the unit at all: that angle is a fifth of the exchange. A step of the control's divergence is close, and correcting it by a fifth of the surviving hop takes the worst row from 11.8 per cent to 4.0.
Fig. 16 The four accounts by mean error. The best of them is a rule with a free parameter and the essay says so where it is scored.

Why a residual is worth an essay

Because a twelve per cent shrug is where a measurement stops being tested. “About one divergence step” was true, useful, and unable to be wrong; the twelve per cent had nowhere to go.

Naming a structure in it — even a structure with four points behind it and no mechanism — turns the shrug into a prediction about lags nobody has measured. That is the trade this collection makes over and over, and it is worth stating that the prediction is the point rather than the fit.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 17 The line, drawn once more, with the honest reading beside it: four clusters, two signs, and a coefficient worth about as much as seventeen rows can buy.

What carries forward

A rule, its denominator, its worst row and its test. The test is a search for a wrecked cut with a surviving lag outside 4, 5, 7 and 8, which would put a fifth cluster on the axis and either confirm the sign or break it.

And a piece of arithmetic worth repeating: the within-lag scatter is a factor of two and a half and is untouched. Any account of the exchange that explained that as well would be a considerably better account than this one, and the place to look for one is the arrangement’s own structure rather than in the cut.

Which chain the backward exception sits on, over the census. Chains are numbered from the removed organ, so chain 0 is the chain the hole was on and chain 2 is two organs along it. The exchange is at the hole's own chain on 10 of the 17 rows that carry one, against 2.8 rows for a chain drawn at random from each row's own period. That is far more often than anywhere else and it is not every row, so the position is a tendency rather than a rule — and the file says so rather than rounding it up.
Fig. 18 Where the exchange sits over the census, which is the other reading of the same pair and has seven exceptions where this one has none.

A second reading of the same rows

The exchange has two readings and they behave completely differently. Its direction is a discrete fact and it is the same on every row, with nothing left over. Its size is continuous and carries a residual with a structure and a scatter inside the structure.

That is not a coincidence about this measurement; it is what discrete and continuous readings usually do. A sign has no error bar and cannot be nearly right. An angle can be nearly right in several ways at once, and the work is separating them.

A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in.
Fig. 19 A row’s chains round its period. The direction between the two exceptions is exact and the distance between them is the quantity this essay corrects.

What this does to the census’s thirteen other rows

Nothing. They carry three or more exceptional chains and have no balanced pair, so they have no size to correct and are refused rather than scored.

Nearly half the census is therefore outside every number in this essay, and an account of the exchange that also said what those rows are doing would be a different and much better result. Nobody has started on it, and the first question — whether three exceptions are a pair plus a stray or a three-cycle — is arithmetic on numbers already computed.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 20 The whole census by how steady its classes are. The thirteen rows with no balanced pair sit through it rather than at one end.

The one line

The exchanged pair’s size is one step of the control’s divergence less a fifth of the surviving hop’s own angle — worst row four per cent, mean one, and the sign right on every one of the seventeen rows.

The seventeen rows carry four distinct hop angles, so the coefficient is fitted to four clusters; inside a cluster the rule says nothing, and the row it fits worst is the row a separate measurement independently calls unreliable.

Where this goes next

Four accounts of one angle, scored on the same 17 rows. Each bar is how far an account of the exchanged pair's size sits from the measurement, as a share of the control's divergence, averaged over the census. A step of the wrecked stem's own divergence is the obvious candidate and the worst of the four. The surviving family's own step is not the unit at all: that angle is a fifth of the exchange. A step of the control's divergence is close, and correcting it by a fifth of the surviving hop takes the worst row from 11.8 per cent to 4.0. Four accounts of one angle The exchanged pair in a wrecked stem is about one divergence step, and about is doing twelve per cent of work. Four candidate units were written down and scored on the same seventeen rows: the cut stem's own step, the surviving family's step, the control's step, and the control's corrected. The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here. A step of one organ The balanced pair inside a wrecked stem's period measures 88.0° to 147.2° against divergences of 99.1° to 138.0° — one organ's step, to within twelve per cent on every row. The residual is not scatter: every stem keeping a 5 or a 7 overshoots and every stem keeping a 4 or an 8 falls short. Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473. Twice the run Five wrecked cuts never reached a pattern inside three hundred organs and one reached it at three hundred and three, which is a number asking to be checked. Run every cut in the census twice as far and three of the thirty change their answer.

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.

  • A window nobody aligned — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic
  • Three rows change sides — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic
  • Two regimes above a hole — both name ablation, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sample size, summary statistic
  • Which chains changed places — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic
  • An onset at the end of the run — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, summary statistic
  • Every rise of a band — both name ablation, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sample size

Named objects

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

AblationCensusClaim testingControlDivergence angleFittingHonest limitsIdentifiabilityMeasurementNegative resultPredictionResidualResolutionRigid hopSample sizeSummary statistic