Cuts of several organs send a stem to six settled divergences and no more. Three of them are the old lattice with a family left standing. The other three are counted 2/6, 4/6 and 3/6 — and a pair whose numbers share a factor is the classic signature of a pattern that arrives several organs at a time.
Remove several organs from a settled stem and let the rule place what comes next.
Most arrangements of the cut never repair, and the stems that do not repair land
on a strikingly short list of places: at one
lattice, over more than three hundred wrecked runs made by cuts of one to five
organs, exactly six settled divergences, with the largest cuts reaching nowhere the smallest had not.
Fig. 1 The list, measured. Every place a cut of one to five organs can send a stem at one lattice, over the whole sweep of arrangements.
Three of the six are understood. They are the lattice the stem was cut from with
one family left standing and a whole
number of turns threaded through
it, and
they sit at 136.99°, 208.80° and 280.43° — steps of 71.81° and 71.63° against
360° divided by five, which is 72°. Each keeps the five-family. Three rungs, one
ladder, one hop.
Fig. 2 The six on one axis. The three marked below are the ladder; the three above are what this essay is about.
The destinations are 175.01°, 189.96° and 235.00°. A counter shown the top two
hundred organs of each returns:
destination
reached by
counted
shared factor
175.01°
two organs removed, three and five places back
2/6
2
189.96°
four organs, five to nine places back
4/6
2
235.00°
four organs, five to ten places back
3/6
3
Fig. 3 The first of the three, drawn from its positions, beside the pattern its count usually indicates. The panels are the subject of the next essay; here only the number underneath matters.
Every one of those pairs has a factor in common. The undisturbed stem is counted
5/8, and five and eight are coprime — as are 3/5, 8/13, 4/7, 7/11 and every other
pair this collection has ever measured on a stem the rule grew one organ at a
time.
Fig. 4 For comparison, the pairs an undisturbed stem gives across a sweep of rises. Consecutive Fibonacci or Lucas numbers, and never two numbers with a factor between them.Fig. 5 The counting done on a head rather than a stem, by the same machinery, for the same pairs. A count with a factor in it is not something this subject produces by accident.
The sweep behind the list is worth describing, because “six destinations” is a
number that depends on how hard anybody looked.
A cut is named by the offsets of the organs removed: [3, 5] takes out the organs
three and five places back, furthest first so that each offset means what it says.
For each size from one organ to five, every arrangement described by a starting
offset and a set of gaps is tried — a hundred and eighty-eight arrangements in
all — and each one is continued for three hundred organs against a control that
shares its history exactly.
Fig. 6 The smallest member of the sweep, drawn at the moment it is made: one organ removed, and the rule left to place what follows.Fig. 7 And what a single removal does immediately, offset by offset. Every larger cut is built out of these, and none of them is the sum of its parts.
The share that never repairs climbs with the size: 0.25 at one organ, then 0.56,
0.71, 0.83 and 0.98 at five. So the intervention is doing what an intervention of
increasing size should. What it is not doing is inventing new places to land — the
destinations reached by the five-organ cuts are the ones the single cuts already
reached, and the total across every size is six.
Fig. 8 The rate against the share of the front removed, which is the axis a dose is naturally read on. It climbs at every step and decides nothing about where a wrecked stem goes.
Two destinations are within a degree of each other and are counted as one place;
the tolerance for that is stated and nothing in the table is near it. The six are
separated by twenty-six degrees at the closest.
A reflex here would be to say the counter has been shown something it was not
built for and its output can be ignored. That reflex is wrong, and it is worth
saying why before the counts are questioned.
The counting machinery on this site is deliberately blind. It is given
coordinates — nothing else — and it finds chains of near neighbours, follows them,
and reports how many run in each direction. It has never been told a divergence
angle, and it does not know whether the pattern it is looking at was grown, cut,
forged or written down from a formula. That is the whole reason a count here is
evidence rather than a restatement of what the pattern was built from.
Fig. 9 And the check that says it is doing that correctly: the divergence recovered from the counts alone, against the divergence the pattern was built at.
So when it returns 2/6 it has found two chains one way and six the other, and
those chains are in the positions. The question is not whether the count is right.
It is what a count of that shape licenses somebody to conclude, and the answer
turns out to be less than the textbooks assume.
This is not an obscure signature. It is the single most-cited structural fact
about phyllotaxis after the Fibonacci sequence itself.
A whorled stem puts k organs on every node at once, spaced a k-th of a
turn apart. The result has k-fold rotational symmetry by construction; every
parastichy family comes in k copies; and the pair a botanist writes down is k
times the pair of the single-organ lattice underneath. A bijugate specimen with a
5/8 lattice hidden inside it is counted 10/16. A trijugate one is counted 15/24.
Fig. 10 Stems the whorled rule grows at one, two and three organs per node, at one rise. The counts scale with the jugacy and the pattern underneath does not change.Fig. 11 The census across the divergence: which pairs appear at which angles, at each jugacy. A shared factor is where a whorled pattern lives.Fig. 12 The arithmetic underneath: a jugate lattice’s pairs are the ordinary lattice’s pairs multiplied through, so the branching structure is the same one scaled.
So a stem counted 2/6, 4/6 or 3/6 is, on the face of it, a stem with two or three
files running up it — an arrangement that arrives several organs at a time rather
than one. And that would be a remarkable thing for a cut to produce, because the
number of organs per node is the one property a spiral count is normally
understood to be about, and it is a property of how the pattern is generated
rather than of where it ended up.
Fig. 13 What the distinction looks like on a stem: organs arriving one at a time against organs arriving together, drawn at a rise where both are available.
The half of the list that is understood is worth laying out beside the half that
is not, because the contrast is what makes the three counts strange rather than
merely unfamiliar.
The three slips settle at 136.99°, 208.80° and 280.43°. Each keeps the
five-family of the old lattice rigid — the angle from an organ to the one five
places above it is unchanged from the control, organ by organ, to a fraction of a
degree — and each sits at a different whole number of turns of that family from
the divergence the stem was cut from.
Fig. 14 What a slip is, measured. One lag on the floor and everything else at sixty degrees and more, in a stem whose own divergence has moved by forty-five.
A counter shown those three returns 5/11, 5/12 and 5/9. Every one of those pairs
is coprime, every one carries the five of the family that survived, and none of
them looks like anything unusual: they are ordinary counts of ordinary lattices
that happen not to be the lattice this stem started on.
Fig. 15 The other route to a pair, from the angles rather than the positions, which agrees with the counter on undisturbed stems and is what makes a disagreement worth noticing.
So the six divide cleanly on two independent measurements at once. Three keep a
lag and are counted coprime; three keep no lag and are counted with a factor. The
two halves of that split were found by different machinery — one by comparing hops
against a control, one by counting positions — and they agree on which three are
which.
Fig. 16 The machinery that finds the first half, at three lattices: the period of the motif a wrecked stem repeats, which is the lag it kept.
Why a cut converting a spiral stem into a whorled one would matter #
Set out plainly, because the claim is worth taking seriously before it is tested.
The rule this collection uses places one organ at a time, at the minimum of a
sum over what is already there. There is a separate rule for whorls, and it places
k at a time. They are different rules. A stem grown by the first has never once
been observed to become a stem of the kind the second produces, and there is no
mechanism in the first by which it could — the organs are still arriving one at a
time, at prescribed heights, after the cut exactly as before.
Fig. 17 The rule doing the placing, before and after any cut. One organ, one height, one minimum of one sum.Fig. 18 The intervention that would have to do the converting: two organs removed from the recent history, with everything else held fixed.
If a large enough cut could nevertheless produce a genuinely whorled pattern, the
consequence would be sharp and would reach outside this collection. It would mean
that jugacy — the property counting is least able to decide and morphologists
have argued about longest — is reachable by damage, and that a whorled specimen
in a herbarium is not necessarily a specimen of a whorled kind.
Fig. 19 The reason the question cannot be settled by counting: two patterns written down from formulas, counted identically, with different symmetry. Counts do not see jugacy.
Three readings can be disposed of before the measurement, because each is checked
by something already in the runs.
They are not transients. Each destination is the mean over a whole number of
periods of the motif the stem repeats, taken from the last hundred and twenty
organs of a three-hundred-organ continuation, and the recovery test has already
refused to call these stems recovered. A stem still moving would not produce the
same four decimal places from arrangements that differ in which organs were taken.
Fig. 20 The evidence: after the transient, a fixed sequence repeating without end, which is what a destination means here.
They are not artefacts of the azimuth grid. The grid is 1,536 steps of 0.234°,
and the motifs at these destinations span hundreds of steps. A motif narrower than
a few steps is a constant the grid could not write down rather than an orbit, and
this collection has been caught by that once; the check has been in the machinery
since and these cells pass it comfortably.
Fig. 21 The check in question, and the reason for it: some of what a period-finder reports is the resolution the azimuths were computed at.Fig. 22 And the same result at a finer grid, which is what says the structure belongs to the rule.
And they are not on the ladder. Every slip’s divergence is the original plus a
whole number of turns of the family it kept; 175.01°, 189.96° and 235.00° are not
at any multiple of 72° from 136.78°, and none of them has a rigid lag at any
period up to twenty-four. Whatever they are, they are not the lattice the stem was
cut from, displaced.
One more thing is worth fixing before the test. The three destinations are not
rare corners of the sweep: the one at 175.01° is reached by a cut of two organs,
which is the smallest intervention that reaches anywhere other than a slip, and
the two at 189.96° and 235.00° by cuts of four. So whatever they are, they are
ordinary outcomes of an ordinary intervention rather than the tail of something.
There are two ways the three counts could have come about, and they make different
predictions about a measurement that has nothing to do with counting.
The first is that the cut really has produced a whorled arrangement. Then the
positions should have k-fold rotational symmetry: rotate the whole pattern by a
half turn, or a third, and every organ should land on another organ.
The second is that the stem is an ordinary one-organ-at-a-time spiral whose
two shortest families happen to share a factor. Then rotating it should land
nothing anywhere, because a spiral lattice with one organ per height has no
rotational symmetry at any order.
Fig. 23 How the counter finds a family, and why it cannot tell the two readings apart: it follows chains of near neighbours and reports how many there are.Fig. 24 The check that the counting machinery is right about jugate patterns when it is shown one: a whorled lattice, counted, and the divergence recovered from the counts.
The two readings are distinguished by one measurement on the positions, made
without reference to counts, and it has been in this collection’s machinery since
the whorled rule was first written — for exactly this reason. The next essay makes
it.
Fig. 25 The sweep the three destinations came out of. More organs removed wrecks more arrangements, and past a single organ the wrecked stems stop being describable as slips.
One more thing is worth fixing before the measurement, because it is the sort of
detail that decides an answer quietly. The symmetry is measured on the interior
of the pattern rather than on all of it. The top and bottom rows of a finite point
set have no partner to rotate onto, so a test applied to every organ reports order
one for every pattern including a genuinely bijugate one — which is a measurement
that cannot fail and therefore says nothing.
Fig. 26 The edge the correction is about: at the top and bottom of any finite pattern, a rotation has nothing to land on, whatever the pattern’s symmetry is.
The tolerance is a fraction of the median nearest-neighbour distance, computed
from the pattern itself rather than stated, so that a coarse arrangement and a
fine one are judged on the same terms. Both of those choices were made when the
whorled rule was written, against patterns whose answers were known, and neither
is being made here for this measurement.
Stated in advance, so that the answer cannot be arranged afterwards: if the
rotational symmetry comes back as two for the stem counted 2/6 and three for the
one counted 3/6, then a cut can change the jugacy of a pattern, and a great deal
of the literature on whorled arrangements has a new failure mode to worry about.
If it comes back as one, then the shared factor is a fact about where the stem’s
divergence landed, the counter is doing exactly what it should and being
misread, and the question becomes what those three divergences have in common.
Essays that name at least two of the same things, and that neither author linked.
What a cut costs a whorl— both name ablation, bijugate, counting blind, honest limits, jugacy, lattice, measurement, parastichy pair, rotational symmetry
Half a turn, four at a time— both name ablation, counting blind, classification, divergence angle, honest limits, lattice, measurement, parastichy pair
One turn per survivor— both name ablation, counting blind, honest limits, lattice, measurement, parastichy pair, rigid hop, slip
The pattern the cut leaves behind— both name ablation, counting blind, divergence angle, honest limits, jugacy, lattice, measurement, parastichy pair
Two accounts of one number— both name ablation, counting blind, claim testing, honest limits, jugacy, lattice, measurement, parastichy pair
A period that is not a count— both name ablation, counting blind, honest limits, lattice, measurement, parastichy pair, rigid hop