Phyllotaxis
the spirals are counted, not admired
Stems and cones

Two files, and a way back

Twelve wrecked runs finish at exactly a half turn, which is a pattern with no spiral in it, and every one is at the coarse end of the ladder. Three finish at the divergence they would have had anyway, after being thrown a hundred degrees off it.
14 min read 20 figures Counted, not admiredState the claim, state the testForm is not mechanism

Worth reading first: Both walls of the slot · The organ that was taken away.

Sixty-three of the slot design’s ninety cells wreck the stem: the pattern above the hole never returns to the arrangement it had. Where those runs finish falls into a small number of places, and two of the places are worth separating from the rest.

Twelve finish at a half turn. Eight of them read exactly 180.000 degrees and the rest sit within 0.35 of it. Three finish at the divergence their own intact stem settles to, after being thrown between seventeen and a hundred and twenty degrees off it by the cut.

The twelve wrecked runs that finish at a half turn. Wrecked runs whose final divergence is a half turn, which is two files of organs rather than a spiral. Every one is on a rung at the coarse end of its branch, where the front is short enough that removing one organ reaches past it. The value is within 0.35 degrees of 180 on eight of the twelve.
Fig. 1 The wrecked runs that finish at a half turn, with each row’s own settled divergence marked.

What a half turn is

A divergence of 180 degrees puts every organ directly opposite the one before it, so the stem has two files running up it and no spiral at all. There is no parastichy pair to count and no contact family in the usual sense — the arrangement is degenerate.

It is a real arrangement rather than a failure to have one, and this site has a whole reading of what a count is worth on one. A stem at 180 degrees is a stem whose placement rule has settled into the simplest packing available, and it is the one every sequence of divergences passes through if it is going anywhere near a half turn.

What it is not is a lattice of the kind this site counts. Everything the counter returns is built on two families of spirals crossing, and two files have neither.

A whorl and a spiral, from one lattice at two divergences. At 180° the nodes fall on 2 rows and the two counts share a factor: the elements of each turn are level with one another. At 137.51° they never are.
Fig. 2 The arrangements a stem can be in, of which two files is the degenerate member.

Where they are

Every one of the twelve is on a rung at the coarse end of its branch: six on the Lucas 1/3, five on the golden 2/3, and one on the Lucas 3/4.

The rises are 0.049 to 0.069, which is the top of the ladder. Nothing at a rise below 0.028 finishes at a half turn.

That is a clean split rather than a tendency. The coarse rungs’ wrecked runs go to two files and the fine rungs’ go somewhere else, with one rung straddling.

Where every wrecked run finishes, rung by rung. One row per rung of the ladder, one mark per cut cell of the slot design, placed at the divergence that run ended on. The open mark on each row is the divergence the intact stem of that rung settles to. On the Lucas 1/3 and golden 2/3 rungs every wrecked run finishes at the same value; on the golden 5/8 they finish at 13 values spanning 215 degrees. 27 cells recover, and each of those finishes at its own settled divergence to within 0.03 degrees.
Fig. 3 Every wrecked endpoint by rung, with the half-turn cluster confined to the coarsest rungs.

Why the coarse end

At a rise of 0.067 the stem is short and fat: consecutive organs are far apart up the axis relative to the cylinder’s circumference, so the neighbourhood a new organ is placed against holds few organs.

How far back a removal still wrecks a stem is a measurement, and at the coarse end it reaches only two or three organs. So a cut there removes a large fraction of what the next placement is deciding against.

With few neighbours and one of them gone, the sum the rule minimises has fewer distinct minima, and the arrangement it falls into is the one with the fewest constraints — which is two files.

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. 4 How far back a removal still wrecks, which at the coarse end is only two or three organs.

The value is exact, and that matters

The azimuth grid is 1,536 samples of the circle, a quarter of a degree a step, and 180 degrees is sample 768 exactly. Eight of the twelve read 180.000.

A run drifting towards a half turn and being rounded there would scatter within a step. These do not scatter: they are the same number.

So the arrangement is one the grid represents without rounding, and the runs are sitting on it rather than near it. That rules out the reading in which the half turn is a display artefact.

The twelve wrecked runs that finish at a half turn. Wrecked runs whose final divergence is a half turn, which is two files of organs rather than a spiral. Every one is on a rung at the coarse end of its branch, where the front is short enough that removing one organ reaches past it. The value is within 0.35 degrees of 180 on eight of the twelve.
Fig. 5 The twelve half-turn endpoints, eight of them at the grid’s own exact 180 degrees.

What the four others do

Two read 179.97 and one reads 180.35, which is one and a half grid steps out. Those are on the golden 2/3 and Lucas 3/4 rungs, at the boundary of the region.

So the cluster is not perfectly sharp, and the imperfect members are the ones at the edge of the stretch of rise where it happens. That is the ordinary shape of a boundary rather than noise in the reading.

Read without handedness the whole group sits between 179.65 and 180.00, which is under half a degree wide over twelve readings from three rungs and two branches.

Wrecked endpoints against the settling table's destinations. The upper lane is the 15 divergences the settling table reaches, grown from intact stems started at nine arbitrary angles across four falloff exponents and eight rises, with no cut anywhere in them. The lower lane is where the slot design's 63 wrecked runs finish, read without handedness so that a run ending at 209 degrees is placed at 151. 29 of them sit within 1 degree of a destination and 34 do not. The two measurements share no run and no design, so the agreement is not a construction.
Fig. 6 The wrecked endpoints folded to remove handedness, where the half-turn group is under half a degree wide.

It is not a destination the settling table reaches

The settling table grows intact stems from a range of starting angles and records where they settle. Its list of fifteen destinations does not include 180 degrees, and the nearest is 157.7 — twenty-two degrees away.

That is not an omission. A stem started at 180 degrees is started at two files and stays there; the settling table’s job is to find where a stem goes, and a run that never leaves its starting angle contributes an uninteresting fixed point.

So the half turn is an arrangement the rule can be driven into by damage and does not find by itself from anywhere else. Half the wrecked runs land on destinations the settling table does reach; these twelve are most of the other half.

Every destination in the settling table, counted. One row per destination, placed across at the divergence the stems settled on and labelled with the parastichy pair a counter returns at the top of the run. 117 settled runs land on 15 destinations when two values within half a degree are called one. five of them sit on the golden or the Lucas sequence and ten do not, and the ones that do not are reached at every falloff exponent including the shallowest. Nothing here refuses to count: the settling test does not admit arrangements this collection would decline to call lattices.
Fig. 7 The settling table’s destinations, none of which is a half turn.

Two files elsewhere on this site

The arrangement has appeared before, from the opposite direction. A whorled stem placing several organs at once can hold a half turn as its own stable pattern, and there the question was whether a counter can tell a whorl from a spiral.

The connection is not established and should not be asserted. A wrecked spiral falling to 180 degrees and a whorled stem sitting at 180 degrees are the same divergence reached by different routes, and nothing here checks whether the arrangements are otherwise alike.

What can be said is that the value is not arbitrary: it is the one divergence on the circle with a symmetry, and two separate threads on this site arrive at it.

A whorl and a spiral, from one lattice at two divergences. At 180° the nodes fall on 2 rows and the two counts share a factor: the elements of each turn are level with one another. At 137.51° they never are.
Fig. 8 The same divergence reached from a different direction, where a whorled stem holds a half turn.

The three that come back

On the Lucas 7/11 rung, two cells finish at 99.21 degrees against their own stem’s settled 99.08. On the golden 8/13, one finishes at 137.83 against a settled 137.84.

Each of those cuts wrecked the stem, which by definition means the pattern above the hole never returned to the arrangement it had. And each finished at the arrangement it would have had.

The first organ placed after the cut moved 17, 18 and 120 degrees respectively. So these are not gentle cuts that barely disturbed anything; the third is one of the largest displacements in the whole table.

The six wrecked runs that finish where they would have. Wrecked runs whose final divergence is the one their own intact stem settles to, read without handedness. Each was thrown well off course by the cut — the first organ placed after it moves between 18 and 120 degrees — and each finished where it would have anyway. So a wrecked stem is a statement about a placement and not about a destination.
Fig. 9 The wrecked runs that finish at the divergence their own intact stem settles to.

Which is what “wrecked” means

A cut is called wrecked when the displacement — the difference between the cut run and its control, organ by organ — does not fall back to nothing and stay there. That is a statement about the two runs being out of step.

Two runs can be permanently out of step and still be at the same divergence. If the cut run is the control’s arrangement shifted by one organ, every organ is displaced and the divergence is identical.

So a wrecked run finishing at its own settled divergence is not a contradiction. It is a run that recovered the arrangement and not the registration — the organs sit at the same angles and the wrong ones sit at each — and the reading was never about registration.

How far every organ moved, 6 places back at a rise of 0.01. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 18 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. six of those levels sit together and two do not.
Fig. 10 The displacement profile a wrecking verdict is read from, which measures being out of step rather than being elsewhere.

Which narrows a word

Wrecked has been doing double duty. In the ablation thread it means the displacement never returns, which is what the census measures and what every claim about surviving families is about.

Read casually it suggests the stem ended up somewhere else, and on three of sixty-three cells it did not. The three are a small minority and they are enough to make the distinction worth stating: wrecking is about the relation between two runs, not about where one of them finishes.

That is the same separation the fine sweep of one rung made between the first organ’s displacement, which is reproducible, and the run’s endpoint, which is not.

The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise.
Fig. 11 The reproducible half of a wrecked run, which is the displacement of the organ placed after the cut.

What a slipped chain would look like

The ablation thread’s own account of a wrecked profile is that one chain of organs has slipped by a place: the displacements fold onto the surviving lag with most classes at a common level and one or two exceptional.

A stem in which every chain has slipped equally is a stem at the same divergence with every organ displaced, which is exactly what these three look like. The check is not made here — it needs the profile rather than the endpoint — and it is one cheap reading away.

If it holds, the three are the clean case of the whole account: a wrecked stem that has done nothing but shift.

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step.
Fig. 12 A wrecked profile folded onto its surviving lag, where a uniform slip would put every class at one level.

Handedness

The three are read with handedness removed, because 209.56 degrees and 150.44 are the same lattice wound in opposite directions and the slot table reports the raw angle.

Two of the three read directly at their own value — 99.21 against 99.08 — so no folding is needed. The third also reads directly.

That is worth checking rather than assuming, because a run that finished at the mirror of its own settled divergence would be a different and more interesting object: a stem that recovered the arrangement and reversed its chirality. None of the sixty-three does that.

Wrecked endpoints against the settling table's destinations. The upper lane is the 15 divergences the settling table reaches, grown from intact stems started at nine arbitrary angles across four falloff exponents and eight rises, with no cut anywhere in them. The lower lane is where the slot design's 63 wrecked runs finish, read without handedness so that a run ending at 209 degrees is placed at 151. 29 of them sit within 1 degree of a destination and 34 do not. The two measurements share no run and no design, so the agreement is not a construction.
Fig. 13 The endpoints folded to remove handedness, where no run finishes at the mirror of its own lattice.

What the two groups have in common

Nothing except being the two ends of the same column. The half turns are the coarse rungs’ answer and they are a degenerate arrangement; the returns are three cells on the two finest rungs and they are the least degenerate outcome available.

Between them sit the other forty-eight, most of which land on a destination the settling table reaches and none of which is remarkable in either direction.

So the endpoint column has a structure: a degenerate attractor at the coarse end, the rule’s ordinary arrangements in the middle, and a handful of runs that went nowhere at all.

Where every wrecked run finishes, rung by rung. One row per rung of the ladder, one mark per cut cell of the slot design, placed at the divergence that run ended on. The open mark on each row is the divergence the intact stem of that rung settles to. On the Lucas 1/3 and golden 2/3 rungs every wrecked run finishes at the same value; on the golden 5/8 they finish at 13 values spanning 215 degrees. 27 cells recover, and each of those finishes at its own settled divergence to within 0.03 degrees.
Fig. 14 The whole column, with the degenerate endpoints at the coarse end and the returns at the fine end.

What is not checked

Whether any of these endpoints holds. The slot design’s runs are three hundred organs, which is short by the settling table’s standards, and a run’s last divergence is not the same as a run’s destination.

The half turns are the safer of the two groups on that count, because eight of them read the grid’s exact value and a run passing through a value by accident does not land on it exactly. The three returns are the less safe: a run near its own settled divergence at organ three hundred might be crossing it.

Sixty-three long runs would settle it and have not been grown.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 15 How long an intact run takes to settle, against which three hundred organs is short.

The rungs that do it, in full

The Lucas 1/3 rung holds three lattices at rises 0.06917, 0.06727 and 0.06542. At every one of them the larger-wall removal and the doubled removal both wreck, and all six runs finish at 180.000. The smaller-wall removal recovers at all three.

The golden 2/3 rung holds three at 0.06594, 0.05737 and 0.04991. The first two wreck at the smaller wall and the doubled cut and finish at 180.00 or 179.97; the third recovers at all three cells and contributes nothing.

The Lucas 3/4 rung’s coarsest lattice, at 0.05521, wrecks at the smaller wall and finishes at 180.35 — the one member of the group more than a grid step out, and the finest rise in it.

The slot interaction at 30 lattices, gathered by rung. One row per lattice, drawn at how much further the next organ moves when both walls of the slot are removed than the two single removals added together account for. Zero would mean the walls act independently. The pale rows are the ones where removing the second wall costs nothing at all, so their value is minus the first wall's own cost and is arithmetic rather than a measurement. Of the 24 rows that are measurements, 13 are strongly positive and 11 are not, and every rung falls on one side or the other with nothing straddling.
Fig. 16 The slot design across the whole ladder, whose coarsest rungs are the ones that wreck into two files.

What the counter says about a half turn

Nothing usable, and that is the point. The counter returns the two lags whose hop across the surface is shortest, and on a stem at 180 degrees the shortest hop is 2 — every organ is opposite its predecessor, so the second-nearest is two places away — and the second shortest is a long way behind.

So a half-turn stem has a counted pair in the formal sense and it is not a pair of spiral families. Anything reading it as a lattice would report a 2 and something arbitrary.

That is why these twelve runs are excluded from every statement in this thread about counted pairs and surviving families. They are wrecked runs whose endpoint is degenerate, and the endpoint is all that is read off them.

The two spiral families a counter finds between 0.43 and 0.67 of the radius21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.21 and 34 spiralscounted, not assumed
Fig. 17 What the counter returns on an ordinary lattice, against which a half turn has no second family.

Why the coarse end is where degeneracy lives

The whole ladder is a sequence of rungs and at the coarse end the rungs are shallow: the counted pair is 1/3 or 2/3, which is two or three spirals, and a pattern with two spirals is one step from having none.

At the fine end the pairs are 7/11 and 8/13 and the arrangement has a great deal of structure to fall back on. A cut removes one organ from a front holding eight or eleven, and the rest carry the pattern.

So the split in the endpoint column is a split in how much redundancy the arrangement has, which is the same quantity the front’s depth measures from a different angle.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.382, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 18 The ladder from coarse to fine, whose pairs grow from 1/3 to 8/13 down its length.

Whether a half turn is stable

Not checked, and it is the obvious next reading. A run at 180 degrees at organ three hundred might be sitting there or passing through, and the two are told apart by growing it further.

The evidence that it is sitting there is that eight of the twelve read the grid’s exact value: a run crossing 180 would be at 180 for one organ and elsewhere at the next, and reading it at organ three hundred exactly would be a coincidence eight times over.

The evidence against is that nothing has grown one of these runs to a length where the settling criterion could be applied. Both readings are available and the first is much more likely.

How many organs settling takes, by falloff exponent. One mark per stem that settles, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. At the exponent every other measurement here uses, the slowest is 290 organs — the number four rounds of this collection have carried as though it belonged to the rule. It belongs to the exponent: at 4 the slowest is 808 and at 5 it is 674. A table of shares cannot see this, because a stem that settles at organ 800 and one that settles at organ 8 are the same entry in it. And it is still not a shortage of time: of 72 pairs of runs grown to 1200 organs and then to 3200, 0 settle at the longer length after failing at the shorter.
Fig. 19 The reading that would decide it: whether a run’s divergence has stopped changing.

Why a degenerate arrangement is worth counting

Because it is a real outcome of the rule and not a failure of the run. A stem at a half turn has been placed by the same minimisation as every other stem here; it has simply arrived somewhere with no spiral in it.

The temptation is to treat it as an error state and drop it, and that would be a mistake of the kind this collection is built to avoid. The rule’s range is what it is, and a reading that silently discards the outcomes it finds uninteresting is a reading of the reader rather than of the rule. Twelve of sixty-three is nearly a fifth of the wrecked runs.

It also carries a fact about where the ladder stops being interesting. Every half-turn endpoint is at a rise above 0.049, which is the top two rungs; below that, no run finishes there. So the coarse end of this ladder is where the pattern has least to fall back on, and a single removal is enough to collapse it.

What the coarse end is like

Worth a paragraph because it is easy to forget when most of the work happens at the fine end. At a rise of 0.067 the counted pair is 1/3 — one spiral one way and three the other — and the front a new organ is placed against holds two or three organs.

Take one of them away and the next placement is decided by one or two neighbours. There is very little for a pattern to be, and the arrangement with the fewest constraints is two files.

At the fine end the pairs are 7/11 and 8/13 and the front holds eight or more. A removal there is one neighbour of many, the rest carry the arrangement, and the run finishes somewhere with a counted pair. That difference is what the endpoint column is measuring when it is read across the whole ladder, and it is a property of the ladder rather than of the cut.

What is not measured about either group

Whether the endpoints hold. Every one of these readings is a run’s divergence at organ three hundred, and a run still moving reads the same as a run that has stopped.

For the half turns the evidence is indirect and good: eight of twelve land on the grid’s exact 180 degrees, and a run crossing a value reads it exactly only by coincidence. For the three returns there is no such argument, and a run near its own settled divergence at organ three hundred might be passing through.

Sixty-three long runs would settle both, and they have not been grown.

What is claimed

That twelve of the sixty-three wrecked runs in the slot table finish at a half turn, eight of them at exactly the grid’s 180 degrees; that all twelve are on rungs at the coarse end of their branch, at rises above 0.049; and that a half turn is two files of organs rather than a lattice with a counted pair.

That three wrecked runs finish at the divergence their own intact stem settles to, after first-organ displacements of 17 to 120 degrees, and that this is consistent with the meaning of wrecked rather than a contradiction of it.

And that none of the sixty-three finishes at the mirror of its own settled divergence.

The twelve wrecked runs that finish at a half turn. Wrecked runs whose final divergence is a half turn, which is two files of organs rather than a spiral. Every one is on a rung at the coarse end of its branch, where the front is short enough that removing one organ reaches past it. The value is within 0.35 degrees of 180 on eight of the twelve.
Fig. 20 The two groups at the ends of the endpoint column: a degenerate arrangement and a return.

Where this goes next

Where every wrecked run finishes, rung by rung. One row per rung of the ladder, one mark per cut cell of the slot design, placed at the divergence that run ended on. The open mark on each row is the divergence the intact stem of that rung settles to. On the Lucas 1/3 and golden 2/3 rungs every wrecked run finishes at the same value; on the golden 5/8 they finish at 13 values spanning 215 degrees. 27 cells recover, and each of those finishes at its own settled divergence to within 0.03 degrees. The column nobody read Every cell of the slot design carries where its run finished as well as how far its first organ moved. One rung's worth had been plotted and called unusable. Reading all eight says the endpoint is exact on two rungs, wanders on six, and is worst on the one it was read on. Wrecked endpoints against the settling table's destinations. The upper lane is the 15 divergences the settling table reaches, grown from intact stems started at nine arbitrary angles across four falloff exponents and eight rises, with no cut anywhere in them. The lower lane is where the slot design's 63 wrecked runs finish, read without handedness so that a run ending at 209 degrees is placed at 151. 29 of them sit within 1 degree of a destination and 34 do not. The two measurements share no run and no design, so the agreement is not a construction. A wrecked run goes somewhere Where a wrecked stem finishes was called unstable. Half of them finish within a degree of a destination measured from intact stems started at arbitrary angles — two tables that share no run, no design and no question. What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise. Half a turn, four at a time At four of the nine coarse rises no wrecked cut reverses. What those stems do instead is stop settling: they repeat 171.09°, 269.53°, 189.14°, 90.23° without end, which adds to two whole turns over four organs. The mean is exactly half a turn and a counter finds four files where the lattice had three.

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.

Named objects

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AblationAttractorAzimuth gridClaim testingDivergenceHandednessHonest limitsMeasurementRotational symmetrySlotTransientWhorled