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.
At a rise of 0.060, thirteen of the thirty-six arrangements never repair. Every
one of them settles into the same thing, and it is not a
divergence. It is a sequence of
four:
171.09°, 269.53°, 189.14°, 90.23°, and then 171.09° again, without end.
Fig. 2 For comparison, wrecked stems at two finer rungs. Those also repeat a motif, and this essay is about what makes the coarse one different.
The four add to 719.99°, which is two whole turns over four organs. The mean is
180.0000° — half a turn — to four decimal places, and it is the same four
numbers at every one of the thirteen cells.
Fig. 3 The four rises where it happens, with the count of cells at each. Thirteen, twelve, seven and three, out of thirty-six arrangements tried at each rise.Fig. 4 What the stem was doing before the cut: the settled divergence across the coarse rung, flat within each rise and sliding steadily from one rise to the next.
Across the whole coarse rung, forty of the
seventy-one wrecked cells end in a cycle whose mean is within half a degree of a
half turn. Twenty-two reverse. Nine
do neither.
It does not, and the test is the one the fine rung’s result was established
with.
For each lag from one to twenty-four, take the angle from each organ to the one
that many places above it, over the last hundred and twenty organs, and compare
its mean with the same lag on the control. A surviving family shows up as a lag
whose spread is a fraction of a degree and whose mean has not moved.
Fig. 5 What a surviving family looks like when there is one: a single lag on the floor, in a stem whose own divergence has swung through eighty degrees.
In a cycling coarse stem there is no such lag. The stem’s divergence is not even
approximately constant — it takes four values in rotation, sixty-three degrees of
spread — so no lag has a steady hop to begin with, and the comparison has nothing
to report.
Fig. 6 The contrast at a rung that does slip: a wrecked stem with a settled divergence, which is what a rigid family produces.
That is the sharpest way to state what the two fates are. A reversal keeps
everything about the lattice except its handedness. A cycle keeps nothing: not the
divergence, not a family, not the counted pair. They are the two extremes of what
a removal can do, and the coarse rung is where both are available.
Fig. 7 The middle case, for completeness: at the finer rungs a wrecked stem keeps one family and slips along it, which is neither of the two things the coarse rung does.
A period reported by a period-finder is worth checking before it is believed,
because a motif narrower than the grid the azimuths are computed on is a constant
the grid could not write down rather than an orbit. This collection has been
caught by that once and the check has been in the machinery since.
Fig. 8 The check’s subject: how much of a reported structure is the pattern and how much is the resolution the azimuths were computed at.
The four-cycle’s motif spans 750 to 767 steps of the azimuth grid. The
mirrored cells’ motifs span zero to four. So the two populations are not near
each other and the distinction is not a matter of where a threshold is put: one
kind of cell is a constant the grid rounds and the other is an orbit that crosses
most of a turn on every pass.
Fig. 9 The same result checked at a finer grid, which is what says the cycle is a property of the rule rather than of the arithmetic.Fig. 10 A cell of the other kind, at a rise where the whole table reverses. Its divergence is a constant, and its motif spans nothing.
The motif is worth looking at closely, because its structure is not arbitrary and
the arithmetic in it is checkable.
The four divergences are 171.09°, 269.53°, 189.14° and 90.23°. Two of them sit
near half a turn and two sit near a quarter and three quarters — 90.23° and
269.53° are within a third of a degree of 90° and 270°. So the cycle alternates
between organs placed nearly opposite the previous one and organs placed nearly at
right angles to it.
Fig. 11 The stem the cycle is drawn from, before the cut. Three organs to a turn and a bit, which is the arrangement the four-cycle replaces.
The sum is 719.99°, which is two turns to within a hundredth of a degree. That
means the pattern closes: organ i and organ i + 4 sit at the same azimuth, four
heights up, which is exactly the statement that the four-hop is vertical and there
are four files.
Fig. 12 The hops of the lattice the stem came from, for comparison. Its shortest steps are the two and the three, and there is no four in it.Fig. 13 And the general fact behind a closing sum: whether a sum over the arrangement converges or runs away is a property of the geometry, and it is what makes some arrangements stable and others not.
At the rise of 0.055 there is a second cycle as well, of period six, averaging
198.91° and counted 3/4. Its motif is 280.6°, 165.9°, 107.1°, 160.1°, 271.9° and
207.9°, summing to 1193.5° — three turns and a hundred and thirteen degrees, so it
does not close, and its mean is nine degrees off half a turn. It is reported here
rather than folded into the count, because a table that folded it in would be
reporting one phenomenon where there are two.
A counter shown the positions of a cycling stem returns 1/4 — or 2/4, or at
one rise 3/4. Every one of those pairs has a four in it, and the lattice the
stem was cut from is counted 2/3.
Fig. 14 The counting machinery, run on a pattern it was not built for. It reports the chains it finds and does not know anything about what was done to the stem.
That is what a mean divergence of half a turn does. Two consecutive steps at 180°
come back exactly to the start, so consecutive organs alternate between two sides
and the pattern has files. The wobble around the mean turns two files into four:
the organs on each side are not quite aligned, so what would have been two rows
splits into a pair of near-rows each.
Fig. 15 The same effect at the finer rung, where a wrecked stem near half a turn is counted with a factor of two. The mechanism is the divergence, not the way the organs arrived.Fig. 16 Exactly half a turn, drawn. Two files and nothing else, which is the arrangement botanists call two-ranked.
The shared factor is doing here exactly what it does at the finer rung: reporting
where the divergence landed, not how many organs arrive at a time. Rotate a
cycling stem by a half turn or a quarter and nothing lands on anything.
Thirteen arrangements at one rise, and every one of them produces the same motif.
That is worth reporting on its own, because it says the cycle is a state rather
than a residue of the particular damage.
The cuts that reach it at a rise of 0.060 remove organs at offsets 1 and 3, 1 and
4, 2 and 3, 2 and 4, 2 and 5, 2 and 6, and so on — arrangements that differ in how
much of the front they take and in where they take it. All thirteen settle on
171.09°, 269.53°, 189.14°, 90.23°, to the resolution of the azimuth grid. Two of
them come out phase-shifted, starting the cycle at a different one of the four,
which is what a period-four orbit reached from a different transient should do.
Fig. 17 The shape of the table the thirteen come from, drawn at a finer rung. Every arrangement of two organs removed, an offset along one axis and the gap to the second organ along the other.Fig. 18 And the axis along which they differ, at the same finer rung: the second organ’s distance from the first, which changes the transient enormously and the destination not at all.
A residue of the damage would look different from cut to cut. A state does not,
and this does not. The same thing is true of the mirrored cells at the rises that
mirror: seven arrangements, one destination, agreeing to four decimal places.
Fig. 19 Both fates behaving the same way in this respect: at each rise, many arrangements and one or two destinations.
Wrecked stems that repeat a fixed cycle are not unusual. The 5/8 rung does it at
thirty of its thirty-two wrecked cells, and the periods those cycles have are
the periods of the lattice they came from — five, eight, and their multiples. That
is the whole subject of the essays on what a wrecked orbit inherits.
Fig. 20 The ordinary case: wrecked stems at four finer lattices, with the period of each one’s motif, and every period a number of the lattice that was cut.
What is unusual at the coarse rung is the cycle’s mean. Forty of seventy-one
coarse cells average half a turn to within half a degree. The nearest any cell at
either finer rung comes to half a turn is 175.01° at 5/8 — five degrees
away — and 187.47° at 3/5, which is seven and a half.
Fig. 21 The finer rung’s destinations on an axis, with half a turn marked. Nothing sits on it.Fig. 22 And the same destinations over the whole dose sweep at that rung, which is more than three hundred wrecked runs and still nothing at half a turn.
So the coarse rung has somewhere to go that the finer ones do not, and the
somewhere is the coarsest arrangement there is.
The three numbers do not add to seventy-one by accident, and the nine that are
neither are worth naming rather than left as a remainder.
Two of them are cells whose divergence sits near half a turn and whose motif the
period-finder could not resolve — the tail repeats something, but not at any
period up to twenty-four with a motif wide enough to be an orbit. Six are the
period-six cycle at 198.91° described above, which averages nine degrees off half
a turn. One is a cell at a rise of 0.075 averaging 179.97°, which is inside the
half-turn band and carries a period of eight rather than four.
None of the nine is a third destination in any interesting sense; they are two
resolution failures and a neighbouring cycle. Reporting them separately rather
than assigning them is the difference between a table that adds up and a table
that has been made to.
At a rise of 0.075, seven arrangements never repair. Four of them reverse onto
the mirror and three fall into the cycle.
The two-organ table this collection draws at finer rungs cannot be drawn here, and
the reason is worth more than the picture. Its own check requires the run of felt
offsets to end at the larger of the two counted numbers; at 2/3 a removal is still
felt four organs back against a larger count of three. The front runs deeper than
the count does, so the instrument refuses the rung rather than reporting from it.
Fig. 23 The fates across the rises either side of it: every arrangement of two organs removed, at each rise from 0.065 to 0.090, with the reversals and the cycles shared out between them.
That single row does more work than any of the others. It says the two fates are
not properties of the rise — not “this rise reverses and that rise cycles” — but
alternatives available to the same stem, chosen by which organs were taken out.
Fig. 24 What varies within that table: where the removals sit relative to each other and to the front, which is the only thing that differs between a cell that reverses and a cell that cycles.Fig. 25 And why the second organ’s position is not a small correction, measured where the front is deep enough to move it through: the second removal passes through the profile the first one deformed and swings the outcome without a trend in it.
Without that row, a table of nine rises would have read as a rise-by-rise
property, and the obvious next question would have been what changes about the
rise. With it, the question is what changes about the cut, which is the same
question the single-organ thread has been asking about which family survives.
One more control, because the obvious remaining explanation is that the cycle is
what a bigger cut does.
It is not. Every cell in these tables removes exactly two organs. What varies
between a cell that reverses and a cell that cycles is where the two organs sat,
not how many there were. At the rise of 0.075 the four cells that reverse and the
three that cycle are the same size of intervention applied at different offsets.
The finer rung tells the same story from the other direction. Cuts of one to five
organs there reach six destinations, the biggest cuts reach nowhere the smallest
did not, and the size of the cut moves the rate of wrecking from a quarter to
almost all while leaving the list of places unchanged. The dose decides whether,
and it does not decide where.
Fig. 26 The dose doing what a dose does, at the finer rung: more organs removed wrecks more arrangements, monotonically.
So the coarse rung’s two fates join a pattern this thread has now seen three
times. Which family a single removal leaves standing is decided by where it
landed. Which of six places a many-organ cut reaches is decided by the arrangement
and not the size. And which of two fates a coarse stem meets is decided the same
way.
The word is worth choosing carefully, because “the stem falls to a coarser
arrangement” is an interpretation and the measurement is narrower than that.
What is measured: a stem counted 2/3 is cut, does not repair, and ends in a
four-cycle whose mean divergence is half a turn and whose positions a counter
reads with a four in the pair. Half a turn is the two-ranked arrangement, which is
below 1/2 on the ladder of rungs — the coarsest thing a stem can be.
Fig. 27 The ladder the stem is on, and the end of it. There is nothing below two-ranked, which is why this outcome is available to the coarse rung and to nothing finer.Fig. 28 And the shape of the coarse end generally: arrangements with a handful of organs to a turn, where the distinctions the fine end depends on stop being available.
What is not measured: that the stem has “gone down a rung” in the sense the rise
sweep means, because the rise has not changed and the arrangement does not settle.
A rung member has a divergence. This has four.
Fig. 29 For the difference: the ladder as the rise sweeps it, where every rung member is a settled divergence and none of them is a cycle.
So the honest form is that a coarse stem, damaged, can lose its divergence
altogether and end up oscillating about the one angle at which a stem has no
spiral structure left. Whether that deserves to be called a rung is a question
about vocabulary. That it is available at a front of three and not at a front of
five or eight is a measurement.
Essays that name at least two of the same things, and that neither author linked.
The stem that changed hands— both name ablation, attractor, counting blind, divergence angle, equilibrium, handedness, honest limits, lattice, measurement, parastichy pair, rung
The block is the count it was cut from— both name ablation, attractor, counting blind, divergence angle, equilibrium, honest limits, lattice, measurement, parastichy pair, rung
A stem on the other branch— both name ablation, attractor, counting blind, honest limits, lattice, measurement, metastability, parastichy pair, rung
A wreck has a short list— both name ablation, attractor, counting blind, equilibrium, honest limits, lattice, measurement, negative result, parastichy pair
The pattern the cut leaves behind— both name ablation, attractor, counting blind, divergence angle, equilibrium, honest limits, lattice, measurement, parastichy pair
The share was not the thing— both name ablation, attractor, equilibrium, handedness, honest limits, lattice, measurement, negative result, rung