Concept

Summary statistic — where it appears

A single number standing for a whole measurement, which discards whatever it is invariant to. A mean side count is the standing example here: Euler's relation fixes it at six, so it is the same for every tissue and reports nothing about any of them.

Named by 83 essays across 8 fields — each of them below, with the objects they name alongside it.

The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

Which junctions say anything

Da Vinci's rule and Murray's law differ by 12% at an even fork and by a tenth of a per cent at a twig. So one even fork settles which is right, and two thousand twigs do not — a factor of two thousand across a tree, decided entirely by the shape of the junction and not by how carefully it is measured.

branching · Exponent
The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

A sample that is confidently wrong

Fifty lopsided junctions from a tree built at an exponent of exactly 3 return 1.7, with an interval that excludes 3 and excludes 2 as well. The sample carrying almost no information does not give a wide answer — it gives a narrow wrong one, and the cause is a selection nobody applies on purpose.

branching · Exponent
Two laws, two tilings, and they disagree about which tiling is tissue. Lewis's law wants disorder: its slope is 0.231 on the random set and 0.009 on the golden head. Aboav's relation wants order: a = 1.18 on the head, 0.59 on the random set.

Two laws that want opposite tissue

Lewis's law and Aboav's relation are quoted side by side as properties of cellular tissue. Measured on the same two tilings they point opposite ways — the ordered head satisfies Aboav's with the textbook value of 1.18 and fails Lewis's completely; the random set does exactly the reverse.

tissue · Cell laws
The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

The second moment is the measurement

The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. On heads of nine hundred organs the mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.

tissue · Sixsides
One geometry, six boxes: 4.642 per cent to 52.81 per cent forbidden. The share of each box that the coiling geometry excludes, across six boxes that differ only in where their edges were put and in how one axis is sampled. The spread is a factor of 11.4, from 4.642 per cent in the widest box to 52.81 per cent in the tightest, and sampling the same two decades of expansion geometrically rather than uniformly multiplies the answer by 4.59 on its own. The forbidden area is the integral of 1/W and grows as a logarithm while a box grows as a line, so the fraction has no value of its own to quote.

A fraction of nothing

Six boxes differing only in where their edges were drawn give between 4.64 and 52.81 per cent for the same geometry, and sampling one axis geometrically rather than uniformly multiplies the answer by 4.60. The share tends to zero as the box widens, because the region under a hyperbola is a logarithm and a box is a line.

shells · Morphospace
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.

A counter that sees no positions

This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.

lattices · Counting
One junction's bias moves 1.36-fold across the range its leverage moves 308,352-fold, at an exponent of 3. The bias a single junction of daughter ratio γ contributes to a least-squares fit, as a multiple of the squared measurement error, drawn against the leverage that junction carries — both computed from the expansion about a true exponent of 3 rather than fitted to anything. Across the whole range from an even fork to a twentieth the bias moves by a factor of 1.36 and the leverage by a factor of 308,352, and the uninformative junction contributes the larger share: 6.00σ² at γ = 0.05 against 4.50σ² at an even fork.

The fragile junctions are the informative ones

That is the obvious worry once the radii are uncertain, and it is false. Across the whole range of asymmetry a junction's contribution to the bias moves by a factor of 1.36 while its leverage moves by a factor of 308,352, so the junction that says nothing damages the answer as badly as the one that says everything — and a sample is spoiled by counting rather than by weight.

branching · Exponent error
The disorder of a head against its divergence angle, 300 points. μ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 300 points inside 86% of the radius. Swept across 1.60° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.142° — which is 360 × 8/21 — it is 0.078; At 137.646° — which is 360 × 13/34 — it is 0.197; At 138.458° — which is 360 × 5/13 — it is 0.060. The ticks along the top are the fractions p/q, placed from arithmetic rather than from the curve.

The disorder is a staircase

Sweep the second moment of a head's side-count distribution across the divergence angle and it is not a curve. It is flat in stretches with sharp steps between them and narrow deep dips wherever a rational falls — so 0.253 is not the golden angle's number, it is the number of every angle from 137.47° to 137.54°.

tissue · Second statistic
The dip at 5/13 — 138.4615° — at three head sizes. Walking the divergence angle off an exact rational, at 300, 600, 1200 points. The floor falls as the head grows — 0.059 at 300, 0.029 at 600, 0.014 at 1200, halving for each doubling — and the dip narrows faster: half-widths of 0.0162°, 0.0048°, 0.0010°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1456, 1726, 1402, which is what makes the width a property of the sample rather than of the angle.

A dip belongs to the head

At an exact rational the disorder halves when the head doubles, and the dip around it narrows by a factor of four. So which angles look ordered is set by how many organs were counted, and a head of three hundred cannot tell 138.4615° from 138.48° while a head of twelve hundred tells it from 138.4625°.

tissue · Second statistic
The neighbourhood of 21/55, and where its background was taken from. μ₂ across nine tenths of a degree either side of 21/55, on a head of 825 organs — 15 in each of its 55 rows. The deep notch at the centre is the dip. The shaded columns are the offsets now used as a background: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 54 here. The marked sample at 0.2° is the one the earlier work used, and it lands 0.007° from 13/34 — inside that rational's own dip. It reads 0.115 against a floor of 0.148, so measured that way the dip is an inversion. The clear offsets give 0.339.

The background is not one sample

The dip in disorder at a rational angle is a comparison against a background, and the background was one measurement taken two tenths of a degree away. At 21/55 that lands seven thousandths of a degree from 13/34 — inside another rational's dip — and the comparison inverts. Fixed, the dip survives to a denominator of 89.

tissue · Second statistic
What the experiment costs, in internodes. The combined sampling band of two autocorrelations falls as one over the root of the sequence length. The difference to be resolved is 0.76 — between noise that arrives before the primordium is placed and noise that arrives after — so the count needed is 56 internodes on a single stem. Every other open question in this collection is priced in tens of specimens.

The test a plant could settle

Every other open question in this collection is priced in tens of specimens, and one of them in a hundred and sixty. This one is priced in internodes on a single stem, and the number is fifty-six — because it is a statistic of one sequence rather than a share of a population.

wrong · Sequence
The coefficient is not one number. n²·w, measured at the two larger of each denominator's three head sizes and averaged, for six Fibonacci fractions. Undivided it runs 1188, 1285, 2999, 7005, 9125, 14297 — a spread of 12.0. Divided by the denominator it runs 149, 99, 143, 206, 166, 161 — a spread of 2.1. So the law is w ≈ 150·q/n² to within a factor of two, and that earlier work's constant of 1,400 to 3,300 was three measurements of the small-denominator end of it.

The width carries the denominator

The earlier work measured three denominators, found the dip's half-width falling as the square of the head size, and could not say whether its coefficient depended on the denominator. Six denominators say it does: the coefficient runs from 1,188 at q = 8 to 14,297 at q = 89, and dividing by q flattens a factor of twelve into a factor of two.

tissue · Second statistic
One wrecked stem, lag by lag — golden, rise 0.005, organ 6 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 55 degrees. The lag-4 hop swings by 0.12 degrees and sits 0.01 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 4, which is the surviving lag and not a coincidence.

A period that is not a count

Eighteen wrecked stems settle into a block whose period is one of their own spiral counts, and one settles into a block of four on a lattice counted 8 and 13. The odd one is not noise. It is the case that shows what the rule is actually conserving, and it is the reason this thread is about lattice steps rather than about spirals.

lattices · Counting
Four fractions of 55, one width. The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 1279 organs — 23 in each of 55 rows. 21/55 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 12/55, 23/55, 17/55 are not. The four agree within a factor of 1.28, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.

Four fractions with one denominator

The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.

tissue · Second statistic
Four fractions of 34, one width. The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 791 organs — 23 in each of 34 rows. 13/34 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 9/34, 15/34, 11/34 are not. The four agree within a factor of 1.14, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.

A width read off a staircase

Two fractions of the fourteen measured return a dip width that moves by a factor of two when the head size changes, where the others hold to three per cent. The cause is not their neighbourhood. It is that the disorder statistic changes only when the tessellation changes, so the curve a half-width is read off is a staircase, and a width narrower than the tread cannot be read at all.

tissue · Second statistic
The measurement is limited by the protractor, not by the plant. The peak falls as the reading error grows, and it falls by an arithmetic factor with nothing fitted: a position error enters two consecutive divergences with opposite signs, adding variance at every lag while the pattern's signal sits at one. At a quarter of a degree the readout is right on all 5 runs; at half a degree on 2; at a degree on 1. Below the dashed floor the peak is the largest of thirty noisy numbers rather than a measurement.

What the protractor has to be

The readout that names the parastichy number costs sixty internodes, which is cheap. It also needs every organ's position measured to better than a quarter of a degree, which is not — and the requirement follows from arithmetic rather than from care, so no amount of averaging relaxes it.

wrong · Sequence
What insisting on a fork angle costs: 43.3 degrees for one per cent of the network. The three ends are held where the optimum wants them, the branch point is moved everywhere inside them, and the cheapest network at each total angle is kept. The optimum sits at 74.93°. Everything within one per cent of the least cost runs 55.9° to 99.2° — a span of 43.3°, which read back as exponents covers 2.44 to 5.34 — and within a tenth of a per cent it still runs 13.6°. The prediction is steep in the exponent and the cost is nearly flat in the angle; they are the same curve read along its two axes.

An optimum too flat to reach

One per cent of a branching network's cost buys forty-three degrees of fork angle, covering exponents from 2.44 to 5.34, while the angle the theory predicts moves only fourteen and a half degrees across every daughter ratio there is. The prediction is steep and the cost is flat, and those are the same curve read along its two axes.

branching · Fork angle
The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.

A dip with no outer edge

The disorder of a head dips at every rational divergence, and how wide that dip is has carried a long argument. Reading the width as a level crossing has a resolution problem, and the obvious repair is to integrate instead. The integral reproduces beautifully across head sizes and never settles on a value, because there is nothing out there for it to settle against.

tissue · Second statistic
Two combs, at a rise of 0.005. The autocorrelation of 760 divergence angles from one stem held at a rise of 0.005. The filled teeth are the lags at multiples of 8; the open teeth are the second comb, at the same spacing offset by 5. Reading the spacing off the first and the offset off the second gives the pair 8 and 13, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.

A disturbance that is not passed on

The disturbance that forges every observable measured here does two things at once — it correlates an organ's error with its contact neighbours', and it hands that error on to be handed on again. Every result about it has been unable to say which half did the work. This is the control that takes the second half away and keeps the first.

mechanism · Noise colour
The memory of a divergence sequence, at 0.75° of scatter. With no noise at all the lag-one correlation is 0.54: the rule corrects itself, so a lattice arrives with a memory in it. Matched at the same recorded scatter, placement noise leaves -0.10, jostle noise leaves 0.66, field noise leaves 0.47. The band is ±0.13, which is what an uncorrelated sequence of this length gives.

The sequence has a memory

Every measurement this collection has made of a stem's divergence angles throws the order away. A spread is invariant to shuffling. Put the angles back in order and there is a large correlation between one and the next — 0.54 with no noise at all — which is the rule correcting itself, and which nothing had looked at.

cylinder · Sequence
The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

What a summary throws away

Four statistics this collection has relied on turn out to be incapable of varying with the thing they describe — one is invariant to shuffling, one is fixed by a theorem, one is a parameter that stopped mattering, one is a fitted number selected into being wrong. In each case the second statistic was free and nobody had taken it.

wrong · Second statistic
How far a shell growing from 2.8 to 3.6 a turn departs from the one spiral a fit gives it. The shell's logarithmic radius along its 4-turn arc, less the straight line a single growth factor fits. The fit returns 3.17490 a turn, which is the geometric mean of the two ends, 3.17490. The largest departure is 0.0836 in the logarithm against the 0.15 the collection refuses a spiral past, so the fit accepts this shell; a change that is steady in its rate leaves ln(W₁/W₀) × turns/12 = 0.0838.

One number for a shell that changes

An animal is under no obligation to grow at one rate from hatching to maturity, and the fit that recovers a shell's growth factor returns one number whatever it is given. Handed a shell whose expansion rises steadily from 2.8 to 3.6 a turn, it returns 3.17490 — the geometric mean of the two ends, exactly — with a residual it accepts. A change of sixty-four per cent over three and a half turns passes as one logarithmic spiral, and at the aperture, where contact is decided, the one number and the last whorl give opposite verdicts.

shells · Spiral fit
Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments.

The pair read from the angles

Checking that a jostled stem is still the lattice its panel is about turned up a disagreement. Counted from the point positions every rule's stems return five and eight spirals; read from the divergence sequence, the deepest rule's stems come back as five and seven at every seed. The points are right, and this site's founding rule is why.

lattices · Counting
The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 55, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0067° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 595 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.

The window is the neighbour

An integral needs a limit, and this one has two conditions on it that pull opposite ways. It has to scale with the dip, so that two head sizes are comparable, and it has to stay clear of the next rational, which is a fixed distance in degrees. Between them there is no stretch where the answer holds still — and the limit that decides it is the crowding.

tissue · Second statistic

What one angle says about the next

A tenth of a degree of placement noise moves a stem's divergence scatter from 0.50° to 0.62°, which nobody would report. It takes the correlation between consecutive angles from 0.54 to below zero. The other two kinds of noise, at scatters where no measurement can separate them, leave it at 0.6.

cylinder · Sequence

What the pair costs

The single parastichy number cost sixty internodes. The pair costs two hundred and fifty, and a protractor error of three quarters of a degree takes it to eleven hundred. The arithmetic that predicts the second of those is right about the shape and wrong about the scale by a consistent factor, which is recorded rather than fitted away.

wrong · Survey spec

A difference forgets a drift

This collection proposed a second observable and priced it as free: if a plant's errors are inherited between touching organs, the divergence sequence should carry a slow wander as well as a comb. The disturbance with the largest wander of any built here leaves none at all in the sequence, because a divergence is a difference and differencing is what removes a drift.

mechanism · Noise transport

Fractions with the same neighbours

Every instrument this collection has for the width of a disorder dip has a free parameter set by how close the next rational sits — which makes a hypothesis about the neighbourhood untestable with any of them. The repair is not a better instrument. It is a set of fractions whose neighbourhoods are identical and whose denominators are not, and the arithmetic supplies twenty-four of them.

tissue · Second statistic

The order carries the count

Take the divergence angles off a stem, throw away every coordinate, and autocorrelate what is left. The result is periodic at the smaller parastichy number — peaks at it and at every multiple of it. A list of angles, with no picture and no position in it, carries the spiral count.

cylinder · Sequence

What the rule does to a drift

A placement rule was supposed to leave no slow wander in a divergence sequence, because its errors are corrections rather than inheritances. Driven by a disturbance that drifts, it leaves a larger one than a lattice with no rule in it at all — while cutting the per-organ scatter by more than half. The rule removes what is relative between neighbours, and a drift is not.

mechanism · Noise colour

The memory was the rise

The earlier work measured a lag-one correlation of 0.54 in a noiseless divergence sequence and called it the sequence's own memory. Hold the rise fixed and there is no sequence at all — every angle identical — and under a disturbance the correlation is negative. The 0.54 belongs to the pattern chasing an equilibrium that is moving under it.

cylinder · Sequence

The second comb

The autocorrelation of a divergence sequence has peaks at the smaller parastichy number and at every multiple of it. It also has a second set of peaks, at the same spacing, offset by the difference of the pair — so a list of angles with no coordinate in it returns both numbers rather than one.

cylinder · Sequence

A harmonic is a step taken twice

The spectrum contains the larger parastichy number, their sum, and echoes of the smaller one, and no ranking of peak heights separates them. What separates them is arithmetic: a harmonic is a multiple of the spacing and a family is not, and the two kinds sit in different residue classes.

cylinder · Sequence

Two readings from one stem

Three note left with the work in a row have recorded that the two statistics of a divergence sequence want opposite plants — one quiet, one disturbed. Measured on the same stems they do not. The conflict was in the interpretation of a sign, and the window in which both are readable is wide.

cylinder · Sequence

What the ratio was hiding

The statistic that says a rule sharpens a drift rises by a factor of nearly four across a sweep of the rule's depth. Undo the normalisation and ask instead how many degrees of drift actually reach the divergences, and the answer changes by a fifth. Nearly all of the effect was in the denominator, and the denominator is the thing the rule is good at.

mechanism · Noise colour

Four ways to count a neighbourhood

How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.

mechanism · Noise colour

The boundary belongs to the pattern

Three kinds of noise, in three incommensurable units, destroy a lattice at the same place — about a degree and a half of divergence scatter. The earlier work measured that of two kinds and called it a scale rather than a constant. With a third it looks less like a coincidence and more like a property of what a lattice is.

emergence · Scatter tolerance

The order belonged to the method

A residual was left over after the two width laws, and it looked ordered: the most crowded fraction gave the widest dip, in all three families, in the direction a measurement artefact would take. Measured again with an instrument that has no level in it, the order changes with the window, disagrees between families, and in one of them comes out backwards.

wrong · Second statistic

A period the grid invented

A wrecked stem was reported as settling into a repeating block of three angles — 219.84°, 220.31°, 220.78° — which is the smaller of its two spiral counts and would have confirmed a standing prediction. Those three numbers are three consecutive samples of the azimuth grid. There is no block; there is a constant the grid cannot write down, and the routine that found the block was working perfectly.

wrong · Instrument ceiling

The corner that does not move

Read as degrees of drift getting through rather than as a ratio, and compared seed by seed, the deep and shallow rules change hands. The share that goes to the deeper rule climbs from twenty-three per cent under white noise to ninety-four at a correlation length of a hundred organs — and the crossing sits at two or three organs whether the two rules differ by a factor of four or sixty-one.

mechanism · Noise colour

The error budget for a nautilus

Every way a growth factor read off a shell section can be wrong has been priced here, one essay at a time. Added up they come to 332.6 per cent in the worst case and 279.6 in quadrature, against a golden-spiral claim that is 114.2 per cent away — so the budget does not refuse the claim at all. One entry decides it: the dividers, at 278.2 per cent on their own, and the dividers are the historical method and the only route measured that pushes a nautilus towards a golden spiral. Set them aside and the budget falls to 54.4 per cent and the claim is refused twice over. What the same budget cannot settle is anything smaller than half: 3.2 against 3.4 is inside it, and stays inside it until six of the seven sources are controlled.

shells · Nautilus

The second statistic was the first

The experiment this collection has been specifying was priced as two readings off one sequence, the second of them free. The two readings turn out to be one function looked at twice, so the specification loses a statistic — and gains a cheaper one, a warning about how observables get priced, and a question it could not previously ask.

wrong · Survey spec

What the sharing costs a lattice

A disturbance inherited from the contact neighbours destroys a stem's lattice at half the displacement independent noise needs, and it moves the comb ratio a fifth of the way to a forgery's. Take the inheritance out and keep the sharing, and the damage stays and most of the ratio shift goes — so the two effects have different causes.

cylinder · Noise transport

Packing, measured against the interior

An earlier reading of these heads reported that no packing criterion singles out the golden angle and that three criteria give three winners. Every one of those readings was divided by a mean cell area that, on a head of 150 organs, was 38.8 where the interior's is π. Divided by the interior's own, the criteria about distance put the golden angle first of 72 angles and the criteria about cells go to rational ones.

tissue · Packing

The band nobody can be placed in

The boundary between shells whose whorls run into one another and shells whose whorls run free was located here to the last bit a double holds. A specimen is not a point on that line, it is a measurement with an error, and carrying the whole measured error budget onto the boundary turns the line into a band running from 1/(W(1+b)) to 1/(W(1−b)). At the budget with the dividers set aside that band covers 48.2 per cent of the box the morphospace figure here is drawn on, and its share runs from 6.7 to 59.5 per cent across the six boxes in use — the same box-dependence the contact region itself showed. The angle criterion carries the same error better above an expansion of 1 + √2 and worse below it, exactly.

shells · Morphospace

The damage has a period

Every wrecked stem in the census has had two numbers read out of its displacement profile and the profile itself read out of none of them. Folded on the lag the stem kept, twenty-five of the thirty are constant inside each residue class to between 0.12° and 6.09°.

mechanism · Damage shape

The residual was the window

After the depth and the q over n squared scale are taken out of a disorder dip, something looked left over and looked ordered by how crowded the fraction's neighbourhood is. Measured on fractions whose neighbourhoods are identical by construction, seven widths across a factor of two and a half in denominator agree to one per cent. There is no residual; there was a comparison made at different effective windows.

wrong · Second statistic

Matching instead of correcting

Two rounds of work failed on one question because every instrument's free parameter was set by the thing under test. The repair was not a better instrument or a model of the bias: it was choosing what to compare so that the confound could not vary. That move is available in four other places here, and three of them have already used it without anybody naming it.

wrong · Instrument ceiling

A second moment that goes to zero

The mean squared departure of a cell's side count from six separates a random tissue from a whorled head by a factor of eighty, on heads of 900 organs. Read at thirty-three head sizes it is exactly the share of cells on the defect rings of a spiral head and falls as one over the square root of the organ count, it falls as one over the count on a whorled head, and the factor is 25 at 300 organs and 677 at 8,000.

tissue · Sixsides

The lengths that name the rule

A real crown has no single length ratio, and giving every fork a spread of daughter lengths does not blur what a sizing rule conserves: Murray's flow rule still conserves three at every junction, and the two mechanical rules keep their mean exponent, moved only as the square of the spread. What the spread adds is a second number. Each junction's exponent follows its daughters' summed log length with a slope of 0.30 under equal stress and 0.86 under equal bending at the planar crown, where the exponents are both two — and a two per cent error in every radius moves neither slope, while it swamps the scatter that looked like the obvious instrument.

branching · Murray

The plateau was a prediction

The search for a reference organ found that the largest displacement above a hole is a plateau rather than a peak, and reported it as a failure. A profile constant on each of k residue classes has exactly k levels, so its maximum is attained by a whole class — a ninth to a quarter of every window, forever.

mechanism · Damage shape

Nothing in the staircase moves

Disorder swept across the divergence angle is a staircase, and every step of it had been read at one head size — which leaves open whether a step is the lattice changing or a ring of defects crossing the rim as the angle moves it. Read again at 539, 900, 1409 and 3690 organs, 52 of the 53 features present at a smaller head are still there at the same angle at the next size up. Not one slides. A bigger head adds steps between the ones already there — 4, 18, 31, 40 — so the staircase belongs to the angle and the head size decides only how much of it is resolved. The one size every other disorder figure here uses turns out to sit three per cent past a ring entry.

tissue · Sixsides

Two regimes above a hole

Below the repeating pattern there is a transient, and the boundary between them is measurable: the first organ from which every class stays at its own level runs from 7 to 303 organs above the hole on twenty-five of thirty cuts, and five never reach it inside the run.

mechanism · Damage shape

Scars with dates on them

A bad season shared by every growing point wrecks the reading of a branching plant by two totals: over eighty seasons the rate and the scar share name a two-season wait for 22 per cent of plants. Date the scars — by position along a shoot, by growth ring — and each season's death chance is read off its own scars, so the bad years stop being noise and become a known input. Running the branching recurrence through the plant's own seasons names the wait for 94 per cent of plants over twenty seasons whether bad years kill a tenth of the points or seven tenths, and for every plant over eighty. What a plant cannot read from its own scars is the climate: the averaged chance comes only as fast as seasons do.

branching · Lsystem

Which chains changed places

A wrecked stem's displacement profile is a set of levels, one per chain, with two of them out of line — equal and opposite, on neighbouring chains. Nothing said which two. They are the hole's own chain and the one below it, on ten of the seventeen cuts that carry a pair.

mechanism · Damage shape

One law counts sides, the other pairs

Lewis's law and Aboav's relation point opposite ways at the two ends of disorder, and the obvious guess is that they are one reading of disorder taken from two sides. Measured on the same moved heads, they are not. Displaced organ by organ, Aboav's a first rises — to 1.45 at 0.15 of a wall spacing, as the first new defects arrive as bound five–seven pairs — and falls half-way to a random set's only at 0.45 of a spacing, where Lewis's law had switched on at 0.2. Between the two a tissue satisfies both. A smooth field, which never switches Lewis's law on, lowers a by pulling the pairs apart without making any new defects. Lewis's law reads how varied the sides are; Aboav's reads whether the defects are paired.

tissue · Cell laws

A frost and a drought in the scars

A branching plant whose buds and apices die at chances of their own, in a climate with bad years, leaves two kinds of scar in every season. Sorted by kind as well as dated, the scars give two shares a season, and the branching recurrence run through both names a two-season wait for 97 to 98 per cent of ten-season plants where one share a season names it for 81 to 83 — even when buds and apices die alike, because a small plant's deaths fall on one kind or the other by chance and the sorted series knows which. The three totals that the two chances were first thought to need fail under either kind of bad year, leaving the bud chance a tenth out however many seasons are read. And the sorted series can say what kind of bad year it had: a frost that strikes buds alone is told from a drought that strikes both at nine plants in ten over forty seasons, and one frost season is seen once the plant carries a few hundred buds.

branching · Lsystem

Two numbers for a tissue, and which two

Lewis's law and Aboav's relation read different things in a tiling — how varied the sides are, and whether the defects are paired — so a tissue has a place on a plane of both. Move a golden head by a smooth field and then displace it organ by organ, over a grid of both, and the tissues fill that plane rather than lying along a line. No single one of the four numbers a tissue is usually reported by places it on both laws: the variance of the side counts reads Lewis's slope to three times the seeds' noise and misreads Aboav's a, the pairing share reads a to two and a half times and misreads Lewis's slope. The variance with either law's own statistic places both to within one and a half times the noise; the variance with the pairing share, which is what a counter of cells records, to about twice. And the only tissues that fail both laws are heads moved by a smooth field of two spacings or more and nothing else.

tissue · Cell laws

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.

mechanism · Damage shape

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.

mechanism · Damage shape

When the second wall is free

On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.

mechanism · Both walls

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.

mechanism · Run length

An onset at the end of the run

One cut reported that its pattern began 299 organs into a 300-organ run. Given twice the room it reports no pattern at all. The reading was the run stopping, not the disturbance ending, and the definition guarantees one at the last organ of every run.

mechanism · Run length

Three rows change sides

Twenty-five of thirty wrecked cuts have a periodic displacement profile over three hundred organs and twenty-six do over six hundred. The count barely moves and the membership does: two rows join, one leaves, and the gap the threshold sits in narrows from 1.69 to 1.27.

mechanism · Run length

A window nobody aligned

Every reading this thread takes of a wrecked stem is taken inside a window, and there are three of them: a run of three hundred organs, a window of a hundred and twenty at its top, and a search window of fifteen to thirty-nine. None was aligned to anything, and one of them turned out to decide its own answers.

mechanism · Run length

A spread that grows with its window

A spread over a sample of a steady quantity does not depend on how big the sample is. These spreads triple when the window triples, on nearly every row of the census, which means the classes this thread calls steady are sliding — slowly, and invisibly at any single window.

mechanism · Two windows

A median that is an exception

Every number in the ablation thread is measured against the value most of a wrecked stem's chains sit at, taken as a median so that a few exceptions cannot move it. On eight of thirty-six rows the median stands on a chain sitting by itself.

mechanism · Damage shape

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.

emergence · Survey spec

No majority and no pair

Whether a wrecked stem's chains agree on a level and whether its exceptions form a balanced pair are computed from different things. Sixteen rows have neither, sixteen have neither, and fifteen rows are in both — and the two they part on are the two most interesting rows in the census.

mechanism · Damage shape

What a count cannot decide

A spiral count is the measurement this whole subject is built on, and it is deliberately blind to everything that varies inside a rung. Four results this collection now holds are results about that blindness rather than about the arrangements.

emergence · Counting

The rule comes back

A rule sorting the awkward rows was right fifteen times in sixteen, with the one failure blamed on a statistic. Recomputing the statistic makes it fifteen of fifteen — and halves the gap the line is drawn in, which is the price.

mechanism · Damage shape

The slide a counter holds constant

Inside one rung the settled divergence moves by more than a degree, monotonically, with no flat stretch anywhere — measured at a thousandth on one rung and at half a ten-thousandth on another. A rung is a plateau in one reported number laid over a geometry that never stops moving.

emergence · Coarse rung

A destination or a refusal

Sixty-three wrecked runs were regrown to twelve hundred organs and put to the settling table's own criterion, unchanged in every tolerance. The prediction written down before the sweep said twenty-nine would settle; seventeen do, and the prediction is wrong on its own side of the table as well as in its total.

mechanism · Settling

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.

emergence · Survey spec

Like with like

Seventeen wrecked runs settle when regrown to four times their length, and every one of them settles at the divergence the short run already reported. The endpoint moves by four thousandths of a degree at worst, which makes the short reading precise about where and silent about whether.

mechanism · Settling

What a run length was hiding

The settling criterion returns an identical verdict on all sixty-three wrecked runs at every length from three hundred organs to twelve hundred, so run length explains nothing. What the sweep does find is that an endpoint is a mean over an orbit, and thirty-four refusers never come within a degree of their own reported endpoint.

mechanism · Settling

The clock a share cannot see

Settling takes nothing to 290 organs, and four rounds of this collection have carried that as a property of the rule. At a steeper falloff the slowest is 808 — and not one of the 72 pairs of runs settles at 3,200 organs after failing at 1,200, so the fine end is still a wall.

emergence · Settling

The level was doing the ordering

Four falloff exponents have been put in order by the rise at which half their runs stop settling, and a half is the only level that order has ever been read at. Read at nine levels the same runs give ten different orderings of the same four numbers, and the one comparison in the whole study that resolves runs the other way.

mechanism · Falloff exponent

Destinations only a steep rule reaches

A settled stem's divergence is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on one grid at every exponent. Exponents 4 and 5 reach 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them.

emergence · Settling

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.

cylinder · Both walls

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.

cylinder · Both walls

A wall that was never measured

Three samplings of the starting angle give three orderings of the four falloff exponents' walls and a spread that does not shrink, while every error bar behind them halves. The reason is that a wall is a crossing of a nearly flat curve, and nobody had asked how well it is located.

emergence · Exponent

The last unmoved setting

Halving a band sweep's step is only a halving if the sweep steps where it says it does, and this one does not: its rises are rounded to five decimal places, so at one handover the grain is 1.65 parts per thousand against a nominal step of two. Moving the last setting nobody had moved found the setting was never what it was called.

cylinder · Sweep resolution

Named alongside it

The objects these essays reach for when they reach for this one.

MeasurementHonest limitsClaim testingArtefactNegative resultDivergence angleSamplingMeasurement errorIdentifiabilityAutocorrelationControlNoise

All concepts