Field

Packing and tiling

How evenly a pattern fills its disc is a statement about cell areas, and it can be measured four ways that disagree. Cells average six sides because Euler's formula leaves them no choice.

Why the average cell has six sides

Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.

Closest pair across 120–155° at 400 organs, read both ways. On the interior's scale the golden angle reads 0.9027 and ranks 1st of 72, against 0.9026 for the best grid angle at 137.5°, with the window running from 0.0668 to 0.9026; counting the rim's cells the golden angle reads 0.7076 and ranks 2nd of 72, against 0.7129 for the best grid angle at 137.5°, with the window running from 0.0331 to 0.7129. The dashed upright is the golden angle, which a grid of decimal degrees never lands on and which is therefore read separately.

Packing, measured four ways

The claim is that the golden angle packs best, and it is measurable. Read on the interior of a head, the two criteria about distance put the golden angle first among the angles near it and the two about cells are won by rational angles — which makes the claim half right, and makes the right half a statement about a class of angles. An earlier reading of the same four criteria, divided by the cells at the head's edge, said the opposite.

How the largest gap behaves as the head fills, on the interior's scale, from 150 to 2000 organs. On the interior's scale, the rational angle's gap grows by a factor of 3.8 over this range, from 2.48 to 9.38 spacings; the golden angle's runs from 0.841 to 0.844, decided at radius 0.871 at every size, and 137.3° reaches 0.863. A rational angle's gap is unbounded and an irrational one's is not, which is a claim about growth rather than about a value at any one head.

The gap that grows

A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of 2.9 between two hundred primordia and sixteen hundred, and 3.8 from a hundred and fifty to two thousand. An irrational one does not. That is the division between rational and irrational angles that survives measurement.

Cell area against side count, at 0% disorder. The dashed line is Lewis's law, (n−2)/4. The fitted slope here is 0.014 against his 0.25, and the side count accounts for 20% of the variation in area.

Lewis's law wants disorder

Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.

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.

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.

The six are the spirals

Label every contact between two cells in a seed head with the difference between the two nodes' placement indices. The labels are the parastichy numbers — 34, 55, 21, 89 — and the six sides Euler forces turn out to be about two from one family, one and a half from the next, and one each from two more.

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

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°.

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°.

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.

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

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.

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

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.

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

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.

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

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.

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

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.

$Seven fractions with one neighbour distance and every denominator. Each member of a matched set drawn on its own stretch of the divergence axis, 0.5° either side of itself, with the nearest other rational marked. The distances are 0.3158°, 0.3158°, 0.3117°, 0.3069°, 0.3117°, 0.3077°, 0.3064° — a spread of 3.1% — while the denominators run 19, 20, 21, 23, 33, 45, 47, a factor of 2.47. That is the construction this thread needed. Every instrument for the width of a disorder dip has a free parameter set by how close the neighbour is, so a hypothesis about the neighbourhood cannot be tested by varying the neighbourhood; on this set the neighbourhood is held fixed and the arithmetic of the fraction is what varies.$

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.

135 fives and 129 sevens among 1631 interior cells. The side counts of every cell strictly inside a golden, 137.508° head of 2400 organs, cut at 86% of the radius. The fives and the sevens are counted apart rather than summed into a spread, because they are opposite charges and the sum hides them. Summed over the interior, 264 cells that are not hexagons carry a charge of +6. Over the whole patch the charge is 294, which is exactly 6 + 2·144 — a number fixed by the 144 cells on the patch's own boundary and carrying nothing whatever about the interior. The defects are not scarce; they are balanced.

An interior that is nearly neutral

Give every cell a charge of six minus its number of sides and the total over a tessellated head is fixed by its own boundary, exactly, with nothing left over for the interior. On a golden head that freedom is spent on 264 exceptions among 1,631 cells which cancel to six.

264 exceptions in a 2400-organ head, and eight circles. A golden, 137.508° head of 2400 organs, tessellated inside 86% of its radius. Every interior cell is drawn, and the 135 five-sided cells and 129 seven-sided ones are marked apart from the 1367 hexagons, and the pale circles are radii computed from the divergence angle through the lattice's third-shortest vector — nothing is fitted. Every defect sits within 0.64 of a cell spacing of one of those eight circles, and between them there is not one exception in hundreds of cells.

The defects lie on rings

The cells in a seed head that are not hexagons are not scattered through it. Every one of 264 sits within a cell of a radius computed from the divergence angle alone, the radii are a factor of φ apart, and between two of them lie 422 consecutive cells without a single exception.

Every five is bound to a seven

A five-sided cell beside a seven-sided one is one object in a crystal and two exceptions in a tiling, and the phyllotaxis literature borrows the crystallographic word without measuring the binding. Measured against a seeded permutation null on a 2,400-organ head, every five in the interior shares a wall with a seven, and the loneliest one in the head is 0.927 cell spacings from the nearest.

The defect rings are not where the counts change — they are √φ further out. A logarithmic radius axis for a golden, 137.508° head. The lower marks are the radii at which the counted parastichy pair changes, where the two shortest lattice vectors change places; the upper marks are the radii at which a cell's neighbours change, where the third-shortest does. They interleave, and the ratio of each ring to the transition inside it is 1.2715, 1.2723, 1.2723, 1.2719, 1.2719, 1.2723 against √φ = 1.27202. Consecutive rungs are a factor of φ apart in radius and √φ is their geometric midpoint, so a defect ring sits exactly halfway between two parastichy transitions. Anyone looking for the defect line at the radius where the counts change will not find it there.

The rings are not the transitions

A seed head has two ladders on it — the radii where the counted parastichy pair changes, and the radii where the exceptional cells sit — and the obvious guess is that they are the same ladder. They are not: the second sits a factor of the square root of phi outside the first at every rung of two different divergence ladders, which is exactly halfway between two consecutive transitions.

A cut-off would have to exceed 2.236 and not exceed 1.441, and nothing does both. Each of the 608 interior cells contributes two marks: its furthest wall, and its nearest partner that is not a wall, both in units of that cell's own shortest lag. A single cut-off would have to sit to the right of every mark of the first kind and to the left of every mark of the second, and the two clouds overlap — the extreme cases are 2.236 at 0.0 per cent of the radius and 1.441 at 60.0 per cent. So the interval is empty by a factor of 1.55, while 606 of the 608 cells have a cut-off that works for themselves.

No cut-off makes them one

Two different relations on a head have both been called neighbour: the shortest index lags a count keeps, and a shared Voronoi wall. A cut-off that turns the first into the second exists for almost every cell taken alone, and for no whole head at any size.

$One cell's six walls, and the two the pair 34 and 55 does not name. The cell of primordium 225, at 50 per cent of the head's radius, with each of its six walls labelled by the index difference across it. Its own counting instrument returns 34 and 55, which names four of them; the two drawn warm are 21 and 21, a family the instrument ranked and discarded. Over the whole head that is 33.80 per cent of the union in dispute, and it is the same fraction in every band.$

Two thirds of a cell

The founding claim of this field is that the six sides Euler forces are the spiral families. Measured against the tessellation it names two thirds of a cell's walls exactly, in every band of a head and at every rise of a stem, and the missing third is the same third everywhere.

Two rankings, one list

An essay in this collection claimed that the four shortest index hops on a seed head and the four largest shares of its cell walls are the same four numbers in the same order, and called the correspondence exact. Measured again from the same points, the two lists hold the same four families and order them differently, and they order them differently in five of the six bands the head can be read in.

A stem is a thread at six rises of the sweep, a ribbon at 23 and a surface at 111. The 140 rises of the sweep, split by how many index families the tessellation carries. On 6 rises above 0.5010 a node has 2 walls and the strip is a thread; on 23 between 0.2187 and 0.4833 it has 4 and is a ribbon; on 111 below 0.2109 it has 6 and is a surface. The comparison means something different in each: two families leave 33.333 per cent in dispute on the surface and 0.00 on the ribbon, and three families leave 0.000 and 33.333.

The third family

On a seed head no threshold makes the counted contacts and the shared cell walls the same relation. On a stem they are the same relation exactly, at every one of a hundred and eleven rises and to three decimal places of nothing, provided the contact cut keeps three families where a count keeps two.

The cell area a 137.508° golden head's packing is divided by, at 10 sizes. Counting every bounded cell the mean area runs from 3.607 to 38.828, the largest at 150 organs, because the cells just inside the edge of the head reach out to circumcentres far beyond it. Leaving out every cell whose polygon crosses the head's own radius, it stays between 3.1425 and 3.1634 at every size drawn, which is π, the area the square-root rule gives each organ.

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.

The largest empty circle of a 2400-organ 137.508° golden head, ring by ring. Each dot is the largest empty circle in one ring of the head, on the interior's scale. The whole head's is 0.8437 at radius 0.871, at the centre; the rings peak at the radii where the lattice flips from one pair of neighbours to the next, and at the 4 flips beyond radius six the head reads within 0.0020 of the closed form. The dashed line is 1/√2 = 0.7071.

One over root two

On the interior's scale a golden head's largest empty circle is 0.8435 of a spacing at every size from 150 organs to 2,000, because one triangle at its centre decides it. Everywhere else it is 1/√2 — a square cell at every ring where the lattice flips — and a closed form in the angle's continued fraction says that only noble angles hold it there.

The empty interval is the rings

No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

The second moment of five arrangements' side counts against the number of organs on the head. μ₂ on logarithmic axes for heads of 300 to 10,000 organs. Golden: 0.455 at 300 and 0.101 at 10,000; Lucas: 0.362 at 300 and 0.086 at 10,000; 137.5°: 0.453 at 300 and 0.045 at 8,000; whorled: 0.070 at 300 and 0.003 at 8,000; Poisson: 1.727 at 300 and 1.749 at 8,000. The Poisson set is a mean over three seeds. The dashed line is 6.83 over the square root of the organ count, the level the golden head returns to just before each defect ring enters the cut.

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.

Four head sizes, one staircase. The same window swept at 539, 900, 1409, 3690 organs, each curve divided by its own median so that the overall fall with head size is out of the way and only the shape is left. The features line up. Across the three steps in size, 52 of the 53 features present at a smaller head are still present at the same angle at the next size up — nothing slides. What a bigger head does is resolve features between the ones already there, which is a statement about the instrument rather than about the arrangement.

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.

The centre of a 900-organ Lucas head, with the band of two thirds of a spacing drawn round every flip ring. The organs of a 900-organ Lucas head out to a radius of 12.5, with a pale annulus 0.66 of a wall spacing either side of each flip ring, the rings keeping 1, 3, 4, 7, 11, 18. Warm: cells with five or seven sides; dark: six-sided cells a three-family contact cut gets wrong. Ringed: the one cell well inside a band that the cut reads exactly, organ 17 at radius 4.12, 0.23 of a spacing from its ring. Where two annuli overlap, a cell can sit well away from its nearest ring and still inside the next one's band.

The blur was at the centre

On a Lucas head the band of disputed cells round each flip ring looked blurred at its inner edge — exact cells as close as 0.23 of a wall spacing, disputed hexagons out to 0.59 where a golden head's stop at 0.43. Read a ring at a time, the two heads carry the same band on every resolved ring, to a hundredth: disputed hexagons within 0.16, exact cells from 0.64, a ring's own number of fives and of sevens and the number before it of hexagons. Every difference is inside a radius of six, where the Lucas rings of 4, 7 and 11 sit closer together than the band is wide, and the one exact cell is organ 17, which has no organ eighteen behind it.

A 900-organ golden head displaced by 0.02 of a wall spacing, with its flip rings and the cells in dispute. Every organ of a 900-organ golden head inside the rim cut, each moved by a seeded gaussian displacement of 0.02 of a wall spacing, with the flip rings the divergence angle puts at radii 11.3, 18.2. Warm: the 150 cells with five or seven sides, 13 of them more than 0.66 of a spacing from every ring. Dark: the 59 six-sided cells with a wall a three-family contact cut gets wrong. Of the cells a spacing or more from every ring, 0 are disputed.

A hundredth of a spacing

Off the flip rings one hop-ratio cut-off turns a seed head's counted contacts into its cell walls, on every head from 900 organs to 9,000. Displace the organs and it is the first thing to go: shut by a fiftieth of a wall spacing on 900 organs and a two-hundredth on 9,000, because it is decided by the worst of thousands of cells. The three-family count survives two to four times further, because each cell only has to beat its own margin, and the rings keep their fives and sevens in between. All three fail from the rim inward, since the margin one spacing from a ring is 9.7 divided by the ring's family number.

The disputed cells round the ring of 55 on a 2,400-organ head twisted by 0.2 radians at the rim. Every organ turned about the centre by 0.2 times its radius over the head's. Bars: the six-sided cells a three-family contact cut disputes (grey-blue) and the five- and seven-sided cells (warm) within two spacings of the ring the untwisted divergence puts at 55, by signed distance from it. There are 34 disputed hexagons, with a median at 0.54 of a spacing, and 110 fives and sevens. The dashed line is where the twisted divergence puts the ring in closed form, 0.54 of a spacing out; the band keeps its cells and its width and sits on it.

The band moves, it does not blur

Displaced organ by organ, a seed head loses its single contact cut-off first, its rings' hold on their fives and sevens next and its three-family count last. Displaced by a smooth field that moves neighbours together, the same head keeps its census — the same 353 disputed cells and 264 fives and sevens at every step up to a third of a spacing — and moves the band instead. A twist moves each flip ring exactly to where the twisted divergence puts its tie, the ring of 55 by 0.53 of a spacing, the ring of 34 the other way. Read against strain, correlation helps the cut-off and not the count, and on a 900-organ head the two fail at the same step: the order was an order of blurring.

Cell area against side count on a 900-organ head, every organ displaced by 0.2 of a spacing. The joint distribution of cell area, as a multiple of the mean, and side count, over the 639 interior cells of a golden head with every organ displaced by 0.2 of a wall spacing, seed one. The dashed line is Lewis's law, a quarter of the mean area for each side; the solid line is the fit, at a slope of 0.113, and the side count explains 30 per cent of the variation in area. Classes: 4 sides, 16 cells, mean 0.75; 5 sides, 159 cells, mean 0.89; 6 sides, 295 cells, mean 1.00; 7 sides, 146 cells, mean 1.11; 8 sides, 22 cells, mean 1.22; 9 sides, 1 cells, mean 1.26.

Lewis's law needs the sides to vary

Lewis's law holds on a random set of points and fails on a golden-angle head. Walked from one to the other by displacing every organ independently, the head's Lewis slope reaches half a random set's at a fifth of a wall spacing and nine tenths by seven tenths, and in between it explains up to 41 per cent of the variation in cell area — more than the 31 per cent it explains in the random set. Moved instead by a smooth field correlated over eight spacings, the head's cell areas become nearly as varied as a random set's and its slope stays at nought, because its side counts stay the lattice's. The law is not about how varied the cells are. It is about how varied their sides are.

The cells of a golden head with every organ displaced by 0.15 of a spacing, five- and seven-sided neighbours joined. A window eleven wall spacings square, about halfway out on a 900-organ golden head with every organ displaced by 0.15 of a wall spacing, seed one. Cells are keyed by side count; every five-sided cell is joined to each seven-sided cell it touches. In the window: 26 five-sided, 75 six-sided, 26 seven-sided and 5 of other counts, with 40 five–seven contacts. Over the whole head, averaged over five seeds: Aboav's a = 1.45, and 91 per cent of five-sided cells touch a seven.

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.

Every crossed tissue on the plane of both laws, and the path one smooth field takes as independent steps are added. Sixty-nine tissues, each a 900-organ golden head moved by a smooth field of 0.32, 0.64, 1, 1.5, 2, 3 wall spacings or none and then displaced organ by organ by 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.75, 1 of a spacing or none, each averaged over ten seeds and placed by its Lewis slope and Aboav's a. The vertical line is half a random set's slope, 0.114, right of which Lewis's law is on; the band is a within 0.35 of 1.2, where Aboav's relation holds. The ordered head sits at 0.009 and 1.18. The joined path is a smooth field of 2 spacings, as the step grows: 0: -0.023, 0.84; 0.05: -0.003, 1.03; 0.1: 0.041, 1.22; 0.15: 0.076, 1.22; 0.2: 0.098, 1.18; 0.3: 0.135, 0.97; 0.4: 0.157, 0.91; 0.5: 0.170, 0.84; 0.75: 0.203, 0.71; 1: 0.202, 0.67.

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.

A tissue that was never shaken

Every tissue whose laws have been read here was disordered by moving its points. A growing tissue also disorders itself by dividing, and a division is a wall no set of points generates. Held as a map and divided cell by cell by three rules, a golden head's tiling switches Lewis's law on once a tenth of its cells have divided, at a variance of side counts lower than any moved tissue reaches the law at, because the commonest single division makes two half-sized fives and two full-sized sevens at once. Dividing the largest cell first reaches the corner of the plane no moved tissue reached — Lewis's law on and Aboav's a above its band, at 1.67 — because the largest cells of a golden head are its sevens. And the two numbers that placed every moved tissue on Lewis's law to one and a half times the noise misplace a divided one by fifteen times it: they were a calibration of how the tissue was disordered, not of tissue.

A gap in the disorder staircase read at a hundredth of its grid. μ₂ on a head of 3,690 organs from 137.1975° to 137.2725°, a gap in which the staircase counts no step, read every 0.00005° — 1500 samples — against the staircase's median; the open circles are the staircase's own samples every 0.005°, which the fine sweep passes through exactly. No change between two fine samples reaches a fifth of the median. The curve is a sawtooth: 13 falls of ten cells or more, 19 at 137.2004°, 21 at 137.2009°, 18 at 137.2054°, 21 at 137.2058°, 14 at 137.2111°, 18 at 137.2115° and more, against 0 climbs that large; climbs average 2.4 cells and falls 4.0.

What lies between the steps

The disorder staircase — the spread of a head's side counts against its divergence angle — gained steps with every larger head, forty at 3,690 organs, and nothing said whether it had steps at every scale. Read again at a hundredth of its grid inside its two widest gaps, it has none: no change there reaches the size it counts as a step, and no dip hides between two of its samples. The steps stop. What the gaps hold instead is a sawtooth — μ₂ climbing a cell or two at a time and falling in teeth of ten to thirty-two cells, five of them exactly twenty-one — and a step, read at the same resolution, is not one event but two runs of flips of fifty-five cells each. How many steps a head has is a statement about where the line is drawn; the steps themselves are finite.

The five kinds of cell across the ring where the 89 spirals begin, with every contact labelled. One cell of each kind from the ring of a 2,400-organ golden head where the family of 89 enters and the family of 21 leaves, drawn with a spoke to each neighbour labelled by the difference of their placement indices. Inside the ring, a hexagon has two contacts each of 21, 34 and 55. A seven has those six and one 89. A traded hexagon has one 21, two 34s, two 55s and one 89. A five has two 34s, two 55s and one 89. Outside, a hexagon has two each of 34, 55 and 89. Read outward, the cells give up the 21s and take on the 89s, and the seven and the five are the two ways of doing it unevenly.

A defect ring is where a family of spirals begins

Label every wall between two cells of a seed head with the difference of their placement indices and a hexagon is two contacts from each of three spiral families. The cells that are not hexagons sit on rings, and read by their labels each ring is a handover: one family of spirals ends there and the next begins. On a golden head the ring where 89 enters holds 34 sevens, 21 hexagons that have traded one contact and 34 fives — 34 + 21 + 34 = 89 — because every chain of the entering family begins at one of its cells and every chain of the leaving family ends at a traded hexagon. Joined by their walls, the fives and sevens make 21 clusters, and the clusters run round the ring in the order of the Fibonacci word.

A quarter of the ring where the 89 spirals begin, on a golden head displaced by 0.02 of a spacing. A quarter turn of a 4,000-organ golden head with every organ displaced by a gaussian of 0.02 of a spacing, unrolled so that angle runs across and radius down, between the radii the exact ring where 89 enters occupies. Each organ is marked by its cell's kind: seven, traded hexagon or five of that ring, a cell whose labels fit no kind, or an ordinary hexagon; and the walls joining a five to a seven are drawn, which is what makes the clusters. Round the whole ring on this head there are 32 sevens, 25 traded hexagons and 32 fives, 89 cells, in 23 clusters — 8 single pairs, 11 double and 4 of other sizes — whose order round the ring is not the Fibonacci word.

A ring keeps its count after it loses its order

On an exact golden head a defect ring holds exactly as many cells as the family of spirals entering it, and its fives and sevens join into clusters that run round it in the order of the Fibonacci word. Move every organ by a two-hundredth of a spacing and the order is gone from the outer rings on every head, while the count has not moved by a cell. The clusters' make-up goes next, at a hundredth or two. The count lasts longest — to a displacement of about three divided by the entering family, five hundredths of a spacing on the ring where 55 enter and one and a half on the ring of 233 — and while it lasts the ring pays for the displacement in kind, trading fives and sevens for hexagons that have swapped a contact, one pair for two.

A quarter of the ring where the 89 spirals begin, on a golden head moved by a drift of 0.1 of a spacing. A quarter turn of a 4,000-organ golden head moved by a drift along its spirals — each organ inheriting nine tenths of the mean error of its two nearest earlier organs, scaled to 0.1 of a spacing per coordinate — unrolled so that angle runs across and radius down, between the radii the exact ring where 89 enters occupies. Each organ is marked by its cell's kind, and the walls joining a five to a seven are drawn. Round the whole ring on this head there are 30 sevens, 31 traded hexagons and 30 fives, 91 cells, in 27 clusters, whose order is not the Fibonacci word. The same head displaced independently by 0.1 of a spacing holds 101 ring cells.

A ring outlasts an error that runs along its spirals

A golden head's defect ring holds exactly as many cells as the spiral family entering it, and displaced organ by organ the count lasts to a displacement of about three over the family. A grown head's errors are not organ by organ; they are carried along the spirals, so neighbours move together. Laid down that way, a drift of the same size keeps every ring's count to a displacement a quarter to twice as large again — a tenth of a spacing on the ring where 55 enter, against a twentieth — and keeps the clusters' make-up and order longer too, losing them in the same sequence. Stated as how far an organ moves against its neighbours, the drift's tolerance falls back into the band the independent heads set: the count law reads relative displacement.

A quarter of the ring where the 76 spirals begin, on a head at the Lucas angle displaced by 0.02 of a spacing. A quarter turn of a 4,000-organ head at the Lucas angle, 99.5°, every organ moved by an independent gaussian of 0.02 of a spacing, unrolled so that angle runs across and radius down, between the radii the exact ring where 76 enters occupies. Each organ is marked by its cell's kind, and the walls joining a five to a seven are drawn. Round the whole ring on this head there are 29 sevens, 18 traded hexagons and 29 fives, 76 cells against the entering family's 76, in 16 clusters, 5 single and 9 double, and 2 of other sizes.

A Lucas ring gives way where a golden ring of its size does

A head at the Lucas angle has defect rings of its own, each holding exactly as many cells as the Lucas family entering it. Displaced organ by organ, every one of them keeps that count to a displacement of about three over its family — 2.5 to 3.8 on the Lucas rings where 47, 76, 123 and 199 enter, against 2.7 to 3.6 on the golden rings of 55, 89, 144 and 233 — and to four to five over it when the displacement is measured against neighbours, for a drift along the spirals as well as for independent error. The order of the clusters goes first and their make-up second, as on a golden head, and at the same displacements for a ring of the same size: the Lucas ring of 76 and the golden ring of 89 keep their order on three heads in ten at a two-hundredth of a spacing and on none at a hundredth. What a ring tolerates is set by how many spirals enter it, not by the angle that made them.

The holes of a 2400-organ head built at 137.3 degrees, and the lattice read from them with no angle given. Each dot is the largest empty circle in one ring a quarter of a unit wide, read from the positions of the organs alone. The line is the lattice of the one divergence that best makes those rings: 137.3003 degrees for a head built at 137.3, with quotients 2, 1, 1, 1, 1, 1, 4, 1 against the true 2, 1, 1, 1, 1, 1, 4, 1. Over 167 rings the fit misses by 0.0047 of a spacing, where the true angle's own lattice misses by 0.0048.

A head's holes spell its angle

A head's largest empty circles follow a closed form in its angle's continued fraction, and a closed form runs both ways. Given only the organs of a 2,400-organ head, the largest hole in each ring and no angle, a reading outward through the rings returns 28 of forty undisclosed angles to a thousandth of a degree, quotient by quotient, and a larger head tightens them a hundredfold. The other nine it finds are other angles, and they are not random: their expansions end as the truth's ends and start differently, because the start of a continued fraction lives at the centre of a head. And most of it is gone once each organ has moved by a hundredth of a spacing.

A Burgers circuit round a dislocation in a crystal and round a five–seven pair on a golden head's ring. Left: a triangular lattice moved by the elastic field of one edge dislocation, its five- and seven-sided cells marked, with the tightest circuit round them, the 8 organs touching either. Rounded to whole lattice vectors the steps sum to -1, 0: the circuit fails by one lattice vector. Right: a five–seven pair on the ring at radius 29.51 of a 4,000-organ golden head and the 8 organs touching either of the two. Its steps sum to 0, 0, and the circuit closes.

A dislocation that closes

A defect ring on a seed head looks exactly like a low-angle grain boundary: a closed chain of bound five–seven pairs one cell thick. The defining test of a dislocation is a loop that fails to close. On an elastic crystal it fails by one lattice vector at every size. On a golden head's rings the tightest loop round a five–seven pair closes every time, 272 pairs out of 272, and one set of lattice labels holds every patch across the outer rings without a single broken edge. No lattice rotates at the ring, where a real boundary of that density would need thirty-two degrees. What a ring actually is sits in one number: its walls are within two over its own count of being replaced by the other diagonal of their cell, on one curve for all four rings.

A head growing outward beside a uniform head shaken until its cells vary as much. Left: an exact golden head of 500 organs with radius growing as the organ's index to the power 0.6; its cell areas have a coefficient of variation of 0.164, all of it a trend from centre to rim. Right: a head with an exponent of a half, whose cells would all be equal, every organ moved by 0.2 of a spacing; its coefficient of variation is 0.176, none of it a trend. Cells larger than the mean are shaded. A reading of the spread gives the two heads about the same exponent.

A spread of cell areas is not a growth law

An earlier essay proposed that a head's spread of cell areas is a measured growth law: a spread of 0.10 means a radial exponent of 0.455 or 0.555. On an exact head that holds to two figures. A uniform head with every organ moved by a tenth of a spacing also has a spread of 0.094, and reads as 0.457 with no gradient in it at all. By a fifth of a spacing it reads 0.42. Fitting cell area against radius instead is untouched on a golden head and collapses on a rational one, and ranking the organs by radius is untouched on both. The spread is law and disorder added in quadrature, which makes it a good reading of disorder once a trend has told it how much law there is.

A random tissue of 700 cells whose size grows outward as a Vogel head of exponent 0.8 does. Points placed at random with the density of a head of exponent 0.8. Cells above the median area are shaded, and they lie outside: 537 cells whose sizes differ far more by where they are than by how many sides they have. A cell's side count does not follow its radius at any setting.

A gradient keeps Lewis's slope and takes its share

Lewis's law is reported as two numbers, a slope and how much of the variation in cell area it explains, and they get read as one. Grow a tissue's cell size steadily across the field of view and the two part completely. Over a gradient that takes cells from one size to several times that, the slope of a random tissue stays at 0.22 and its explained share falls from 33 per cent to 15. Both are predicted by a line of arithmetic from the flat tissue's own numbers, to within a percentage point or two. Dividing the field into twelve bands of radius restores nearly all of it. So a weak Lewis correlation in a published field may be a statement about the field.

Lewis's slope as a tissue divides, at five rates of growth. Three seeds a point, a vertex model with line tension 0.05. No growth: 0.163, 0.255, 0.365, 0.416. Regains 3% a round: 0.137, 0.164, 0.123, 0.085. Regains 10% a round: 0.098, 0.075, 0.040, 0.030. Regains 30% a round: 0.036, 0.021, 0.013, 0.011. Regains all a round: 0.004, 0.005, 0.005, 0.006, at a tenth, three tenths, one and two times the head's measured cells.

A tissue that grows forgets Lewis's law

A tiling divided cell by cell switches Lewis's law on within a tenth of the cells, because the commonest division makes two half-sized fives, and nothing ever restores them. Give the walls a vertex model and each cell a target size that recovers between rounds, and the law becomes a memory that fades. A tissue regaining a tenth of its loss each round never switches it on. One regaining three per cent reaches 0.16 and is falling again by the time it has twice the cells. One regaining all of it reads 0.006 at a variance of side counts of 1.8, a place no moved tissue goes. Aboav's a hardly notices, and the variance of the side counts, as a map counts them, barely more. So the slope on a growing tissue measures how recently its cells divided, set against how fast they grow.

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