Displacement — where it appears
Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.
What the centre costs
The fit that recovers a shell's growth factor needs a centre, and no shell has one marked. Displacing it by a quarter of the innermost whorl's radius moves the answer by 4.56 per cent at two turns, which is about five times the figure published earlier.
How far a centre must move
Four hundred and eighty-two thousand assumed centres, at nineteen spans and a hundred and eighty directions each, asked whether a spiral drawn at 3.2 can be made to read as the golden 6.854. It can, at every span up to 1.15 turns and at none from 1.2 upward, and every centre that manages it is refused twice over.
A centre that invents a life history
The collection's advice for a shell that might have changed how it grew was to fit it twice, over different arcs, and compare. On a spiral that does not change at all, a centre displaced by a quarter of the innermost radius splits the two halves by 4.09 per cent — the split a genuine 8.35 per cent change from apex to aperture produces — in either sign, depending only on which way the centre is wrong. Point noise of the same size splits them by less than half as much, and averages away where the centre does not. The floor under the test is the centre, not the noise.
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 positions read the twist's shape
A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.
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.
The centre the spirals give
Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.
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.
A head photographed from the side
A seed head photographed off its axis is squashed by the cosine of the tilt across one bearing, and the squash turns every organ's angle by an amount that goes round the head twice. Read square on, a 900-organ golden head tolerates twelve to fourteen degrees of tilt before its two annuli disagree by more than any untwisted head's do, and the spiral counts never notice at all. Past that, the reading is not a smooth drift but a scatter of heads misread one at a time. Stretching the head back along the short axis its own second moments find restores the square-on reading exactly to thirty degrees, even though the moments cannot see a tilt under three or four degrees. And the square-law twist stays hidden: it is a changed angle organ for organ, so no camera, at any angle, sees it differently.
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.
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 head seen in perspective needs a camera, not an ellipse
A seed head photographed off its axis from far away is squashed, and stretching it back along the short axis of its own second moments restores its reading exactly. Photographed from close up, the near side of the head is larger than the far side, and the stretch stops working: at twenty degrees off the axis it reads every head right from ten head radii and none from five. The organs' centroid drifts towards the near side as the camera comes in, but that is not why — the same stretch about the head's true centre fails at the same distances. What fails is the taper. A camera fitted from the organs alone, its tilt, bearing, distance and the head's centre chosen so that the head it un-projects is round, centred and evenly packed, reads seven to ten heads in ten right at every distance down to two radii.
A dome is read as a nearer camera
A camera fitted from a photographed seed head's organs alone restored the reading of a flat head seen in perspective from as close as two radii. Lay the same heads on domes and the camera is fooled in a particular way: it finds the tilt, its median within about a degree, and puts itself nearer than it is, by an amount set by the dome's height, so that a dome a fifth of the head's radius high photographed from infinitely far is read as a flat head photographed from seven radii. The reading it hands on is wrong. Off the axis, a domed head is misread once its height times the sine of the tilt passes about four hundredths of the radius — the same number at which a flat head's perspective broke the old correction — and only a photograph taken square on is safe.
Displacing a disc's organs makes no new near misses
A Fibonacci census's null, drawn from Vogel discs counted in a surveyor's band, holds a few tenths of a per cent of heads that a capped recount could move. Every disc was exact, and a real head's organs sit a fraction of a spacing off the lattice, where a band can read a family the lattice holds only marginally. Displaced by a twentieth, a tenth and a quarter of a spacing, the discs' readings do change — at up to one angle in six — and on large heads mostly to a pair the exact band already reads at one of its edges. But near the golden angle they change no more often than anywhere else, almost never into a Fibonacci pair, and the heads within a capped aim's reach move by under six hundredths of a percentage point. The census still needs three kept specimens.
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 drift along the spirals changes fewer counts
A grown head's errors are not independent: an organ placed against the organs already there inherits their errors, so whole stretches of a spiral move together. Laid down that way over the census's Vogel discs, at the same size as the independent displacement measured before, the drift changes fewer of the band's readings, not more — a sixth to a quarter fewer at a quarter of a spacing — never more often near the golden angle, and at most two into a Fibonacci pair. A count reads an organ against its neighbours, and a drift moves neighbours together: at a strong correlation an organ moves against its four nearest by just over half what an independent displacement of the same size moves it, and the readings changed are about what an independent displacement of that relative size changes, plus a little for the coherent part.
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 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.
A streak is a shear the count can see
Let each organ of a census disc inherit its error from its nearest earlier neighbour alone, rather than from the two it sits against, and the error travels along one family of spirals: streaks instead of patches. Measured against its four nearest neighbours a streak moves an organ exactly as much as a patch does — just over half what an independent displacement of the same size moves it — and yet it changes more of the band's readings at every size and in every draw, and on heads of three and five hundred organs as many as the independent displacement itself, or more. Its movement is nearly all across its own streak and hardly any along it, so it shears one family's chains against the other's, and the relative size that predicted a patch's effect underestimates a streak's by up to three points. Near the golden angle it changes no more readings than anywhere else, and the census still needs three kept specimens.
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 count weighted by direction closes a third of the gap
A streak moves an organ against its neighbours about as far as a patch does and changes far more of a count's readings, so the size of an error, averaged over directions, was the wrong number. Measured the way the count reads it, along each of the two families it is counting, the larger of the two closes between a quarter and under half of the gap, on thirty combinations of three errors at once: independent, patch and streak. It still leaves a strong streak a fifth short. The wobble of each family's spirals, which can be measured from a photograph with no knowledge of where anything belonged, does as well. And the two families are not lopsided on the heads where the gap is widest, so direction was not the whole of what the average missed.
A transition and not a slope
The question was whether a fourth cell's cost declines smoothly to nothing or falls in one step. It falls in one step, and the answer decides whether a word in the collection names something or is a threshold on a continuum.
The end of a wrecked run
The obvious follow-up to a transition in where one organ goes is whether the stem also finishes somewhere different. On this rung the question has no answer: a wrecked run's final divergence takes four values and changes between rises a thousandth apart, three times inside a nine-rise sweep.
Where a slot loses a wall
Two rungs were reported to go free at 84 and 85 per cent of themselves — the second removal stops costing anything over the larger one alone. Three samples a rung cannot say whether that is a transition or a slope, and twenty-nine more say it is a transition one grid step wide.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsClaim testingParastichy pairMeasurement errorVoronoi cellsResolutionDisorderIdentifiabilityVogel's modelDefect ringDivergence angleGolden angle