Defect ring — where it appears
Named by 16 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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 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 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 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.
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.
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 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 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 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 whorl moves a ring out and multiplies it
A head grown k organs at a time is an ordinary lattice wrapped k times, which predicts its defect rings with no measurement at all: each of an ordinary head's rings should appear at k times the radius, holding k times the cells, one copy in every sector. On eight-thousand-organ heads of two to five organs a whorl every ring read holds exactly that count, 110 fives where 55 would be and 105 where 21 would be, at the predicted radius to a tenth of a unit. Stagger the members of each whorl by the single organ's radius that Vogel's rule would give and the counts do not change, while a fifth to a quarter of the defects lose their partner one sector on.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsVoronoi cellsDelaunayTopological chargeTopological defectClaim testingDisorderRim effectBound pairDisplacementLucas numbersParastichy