Topological defect — where it appears
Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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 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 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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsVoronoi cellsDefect ringDelaunayBound pairClaim testingDisplacementParastichyTopological chargeAboav–WeaireDisorderLewis's law