A ring keeps its count after it loses its order

A defect ring is where a family of spirals begins labelled every wall between two cells of a tessellated golden seed head with the difference of the two organs’ placement indices — the labels the six sides of a hexagon had turned out to be spiral families — and read the rings of five- and seven-sided cells by their labels. Each ring turned out to be a handover. Across it one family of spirals ends and the next begins, and the cells come in five kinds in a fixed order outward: hexagons of the old families, sevens, hexagons that have traded one old contact for a new one, fives, and hexagons of the new families. Two laws came with the reading.

The first is a count. A ring holds exactly as many cells as the family entering it — sevens, traded hexagons and fives together — because every chain of that family begins at one of its cells; on the ring where 89 enters, 34 + 21 + 34 = 89. The second is an order. Joined by the walls between a five and a seven, the ring’s defects make clusters of one pair and of two, and round the ring the two sizes follow each other in the order of the Fibonacci word.

Both were read on exact heads, every organ at its Vogel position, and the essay that found the rings had already shown displacement scattering them. That leaves the question this one answers: under displacement, ring by ring, which law goes first — and does the ring lose its Fibonacci order before or after it loses its count?

Ten heads at each displacement #

The head is the earlier one: a golden Vogel head of 4,000 organs, tessellated, its interior cells outside the crowded centre classed by the families of their walls exactly as before. Every organ is moved by a gaussian whose standard deviation in each direction is a stated fraction of a spacing — from a two-hundredth to a tenth, with the spacing taken as $\sqrt{\pi}$, the unit a head displaced before it is counted first used — on ten independent seeds at each displacement.

The rings read are those where 55, 89, 144 and 233 enter. The ring where 34 enters sits too close to the centre on an exact head to separate from its neighbours, and is left out as it was before. Three things are asked of each ring on each head: whether its clusters still spell the Fibonacci word round the ring (its order); whether it still has the exact ring’s numbers of single and double clusters and no cluster of another size, in whatever order (its make-up); and whether it still holds exactly as many cells as the family entering it (its count).

A quarter of the ring where 89 enter #

Unrolled, a quarter of the exact ring is a row of fives and sevens joined into short zigzags — a single pair, or two pairs sharing a cell — with traded hexagons between them and ordinary hexagons above and below. Round the whole ring there are 8 single clusters and 13 double ones, and their order is the Fibonacci word.

Displaced by two hundredths of a spacing on the first seed, the same quarter looks almost the same, and is not. Some zigzags have broken, a seven stands alone where a pair was, and a few cells fit none of the five kinds. Round the ring the clusters are 8 single, 11 double and 4 of other sizes, and their order is gone. The ring still holds 89 cells. Turned to four hundredths, it still holds 89, now in 7 single clusters, 6 double and 14 of other sizes.

Three ways to keep a ring, three ways to lose it #

The three go in a fixed order on every ring.

The order goes first, and it goes at displacements too small to see. At a two-hundredth of a spacing — the organs moved by under one per cent of the distance between them — the rings where 144 and 233 enter have lost their Fibonacci order on all ten heads. The ring where 89 enters keeps it on three heads of ten and on none at a hundredth. The ring where 55 enters is the most stubborn, keeping it on eight heads at a two-hundredth and four at a hundredth, and by four hundredths on none.

The make-up goes next. At a two-hundredth the ring of 233 has already lost it on every head and the ring of 144 on six of ten, while the ring of 89 keeps it on nine. At a hundredth the ring of 89 keeps it on five and the ring of 55 on all ten; by two hundredths the ring of 55 keeps it on four and the others on none.

The count goes last, and long after. Every ring keeps its count on all ten heads at a two-hundredth and at a hundredth. The ring of 233 keeps it on every head to one and a half hundredths and on four at two; the ring of 144 on every head to two hundredths, eight at three and one at four; the ring of 89 on every head to four hundredths, seven at five and one at seven and a half; the ring of 55 on every head to five hundredths, and still on four at a tenth.

A wall flips and a pair glides #

Set cell by cell beside the exact head, the head displaced by a two-hundredth of a spacing on its first seed shows what happens. Across the whole head 84 cells have changed kind, and every one of them has changed its neighbours. On the ring where 89 enters, one change runs along four consecutive cells of a spiral. The wall between organs 320 and 341 has gone and a wall between 286 and 375 has appeared, and with it organ 286, a hexagon, has become a seven; 320, a seven, has become a traded hexagon; 341, a traded hexagon, a five; and 375, a five, a hexagon.

That is one wall exchanged for the other diagonal of a quadrilateral, and its effect is to move a five–seven pair along the ring by one cell. The ring still holds the same number of sevens, traded hexagons and fives — 34, 21 and 34 on this head — and the same number of single and double clusters, 8 and 13. What has changed is which organs hold them, and so which pairs sit next to which: one pair has slid from one cluster’s neighbourhood towards another’s, and the sequence of single and double clusters round the ring is no longer the Fibonacci word.

The flip happens where it does because four organs on an exact ring sit almost on one circle. A ring is where the lattice passes through a tie between two families, and a near-tie in the lattice is a near-tie between the two diagonals, so a two-hundredth of a spacing is enough to choose the other one. The order is a statement about which defect is next to which, and that is exactly what such a flip changes; the count is a statement about how many of each, which it does not.

How many pairs move #

A glide changes the roles of four cells, so counting the cells whose role on a ring differs from the exact head’s counts glides in fours. On the median of five heads displaced by a two-hundredth of a spacing, 4 cells have changed role on the ring where 55 enters, 8 on the ring of 89, 28 on 144 and 48 on 233 — about one, two, seven and twelve glides. At a hundredth the counts are 8, 20, 52 and 116; at two hundredths 16, 38, 85 and 179; at four hundredths 27, 60, 132 and 282.

Two things follow. The outer rings take far more glides at every displacement, more than in proportion to their size — at a two-hundredth of a spacing the changed roles are 0.07, 0.09, 0.19 and 0.21 of each ring’s cells from the ring of 55 outward — which is why the ring of 233 loses its order and its make-up at the smallest displacement read. And the ring of 89 at two glides has already lost its order on seven heads of ten: a word of 21 clusters is respelled by moving one pair one cell. The order was never going to survive a measurement.

What a ring that keeps its count is made of #

Keeping its count does not mean a ring keeps its make-up of cells. The ring where 89 enters holds 34 sevens, 21 traded hexagons and 34 fives exactly, and keeps that on its median head to one and a half hundredths of a spacing. At two hundredths it holds 33, 23 and 33; at three hundredths 31, 27 and 31 — still 89 cells. The ring where 233 enters goes from 89, 55 and 89 to 85, 63 and 85 at a hundredth and 80, 73 and 80 at one and a half hundredths: 233 each time.

The sevens and fives fall together and the traded hexagons rise by twice as many. A ring under displacement pays for it by turning five–seven pairs into pairs of traded hexagons. The trade is exact in labels: a seven carries the leaving family’s contact twice, each kept family’s twice and the entering family’s once, a five the kept families’ twice and the entering family’s once, and two traded hexagons carry between them exactly that — so the trade leaves the number of ring cells, and the number of chains of the entering family beginning there, exactly where they were. On every head and at every displacement where a ring keeps its count, its sevens equal its fives and every chain of the entering family begins at one of its cells.

When the count goes #

The count fails when the ring’s run of organs starts to hold cells whose labels fit none of the five kinds — a cell with, say, two contacts of the entering family and none of one kept family, which no ordered head contains. They appear a step before the count fails: on the ring of 233 a median of 2 at a hundredth of a spacing and 10 at one and a half, where every head still keeps the count, then 19 at two hundredths, where four heads do; on the ring of 144, 4 at two hundredths and 12 at three; on the ring of 89, 2 at three hundredths, 6 at four and 10 at five. Once enough of them sit in a ring, some chain of the entering family begins outside it or two begin at one cell, and the count is off — first by a few, then by dozens: the ring of 233 reads 247 cells at three hundredths and 454 at a tenth.

The count lasts three over the family #

The four rings fail at four displacements, and the displacements fall as the entering family grows. The largest displacement every head survives is five hundredths of a spacing for the ring of 55, four for 89, two for 144 and one and a half for 233. Multiplied by the family, those are 2.75, 3.56, 2.88 and 3.50: the count lasts to about three divided by the family entering the ring, in spacings. And by about twice that displacement, at least half the heads have lost it.

The same arithmetic turned up for a different instrument. A hundredth of a spacing measured how far a six-sided cell one spacing from a ring sits from the tie that defines the ring, and found the margin to be about 9.7 divided by the ring’s family number — because rings are a factor of $\varphi$ apart in radius, and a spacing is a smaller share of an outer ring’s radius. A margin that shrinks as one over the family predicts a tolerance that shrinks as one over the family, and the count law’s tolerance does. That essay measured in a slightly larger spacing, the median wall, about 1.9 in the head’s units against $\sqrt{\pi}$'s 1.77 here; the proportionality does not depend on the choice.

Order, make-up, count — and the rim first #

So the three go in one order on every ring, and every ring goes in one order across the head. The order goes at a two-hundredth of a spacing or less, the make-up at a hundredth or two, and the count at about three over the family. Within each, the outer rings go first, the ring of 233 before 144 before 89 before 55 — the same rim-inward order the earlier essay found for its three instruments, arriving by a third route.

What survives longest is the most topological statement: that a family’s chains begin on a ring, and how many cells that takes. What goes first is the most geometric: which defect shares a wall with which. Between them, the clusters’ make-up, which depends on walls but only through how many pairs join, lasts a little longer than their order.

What it says about the Fibonacci word #

The ring essay found the clusters’ Fibonacci order on every ring of an exact golden head and offered it as a reading of the handover. It is a reading of an exact head. On a head whose organs sit a two-hundredth of a spacing off the lattice — a precision no grown or photographed head has — the order is gone from every ring larger than 89 entering. Whatever the word says about the geometry of a handover, it cannot be counted on a real head, and a bound five–seven pair, the unit the word is spelled in, is a statement about walls that the first displacement rewires.

The count is the reverse. It needs no exact head to be read: on a head displaced by a few hundredths of a spacing, every ring out to the one where 144 enter still holds exactly as many cells as its entering family, and on a head of 2,400 organs, whose outermost resolved ring is the one of 89, that is four hundredths. The count law is the part of the handover a measurement could test.

Gaussians and golden heads #

Every displacement here is independent from organ to organ. The rings were first found scattered under a different shake — each organ moved uniformly within a small square — so that essay’s fractions of a spacing and these are not the same numbers, and no comparison here is made between them. A grown head’s errors are correlated along its spirals, which might move whole rows of cells together and keep more walls than an independent shake does; nothing here grows a head. Every head is golden and of 4,000 organs, and only four rings are read. The ring where 34 enters, and the centre inside it, are left out on exact heads already, and nothing is said about them.

The classification is the earlier one, unchanged, and a cell it cannot class is counted as fitting no kind rather than forced into one; a different classifier, more forgiving of cells with an extra contact, would move where the count is said to fail without moving where the order goes.

Counts that would put the order back #

A displaced head at a two-hundredth of a spacing on which the ring of 144 or 233 keeps its Fibonacci order. A ring that keeps its count while its sevens and fives differ, or while a chain of its entering family begins elsewhere. A ring whose count fails before its make-up does. A ring whose family times its last surviving displacement falls outside 2.7 to 3.6. Each is checked on the displaced heads whenever they are read.

Labels outlast walls #

A defect ring’s Fibonacci order is decided by which defect shares a wall with which, and a two-hundredth of a spacing rewires enough walls to remove it from every ring beyond the one where 89 enter. The ring’s count — as many cells as the family entering it — is decided by each cell’s own labels and survives to about three over the family, five hundredths of a spacing on the ring of 55; while it holds, the ring pays for the displacement by turning five–seven pairs into traded hexagons, two for two, and keeps every chain start. The rings fail from the rim inward, as one over the family, which is the margin a cell one spacing from a ring has always had.

Still open: the count law on a grown head #

Every displacement here shakes each organ independently. A head grown by a placement rule places each organ against its neighbours, so its errors run along its spirals, and a family’s chains might drift together and keep their walls, or tip together and lose a whole ring. The next measurement reads the rings of heads grown on a disc rather than placed and shaken, and asks whether the count law holds on them at all, whether the order survives a correlated error better than an independent one, and whether the tolerance still falls as three over the family when the error is the rule’s own.