A ring outlasts an error that runs along its spirals

A defect ring is where a family of spirals begins read the rings of five- and seven-sided cells on a tessellated golden head — the rings the defects lie on — by the labels of their walls and found two laws. 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. And its fives and sevens join into clusters of one pair and two that run round the ring in the order of the Fibonacci word.

A ring keeps its count after it loses its order displaced every organ of a 4,000-organ golden head by an independent gaussian and found the order going first, then the clusters’ make-up, and the count last: every one of ten heads kept every ring’s count to a displacement that falls as about three over the entering family — five hundredths of a spacing on the ring where 55 enter, a hundredth and a half on the ring where 233 do.

That essay ended on the errors a grown head actually has. An organ placed against its neighbours carries their errors, so a family’s chains might drift together and keep their walls, or tip together and lose a whole ring. It asked whether the count law holds under such an error, whether the order survives it better, and whether the tolerance still falls as three over the family.

The same head, moved together #

The head is the same 4,000-organ golden Vogel head and the rings are read in exactly the same way. Only the movement changes. Instead of an independent gaussian, each organ, in the order the head is laid down, inherits a share ρ of the mean error of its two nearest earlier organs and adds a fresh error of its own; the finished field is scaled so that its size per coordinate is σ spacings, the size of the independent displacement it is compared with. A drift along the spirals changes fewer counts used the same drift on the census’s discs.

The two nearest earlier organs are the ones an organ sits against, one on each family crossing there, so the drift runs along both families at once. At ρ = 0.9 it moves an organ against its tessellation neighbours by 0.62 of what an independent displacement of the same size does, averaged across the rings’ runs; at ρ = 0.6, by 0.85. Both ratios hold to within a hundredth at every size from a two-hundredth of a spacing to fifteen hundredths.

Ten heads are drawn at each of ten sizes for each correlation, two hundred heads in all, out to fifteen hundredths of a spacing — further than the independent heads were taken, because the drifted rings last further.

A quarter of the ring where 89 enter #

At five hundredths of a spacing the drifted ring still reads as a ring: sevens above, traded hexagons between, fives below, the walls between fives and sevens drawing short clusters along the band, and 89 cells round the whole ring. At a tenth of a spacing it holds 91 cells on this head, where the same head displaced independently by a tenth holds 101 — the independent ring has swollen with cells that are not part of the handover, and the drifted one has barely started to.

The drawing shows why. Under the drift neighbouring organs have moved together, so the walls between them — which are what the cell kinds are read from — have mostly not changed.

The count lasts longer #

Under the strong drift every ring keeps its count longer. Every one of ten heads keeps the count of the ring where 55 enter to a tenth of a spacing, against a twentieth under independent displacement; the ring of 89 to five hundredths, against four; the ring of 144 to four hundredths, against two; the ring of 233 to two hundredths, against a hundredth and a half. That is a quarter again to twice as far.

The weak drift is not told apart from independent displacement on this grid. Its rings keep their counts to the same sizes, five, four, two and a hundredth and a half, though they lose them more slowly past there — on the ring of 144 at three hundredths nine heads of ten keep the count, against eight independently. The gain it should give, by its 0.85, is smaller than the steps between the sizes measured: divided by 0.85 the independent tolerances become 0.059, 0.047, 0.024 and 0.018 of a spacing, and every one of those falls between the size the weak drift was last kept at and the next size measured. The grid cannot see a gain of fifteen per cent, and a finer one is what would test it.

The order and the letters last longer too, and still go first #

The clusters’ make-up — how many single pairs and how many double — goes before the count under every kind of displacement, and under the strong drift it goes later. On the ring of 144 at a two-hundredth of a spacing all ten drifted heads keep it against four of ten independently; on the ring of 55 at two hundredths, eight against four; on the ring of 89 at a hundredth, eight against five.

The Fibonacci order goes first everywhere, and it is the most fragile property of a ring under either kind of error: no ring keeps it on every head at any size measured. But the drift loses it more slowly — on the ring of 89 at a two-hundredth of a spacing, six heads of ten keep the order under the strong drift against three independently. So the sequence the earlier essay found holds under a drift exactly: order, then the letters, then the count, on every ring at every correlation. Only the sizes at which they go have moved out.

What a ring is made of while it counts #

Up to the size at which every head keeps it, a drifted ring that holds its count is balanced as the exact ring is: as many sevens as fives, and a chain of the entering family beginning at every one of its cells. Its make-up moves the same way independent displacement moved it, trading dislocations for hexagons without losing a cell: on the ring of 55 the median drifted head holds the exact 21 sevens, 13 traded hexagons and 21 fives at five hundredths of a spacing, and 19, 17 and 19 at a tenth.

The drift is quieter off the ring as well. A chain of a family that begins anywhere but on its ring is a stray start, a defect the displacement has made. The exact head has none — 55 starts of the 55 family, all on its ring. At five hundredths of a spacing the independent heads have 140 starts of that family in the median and the drifted ones 102; at a tenth, 241 against 176. The outer families stray less under either kind of error at these sizes: the 89 family has no stray start in the median at five hundredths under either, and the 233 family 29 independently against 7 drifted.

How the make-up drains #

Inside a ring that keeps its count, displacement trades a five and a seven for two traded hexagons, as every five is bound to a seven would lead one to expect of a bound pair that unbinds. The drift trades fewer at every size. On the ring of 89 at three hundredths of a spacing the independent heads hold 31 sevens, 27 traded hexagons and 31 fives in the median — three pairs traded — and the drifted heads 33, 23 and 33, one pair. On the ring of 233 at two hundredths the independent heads hold 79, 79 and 77, already unbalanced and no longer 233 cells, and the drifted heads 80, 73 and 80: nine pairs traded and the count still exact.

The trading is the ring’s way of absorbing an error without losing a cell. Under a drift of the same size there is less to absorb, because less of the error falls between neighbours. A hundredth of a spacing found the same ordering on a different reading of the same heads: the most fragile reading of a displaced head is the one decided by the worst of thousands of cells, and the readings decided cell by cell survive two to four times further. A ring’s order is decided by its whole run of clusters, its count cell by cell; the drift moves both out without changing which goes first.

The strong drift, size for size #

Put side by side at the same size of error, the two kinds of displacement look like two different amounts of it. At five hundredths of a spacing the strongly drifted ring of 55 keeps its count and its exact make-up in the median, with 47 stray starts of its family where the independently displaced head has 85 and has begun to trade. At a tenth the drifted ring of 55 still keeps its count on every head, with two pairs traded; the independent heads have lost it on six of ten. On the ring of 144 the drift keeps the count to four hundredths on every head, where independent displacement keeps it on one head in ten.

That is what the relative displacement says to expect. The strong drift at a tenth of a spacing moves an organ against its neighbours as much as an independent displacement of about six hundredths does, and an independent displacement of six hundredths keeps the ring of 55 — whose tolerance is five — on some heads and not all. A tenth of drift is, to the ring, a little more than a twentieth of noise.

Past the tolerance, a count by coincidence #

Past the size at which every head keeps the count, the count is no longer a law, and one drifted head shows what that means. At fifteen hundredths of a spacing the sixth head’s ring of 55 holds exactly 55 cells — 17 sevens, 20 traded hexagons and 18 fives. It has the count and has lost the balance: one five too many, one traded hexagon in place of a seven. The count of 55 there is the arithmetic of a ring that has come apart landing on the right total.

That is the reason the tolerance is read as the size every head survives rather than the size most do. A ring keeping its count by coincidence is not reading its family.

Three over the family, and five #

Independently displaced, family times tolerance comes to 2.75, 3.56, 2.88 and 3.49 on the four rings — the “about three over the family” the earlier essay found. Under the strong drift it comes to 5.50, 4.45, 5.76 and 4.66, about five over the family. So the tolerance still falls as one over the family, and the constant has grown by about half.

The family’s part of the law is the same under both kinds of error: whatever makes a larger ring more fragile, the drift does not change it. A larger family’s ring is longer, and the earlier essay measured the fall rather than deriving it. What the drift does change is the constant, and the constant is the part a second kind of error can explain.

The constant is relative displacement #

Restate each tolerance as how far an organ moves against its tessellation neighbours — the plain size times $\sqrt{2}$ for an independent displacement, times $\sqrt{2}$ and 0.62 for the strong drift — and the two kinds of error come back together. Independently the products are 3.89, 5.03, 4.07 and 4.94; under the strong drift, 4.85, 3.92, 5.08 and 4.11. The drift’s four fall inside the band the independent heads set, to within a twentieth at the top. In plain displacement every one of the strong drift’s products is above every independent one.

So the count law’s tolerance is a statement about relative displacement. A ring keeps its count while organs move against their neighbours by less than about four to five over the family, in spacings, whatever the arrangement of the error that moves them; a drift buys its extra reach only by moving neighbours together. The weak drift’s products in relative displacement, 3.30 to 4.27, fall below the band only because its gain is smaller than the grid’s steps — it was measured at the same sizes as the independent heads.

Two instruments, one rule #

A drift along the spirals changes fewer counts found the same thing in a different instrument: a parastichy count on a Vogel disc, read in a surveyor’s band, changed about as many readings under a drift as under an independent displacement of the drift’s relative size. That instrument reads which organs are each other’s neighbours along chains; this one reads which cells share a wall, whose labels the six sides of a hexagon showed are the spiral families. They are two readings of the same neighbour graph, and both answer to the same thing: what the error does to an organ against its neighbours, not what it does to the organ.

That is why the question as asked has a simpler answer than it seemed to need. Whether a family’s chains drift together or tip together is a question about the relative part of the error. A drift that moves neighbours together keeps the walls and the count; one that tips them does not, and the measurement here says which this drift is.

What a grown head would need to show #

A head grown by a placement rule, rather than placed and moved, would carry its own errors. If they are like this drift, its rings should keep their counts to a larger displacement than independent noise of the same size, and the relative displacement should predict how much larger. A grown head whose rings lose their counts at a smaller relative displacement than these would be a head whose errors tip chains rather than carry them.

That gives a grown head a number to be held to. Measure how far its organs sit from the lattice and how far they move against their neighbours, take the ratio of the second to $\sqrt{2}$ times the first, and the rings’ tolerance is about four and a half over the family divided by that ratio and by $\sqrt{2}$: for the ring of 89 on a head whose errors move neighbours apart by half the independent amount, about seven hundredths of a spacing, against four for independent noise.

No head here is grown. The drift is a stand-in with one property of growth — that an organ inherits its neighbours’ errors — and nothing else.

A drifted ring that counts less would end it #

A ring that keeps its count to a smaller displacement under the strong drift than independently. A drifted ring that keeps its count, on every head, with its sevens and fives unequal. A relative tolerance under the drift far outside the band the independent heads set. A ring that loses its count before its order under any drift. Each is checked against every head read above.

Still open: a ring on a head that is not golden #

Every head here and in the essays before it is placed at the golden angle, whose rings hand over between consecutive Fibonacci families. A head at another noble angle — the Lucas angle, whose families run 4, 7, 11, 18, 29 — has its own rings at its own transitions, and the count law should hold on them with the Lucas family in place of the Fibonacci one. The measurement is the same reading on a Lucas head: whether each ring holds exactly as many cells as its entering Lucas family, whether its clusters spell a word of their own, and whether the tolerance falls as the same four to five over the family in relative displacement.

Measured, a Lucas ring gives way where a golden ring of its size does: on a 4,000-organ head at the Lucas angle the rings where 47, 76, 123 and 199 enter keep their counts to 2.5 to 3.8 over the family organ by organ and 3.4 to 5.4 over it against neighbours, displaced or drifted, inside the golden band; they lose order, letters and count in the golden sequence, and the ring of 76 gives way at the displacements the golden ring of 89 does.