A Lucas ring gives way where a golden ring of its size does

A defect ring is where a family of spirals begins read the rings of five- and seven-sided cells on a tessellated head by the labels of their walls and found a count law: each ring holds exactly as many cells as the spiral family entering it — sevens, traded hexagons and fives — because every chain of that family begins at one of its cells. Two essays then shook golden heads to see how long the law lasts. A ring keeps its count after it loses its order moved every organ independently and found the count surviving to about three over the entering family, after the clusters’ order and their make-up had gone. A ring outlasts an error along its spirals carried the error along the spirals instead and found the count lasting longer, but no longer once the displacement was measured against an organ’s neighbours: four to five over the family, either way.

Every displaced head in those essays was golden. The count law itself was already known to hold on a head at the Lucas angle, 99.5°, whose families run 4, 7, 11, 18, 29, 47, 76, 123, 199. What was not known is whether its tolerance is a property of the golden angle — of the particular rings a golden head grows, at the radii it grows them — or of the count itself. The question the last essay left was the same reading on a Lucas head.

The Lucas rings, exact #

On an exact head of 4,000 organs at the Lucas angle, four rings stand apart from the crowded centre: those where 47, 76, 123 and 199 enter. Each holds exactly its entering family. The ring of 76 holds 29 sevens, 18 traded hexagons and 29 fives; the ring of 199 holds 76, 47 and 76. The traded hexagons number the family leaving — 18 where 76 enters, 47 where 199 does — and the sevens and fives the family between. It is the golden arithmetic with Lucas numbers in it, as the essay that found the law already reported.

Unrolled, a quarter of the ring of 76 shows the same structure a golden ring does: sevens on the inside, fives on the outside, each five walled to a seven, the pairs and double pairs strung along the ring with traded hexagons between. At a fiftieth of a spacing of displacement on this head the ring still holds 76 cells, but two of its clusters have grown to sizes the exact ring never has.

Its clusters, in order #

The essay that found the count law also read the order of the clusters round each ring and found it, on a golden head, to be a rotation of the Fibonacci word, and on a Lucas head to be balanced — any two stretches of the same length differing by at most one single pair. Set beside each other the two heads make the connection plainer than either alone. The Lucas rings carry 4 and 7, 7 and 11, 11 and 18, 18 and 29 single and double clusters; the golden rings 5 and 8, 8 and 13, 13 and 21, 21 and 34. Every word on both heads is balanced, and for two letter counts that share no factor there is exactly one balanced cyclic word. The Fibonacci word is not a separate fact about golden heads; it is the one balanced word with Fibonacci counts, and a Lucas ring spells the one with Lucas counts.

So each ring carries three properties to lose: its order, its letters — how many single and how many double clusters — and its count.

What goes first #

Displaced organ by organ, on ten heads at each of ten sizes from a two-hundredth of a spacing to three twentieths, the Lucas rings lose their properties in the golden order. The clusters’ order goes first: the ring of 47 keeps it on nine heads in ten at a two-hundredth of a spacing and on none at three hundredths; the ring of 76 on three heads at a two-hundredth; the rings of 123 and 199 on one and none. The letters go next, then the count, which every ring keeps on every head well after both — the ring of 47 to 0.075 of a spacing, of 76 to 0.05, of 123 to 0.02, of 199 to 0.015.

On no ring is the order lost later than the letters, or the letters later than the count. That was the golden sequence, and on a different angle with different families it is the same.

A ring of the same size, at the same displacements #

The dashed lines in that figure are the golden rings of the nearest family — 55 beside 47, 89 beside 76, 144 beside 123, 233 beside 199 — and each pair of curves nearly coincide. The closest pair is closest by number: 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, and their letters on nine heads and then four or five. The Lucas ring of 47 keeps its order slightly longer than the golden ring of 55 — on six heads at a hundredth of a spacing against four — and has eight fewer cells.

A ring loses its order by walls flipping between neighbouring cells, which slides a five–seven pair along the ring one cell at a time, as a ring keeps its count after it loses its order traced. How easily that happens depends on how close the ring’s organs sit to one another’s circles, and in a ring of n cells laid round a band of fixed width, that is set by n. The angle that made the ring does not appear.

How far the count survives #

The count’s tolerance falls with the entering family on both heads, along one line. Organ by organ, the family times the largest displacement every head survives is 3.52, 3.80, 2.46 and 2.98 on the Lucas rings of 47, 76, 123 and 199, and 2.75, 3.56, 2.88 and 3.49 on the golden rings of 55, 89, 144 and 233. Both sets lie between 2.4 and 3.9, the band the golden essay summarised as about three over the family.

The drifted heads — each organ inheriting nine tenths of the mean error of its two nearest earlier organs, scaled to the same size — survive larger displacements on both heads: the Lucas ring of 47 to a tenth of a spacing, 76 to three fortieths, 123 to a twenty-fifth, 199 to a fiftieth. Again the golden rings beside them survive almost the same: 55 to a tenth, 89 to a twentieth, 144 to a twenty-fifth, 233 to a fiftieth.

Measured against the neighbours #

The golden drift essay’s finding was that the drift’s extra tolerance is not extra: a drift moves neighbours together, and stated as how far an organ moves against its neighbours the drift’s tolerance falls back into the band the independent heads set. On the Lucas head the drift moves an organ against its neighbours by 0.62 of what an independent displacement of the same size does, the golden drift by about the same, and with that taken out every product on both heads, for both kinds of error, lies between 3.4 and 5.4. The four-to-five-over-the-family law of the golden essays is a law of the count on either head.

What the count reads is how far an organ has moved relative to the organs it is walled to — the six it touches — and how many walls the ring has to keep. Neither knows which angle laid the head down.

While it holds, it trades #

A golden ring that keeps its count pays for the displacement in kind: its sevens and fives fall together and its traded hexagons rise by twice as many, so the total stays the family. The Lucas rings do the same. The ring of 199 at a fiftieth of a spacing holds a median of 67 sevens, 65 traded hexagons and 67 fives — nine fewer of each defect, eighteen more traded hexagons than exact — and 199 cells. The ring of 76 at a twentieth holds 25, 26 and 25.

Up to a twentieth of a spacing, every Lucas ring that keeps its count keeps its sevens equal to its fives and every chain of its family beginning on it, displaced or drifted, on every head. The trade is exact while the law holds.

Where half the heads give up #

The tolerance above is the last displacement at which all ten heads keep the count, which is a statement about the luckiest head’s opposite. The other end is where half of them have lost it. On the Lucas head that is three twentieths of a spacing for the ring of 47, three fortieths for 76, a twentieth for 123 and three hundredths for 199; on the golden head a tenth for 55, three fortieths for 89, a twenty-fifth for 144 and a fiftieth for 233. Times the family, the Lucas half-way points are 7.1, 5.7, 6.2 and 6.0, the golden 5.5, 6.7, 5.8 and 4.7 — the same spread around six, about twice the last displacement every head survives.

Under the drift the half-way points move out together: three twentieths for the Lucas ring of 76 against a tenth for the golden ring of 89, three fortieths for both 123 and 144, a twenty-fifth for both 199 and 233. At no ring on either head does the drift bring the half-way point in.

The smaller ring, every time #

In each pairing the Lucas ring is the smaller: 47 against 55, 76 against 89, 123 against 144, 199 against 233, between eight and thirty-four cells fewer. In every reading of the count — the last displacement all heads survive and the one where half give up, displaced and drifted — the Lucas ring is as tolerant as the golden ring beside it or more, never less. The ring of 47 keeps its count on half its heads to half again the golden 55’s displacement; the ring of 199 to half again the golden 233’s. Under the drift the clusters’ order lasts slightly longer on the Lucas rings too — on seven heads in ten for the ring of 76 at a two-hundredth of a spacing, against six for the golden ring of 89, and on two at a hundredth against none.

That is the direction a size law predicts and an angle law does not. If the golden angle conferred some extra robustness on its rings — the property that makes it the hardest angle to approximate, which what a head can mean by most irrational found is shared by the Lucas angle over every count a head shows — the golden rings would outlast Lucas rings of the same size. They do not; they behave like slightly larger Lucas rings.

A count without its clusters #

How much of a ring is left by the time only its count remains is easiest to see in the clusters. On the heads that keep the ring of 76’s count, the median number of clusters of a size the exact ring never has — neither one five with one seven nor two with two — is nought at a two-hundredth of a spacing, one at a hundredth, three at a fiftieth, six at three hundredths and eleven at a twentieth, by which point the ring holds 22 clusters where the exact one held 18, and three of the cells in its run of organs fit none of the five kinds. Every one of those heads still holds 76 ring cells with its sevens equal to its fives. On the ring of 199 at a fiftieth of a spacing, the nine heads that keep the count carry a median of 33 odd clusters among 59, and fourteen unclassable cells in the ring’s run.

So “keeps its count” is a narrow statement. By the time a ring has lost its clusters’ order and make-up, it is a band of defects walled together in no particular way, and what survives is only the arithmetic: as many cells with one entering contact as the family entering, which is a statement about where the chains of that family begin and nothing else.

A count kept by coincidence #

Beyond a twentieth the smallest ring does something the law does not account for. On eleven heads — two displaced by three fortieths of a spacing, four by a tenth, five drifted by three twentieths — the ring of 47 holds exactly 47 cells with its sevens and fives one apart: 14 sevens, 18 traded hexagons and 15 fives, or 13, 19 and 15. A ring whose defects no longer balance has not kept the law; it has landed on the right total by a cell gained in one place and lost in another. No larger Lucas ring does this at any displacement read, and the golden essays met the same thing once, on the ring of 55 at three twentieths of a drift.

So “every head keeps the count” at three fortieths of a spacing on the ring of 47 overstates what is kept: the count survives there, the balance does not. The tolerance of 3.52 for that ring is the count’s, read honestly as the number of cells; the law’s own tolerance — count and balance together — is a twentieth of a spacing, a product of 2.35, just below the band. Read that way the Lucas ring of 47 is no more tolerant than the golden ring of 55.

What the Lucas head does not test #

Both angles here are noble: their continued fractions end in an unbroken run of ones, so every ring drops the smallest of the three families crossing it, and every ring’s clusters come in two sizes. A head at an angle that is not noble, such as 137.3°, has rings, as the essay that found the count law found, that drop the middle family and clusters of a third kind, and nothing here says how its rings give way. Nor does anything here touch rings nearer the centre than the one where 47 enters: the ring of 29 holds sixteen cells on the exact head, inside the radius where rings cannot be told apart. Ten heads are drawn at each displacement, so a tolerance is good to the spacing between the displacements read.

The law in one line #

On either noble head a defect ring holds its entering family, spells the one balanced word its letter counts allow, loses its order first and its count last, and keeps that count to about three over its family organ by organ and four to five over it against its neighbours. A ring’s size decides how much it can take; the angle decides only which sizes there are.

What would show it is the angle after all #

A Lucas ring whose tolerance, times its family, fell outside the band the golden rings set. A Lucas ring losing its count before its order, or keeping its order much longer than the golden ring of its size. A counted Lucas ring with unequal sevens and fives inside a twentieth of a spacing. Each is checked against every head above.

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

The two heads here differ in their numbers and agree in their arithmetic, because both angles’ continued fractions end in ones. At 137.3° the essay that found the count law met rings that drop the middle family rather than the smallest, and clusters of a third size; its rings do not alternate two letters, so there is no balanced word to lose and perhaps no single sequence in which a ring gives way. The measurement is the same displacement on a head at 137.3°: whether its rings keep a count law at all under displacement, whether their tolerance still falls as about three over the entering family, and whether a ring that drops its middle family is more or less fragile than one that drops its smallest.