A defect ring is where a family of spirals begins

The six are the spirals labelled every contact between two cells of a tessellated seed head with the difference between the two organs’ placement indices, and found that the labels are the parastichy numbers: a cell’s six walls are contacts along the spiral families a person would count. The defects lie on rings found that the cells that are not hexagons are not scattered but sit on rings a factor of φ apart, and every five is bound to a seven found each five sharing a wall with a seven.

Those were three findings about the same cells, and none of them said what a five or a seven is in the language of the first. This essay reads the defect cells by their contact labels. The answer turns out to be exact, and to explain the other two.

A hexagon is three families twice #

The heads are Vogel heads of 4,000 organs — organ $k$ at radius $\sqrt{k}$ and $k$ times the angle round — tessellated by the same Voronoi cells the earlier essays used, with the outermost tenth of the radius cut away and the first few rings near the centre set aside, since they are too close together to separate. Every wall of every interior cell gets the difference of the two organs’ indices.

Away from the defect rings every cell is a hexagon with two walls from each of three families. Between the rings where 55 and 89 enter, on a golden head, that is two walls labelled 21, two labelled 34 and two labelled 55: the three shortest lattice vectors, each taken in both directions. Outside the next ring it is 34, 55 and 89. The families step along the Fibonacci numbers, and each ring is where one step happens.

Five kinds of cell across a ring #

Read across the ring where 89 enters and 21 leaves, the cells fall into exactly five kinds, and no cell on any ring read here is anything else.

A hexagon of the old families has two walls each of 21, 34 and 55. A seven has those six and one 89: the old cell with one new contact added. A traded hexagon has one 21, two 34s, two 55s and one 89: one old contact given up for one new. A five has two 34s, two 55s and one 89: both old contacts gone and only one new one gained. A hexagon of the new families has two each of 34, 55 and 89.

So a seven and a five are not exceptions to the lattice so much as the two uneven ways of making the same trade. The hexagon trades one for one. The seven takes the new contact before giving up the old; the five gives up both before it has taken the second.

The three layers, in order #

Unrolled, a ring is three layers. On the golden head the sevens of the 89 ring lie between radii 17.0 and 18.0, the traded hexagons between 18.0 and 18.5, and the fives between 18.6 and 19.4, and no cell of one kind sits among another kind. The 21-walls run in from the centre and stop in the traded layer; the 89-walls start at the sevens and run outward; the 34s and 55s pass straight through.

The same order holds on every ring: sevens, then traded hexagons, then fives, outward. And the radius at which the ring was predicted from the divergence angle alone — 11.274, 18.238, 29.513 and 47.758 for the rings where 55, 89, 144 and 233 enter — falls inside the traded layer every time, between 11.05 and 11.58, 18.00 and 18.55, 29.26 and 29.82, and 47.49 and 48.05. The predicted ring is the trade itself, with the sevens a little inside it and the fives a little outside.

How many of each #

Counted, the rings obey one arithmetic. The golden ring where 55 enters and 13 leaves holds 21 sevens, 13 traded hexagons and 21 fives, 55 cells. Where 89 enters and 21 leaves: 34, 21 and 34, which is 89. Where 144 enters: 55, 34 and 55. Where 233 enters: 89, 55 and 89. On a head at the Lucas angle, where the families are Lucas numbers, the ring where 76 enters holds 29, 18 and 29, and the ring where 199 enters 76, 47 and 76.

Three statements, all exact on every ring read on its own: a ring holds as many cells as the family entering it; it holds as many traded hexagons as the family leaving it; and the rest are sevens and fives in equal numbers. The earlier finding that each ring holds equal numbers of both kinds is the third of these, and the first two say what the number is.

Why the entering family sets the count #

The arithmetic has a reason that the labels make plain. A family of n spirals is n chains of organs, each joined to the organ n places further on. Inside the ring there is no family of 89, so every one of its 89 chains has to begin somewhere, and a chain begins at an organ joined to the organ 89 places later but not to the one 89 places earlier — an organ with exactly one 89-wall.

Every cell of the ring, seven, traded hexagon or five, has exactly one 89-wall, and outside the centre no other cell has one until the family of 89 itself leaves, two rings further out. So the 89 chains begin one at each ring cell, and the ring has 89 cells. The family of 55 shows the whole life of a family on one head: its 55 chains begin at the 55 cells of the ring where it enters and end at the 55 traded hexagons of the ring where 233 enters, and between the two no cell carries a single 55-wall. Counted directly, on the golden and Lucas heads, every chain of the entering family that begins outside the centre begins at a cell of its ring: 55 of 55, 89 of 89, 144 of 144, 233 of 233, and on the Lucas head 47, 76, 123 and 199 of the same.

The leaving family is the same argument run backward. Its 21 chains must each end, and a chain ends at an organ with one 21-wall. The sevens have two and the fives none; only the traded hexagons have one. So there are 21 traded hexagons. What remains, 89 − 21 = 68, has to be shared between sevens and fives, and Euler’s formula — which leaves the interior no net charge to spend — shares it evenly, 34 and 34.

A ring is a run of consecutive organs #

The chain argument has a simpler face. A family of n spirals is the n residues of the placement index divided by n: one chain holds organs 0, n, 2n and on, the next 1, n + 1, 2n + 1. If all n chains begin on one ring, they begin at n consecutive organs, one of each residue. Read in placement order instead of by radius, the ring is exactly that.

On the golden head the ring where 89 enters is organs 290 to 378, eighty-nine in a row with nothing else among them: organs 290 to 323 are its 34 sevens, 324 to 344 its 21 traded hexagons, 345 to 378 its 34 fives. The ring where 55 enters is organs 101 to 155, where 144 enters 801 to 944, where 233 enters 2,166 to 2,398. The Lucas head’s rings are 206 to 281, 576 to 698 and 1,566 to 1,764, and the rings at 137.3° that stand apart are runs of 34, 118 and 215 organs in the same order. A golden head of 2,400 organs has its rings at exactly the same organs as one of 4,000; a head’s size decides only how many rings it reaches.

So the five kinds of cell are not a radial pattern that happens to line up with the indices. They are consecutive stretches of the placement order: a run of sevens as long as the smaller kept family, a run of traded hexagons as long as the leaving family, a run of fives as long as the first. The organ at which the ring starts is the first organ whose cell holds a wall to the organ n places later — the first place the new family fits.

The fives and sevens come in clusters #

Every five on these rings shares a wall with a seven, as the earlier measurement found, and the labels say which wall: every five touches a seven across the larger of the two families it keeps — 55, on the ring where 89 enters — and some touch a second seven across the smaller, 34. On that ring 21 fives touch one seven and 13 touch two. On the ring where 144 enters, 34 touch one and 21 touch two.

Joined by those walls, the fives and sevens of a ring make clusters, and the clusters are of two sizes only: a single five and seven, or two fives and two sevens. On the ring where 89 enters there are 8 single pairs and 13 double pairs — 21 clusters, as many as the leaving family, so that the clusters sit one between each pair of traded hexagons, and every traded hexagon touches three of the ring’s fives and sevens. On the ring where 144 enters, 13 single and 21 double; where 233 enters, 21 and 34.

In the order of the Fibonacci word #

Round the ring the two sizes alternate in a fixed pattern. No two single pairs are ever adjacent, and on every golden ring the sequence, read in order of angle from any starting cluster, is a rotation of the finite Fibonacci word of its length — the word built by appending each word to the one before, which is also the sequence of long and short gaps the golden angle leaves round a circle. On the Lucas head the words are not Fibonacci words, since their lengths are Lucas numbers, but every one is balanced: any two stretches of the same length hold single pairs differing in number by at most one, which is the defining property of the words a line of irrational slope cuts.

So a defect ring of a golden head is a one-dimensional quasicrystal laid round a circle, with the same arithmetic in it that put the organs where they are.

The run of consecutive organs suggests why. A ring is n consecutive multiples of the divergence angle, and the three-gap theorem says that n consecutive multiples of an irrational angle cut the circle into gaps of at most three lengths — for the golden angle and a Fibonacci n, two lengths, arranged round the circle in the order of the Fibonacci word. The clusters’ word has the same length, the same counts and the same order. That the clusters are those gaps, cell for cell, is a reading this measurement suggests and has not traced.

A head at an angle that is not noble #

The golden and Lucas angles are noble, and their families always drop the smallest of three. At 137.3°, which is not noble, some rings drop the middle family instead. The ring where 118 enters keeps 21 and 97 and drops 76; it holds 21 sevens, 76 traded hexagons and 21 fives, which is 118, by the same arithmetic. The ring where 215 enters keeps 97 and 118 and drops 21: 97, 21 and 97. Where the rings stand apart the count law does not care which family leaves.

What breaks is the separation. The rings where 55 and 76 enter abut within a few hundredths of a unit of radius, and the ring where 76 enters has traded hexagons naming more than one leaving family — two transitions run together — so their cells cannot be assigned to one handover, and the count fails there. And the clusters are not the golden head’s: on the ring where 215 enters they are runs of eight and ten cells, and four fives on the whole head share no wall with a seven, which no golden or Lucas head shows.

What it says about the earlier findings #

Three earlier results become one statement. The defects lie on rings because a ring is where a family of spirals begins. Each ring holds equal fives and sevens because the cells that neither begin a new chain alone nor end an old one must balance each other’s charge. And every five is bound to a seven because a five and a seven are the two halves of a trade made unevenly, sitting one layer apart across the same walls.

The rings are also why the second moment of the side counts falls as it does with head size: a ring holds a Fibonacci number of defects, so the share of defect cells is set by how many rings fit and how far apart, which the angle decides.

Exact heads, with the centre left out #

That a real capitulum has these rings. The earlier measurement found the rings destroyed by a twentieth of a cell spacing of displacement, finer than any photograph could place an organ, and every head here is exact. The pairing survived far more displacement; whether the clusters and their order do has not been measured.

Nor does it reach the centre, where rings crowd within a cell’s depth of each other and the first few cells are not a lattice at all, or the rim, cut away here as before. The innermost ring shows where the cut falls rather than where the law fails: on the golden head the ring where 34 enters begins at organ 33, at a radius of 5.7, just inside the radius of six the measurement starts from, so only 31 of its cells and 31 of its chains are counted there. Read whole, it is organs 33 to 66 — 13 sevens, 8 traded hexagons and 13 fives, in that order, 34 cells — the same law one transition further in, where the cells are only a few spacings from the centre. The heads are Vogel’s model, and the labels are placement indices, which a photograph does not have — though a counter that needs no index can trace the same families from positions.

Counts that would break the law #

A ring, read on its own, whose cells do not number the entering family, or whose traded hexagons do not number the leaving one. A ring cell with no wall, or two walls, of the entering family. A golden ring whose clusters are not a rotation of the Fibonacci word, or a Lucas ring whose clusters are not balanced. Each is checked on every ring of the three heads.

A ring, counted by its walls #

Read by their contact labels, the cells of a seed head’s defect ring are five kinds in a fixed order outward — hexagons of the old families, sevens, traded hexagons, fives, hexagons of the new — and the ring holds exactly as many cells as the family entering, exactly as many traded hexagons as the family leaving, and equal sevens and fives with the rest, because every chain of the entering family begins at a ring cell and every chain of the leaving family ends at a traded hexagon. The fives and sevens join into one cluster between each pair of traded hexagons, single and double pairs in consecutive Fibonacci numbers, in the order of the Fibonacci word.

Still open: whether the word survives displacement #

Every head here is exact, and the earlier measurement found that a twentieth of a spacing of displacement breaks the rings into scattered defects while the pairing holds. The handover reading gives that breakdown something precise to watch. The next measurement displaces heads by a hundredth to a tenth of a spacing and asks, ring by ring, when the count law first fails — whether a displaced ring still holds as many cells as its entering family, with chains beginning elsewhere — and whether the clusters lose their Fibonacci order before or after the ring loses its shape.