Lattice vectors — where it appears
Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.
Counting without an index
A person counting spirals on a cone puts a finger on one scale, follows a family round, and counts how many distinct chains there are. That needs no order of arrival — and building it turns out to be strictly more general than the counter that reads the order of arrival, and to find a bug in the counting of a bijugate stem that nothing had caught.
The rings are not the transitions
A seed head has two ladders on it — the radii where the counted parastichy pair changes, and the radii where the exceptional cells sit — and the obvious guess is that they are the same ladder. They are not: the second sits a factor of the square root of phi outside the first at every rung of two different divergence ladders, which is exactly halfway between two consecutive transitions.
A counter that cannot be slid
The counter that needs no order of arrival follows each family into chains and counts them, and on an ideal lattice it agrees with the counter that does. On a stem whose rise falls it works only inside a band of widths, and outside the band it returns a pair rather than refusing: the rung below when the band is too narrow for the larger count, a pair on no rung when the band spans more than about a third of a rung of rise. The upper edge moves with the rate, so a width that is right on one stem is wrong on another, and on the fastest bijugate stem measured no width works at all.
One over root two
On the interior's scale a golden head's largest empty circle is 0.8435 of a spacing at every size from 150 organs to 2,000, because one triangle at its centre decides it. Everywhere else it is 1/√2 — a square cell at every ring where the lattice flips — and a closed form in the angle's continued fraction says that only noble angles hold it there.
The positions read the twist's shape
A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.
The centre the spirals give
Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.
A head is a lattice two dozen organs at a time
Everything said about a seed head leans on the head being a lattice, and nobody states how much of one it is. Fitted by geometry alone, a straight lattice misses a patch of radius L at radius r by L squared over four r, to within a few per cent, whatever the divergence. So a lattice good to a tenth of a spacing holds about three quarters of r organs: twenty-three at organ 900, thirty-seven at the rim of a 2,400-organ head. At about twice that, two organs land on one site and the patch has no straight lattice at all. The coordinates that straighten a cylinder do not help. Azimuth against radius squared makes the whole head exact, to the last bit a double holds.
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.
Most of the Fibonacci share was one family read twice
The census behind How often is it Fibonacci takes the two shortest hops at each divergence and calls them the pair. A pair of spiral families has to be two families, and the second-shortest hop is often the first taken twice in the same direction. That is the whole of the bucket once marked whorled, and it is most of the Fibonacci share too, since the pair 1/2 is one family read twice. Read by the lattice's own two families, the share at a rise of 0.008 is 3.2 per cent, not 14.7; at 0.12 it is 32 per cent, not 67.5; and Fibonacci's tenfold lead over Lucas at the finer rises is gone, 3.2 against 3.0.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsParastichy pairDefect ringRiseRound tripVogel's modelCounting blindChain countingCylinderDelaunayDislocationDisplacement