Phyllotaxis
the spirals are counted, not admired
Concept

Delaunay — where it appears

The triangulation dual to a Voronoi tessellation, joining two points exactly when their cells share a wall. It supplies the neighbour relation every claim about shared errors depends on, computed from positions rather than assumed from the order of arrival.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

A cell's neighbours are its spiral families. Left: part of a 900-point golden, 137.508° head, with every contact between two cells drawn and coloured by the difference between the two nodes' placement indices. Right: the share each difference takes, across all 1903 contacts between interior cells. They are the parastichy numbers — 34, 55, 21, 89, 13, 8 — and the pair a person would count is 34 and 55. The six sides Euler forces are shared out among four of them, 5.72 edges per cell.

The six are the spirals

Label every contact between two cells in a seed head with the difference between the two nodes' placement indices. The labels are the parastichy numbers — 34, 55, 21, 89 — and the six sides Euler forces turn out to be about two from one family, one and a half from the next, and one each from two more.

tissue · Contact network Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.

Every family but two is a sum

A seed head has six spiral families and everybody reports two. That looks like a convention hiding information and it is the opposite — every family but the two smallest is the sum of two others, so a third count is a prediction rather than a measurement, and a check that catches a wrong pair.

lattices · Counting 135 fives and 129 sevens among 1631 interior cells. The side counts of every cell strictly inside a golden, 137.508° head of 2400 organs, cut at 86% of the radius. The fives and the sevens are counted apart rather than summed into a spread, because they are opposite charges and the sum hides them. Summed over the interior, 264 cells that are not hexagons carry a charge of +6. Over the whole patch the charge is 294, which is exactly 6 + 2·144 — a number fixed by the 144 cells on the patch's own boundary and carrying nothing whatever about the interior. The defects are not scarce; they are balanced.

An interior that is nearly neutral

Give every cell a charge of six minus its number of sides and the total over a tessellated head is fixed by its own boundary, exactly, with nothing left over for the interior. On a golden head that freedom is spent on 264 exceptions among 1,631 cells which cancel to six.

tissue · Topological charge Every five is a cell away from a seven, and the loneliest is 0.927 spacings out. How far a five-sided cell is from the nearest seven-sided one, in cell spacings, on a golden, 137.508° head of 2400 organs. The measured bar runs from the closest five to the loneliest — 0.833 to 0.927, with a median of 0.919. It stops a single cell out, so there is no unpaired tail at all rather than a small one. The nulls are seeded permutations over 200 draws: relabelling which defects are fives puts the average five 1.254 ± 0.064 spacings away, and scattering the whole multiset over the interior cells puts it 1.786 ± 0.092. 100.0% of the fives share a wall with a seven against 68.2% for the strong null, z = 7.3.

Every five is bound to a seven

A five-sided cell beside a seven-sided one is one object in a crystal and two exceptions in a tiling, and the phyllotaxis literature borrows the crystallographic word without measuring the binding. Measured against a seeded permutation null on a 2,400-organ head, every five in the interior shares a wall with a seven, and the loneliest one in the head is 0.927 cell spacings from the nearest.

tissue · Topological charge A cut-off would have to exceed 2.236 and not exceed 1.441, and nothing does both. Each of the 608 interior cells contributes two marks: its furthest wall, and its nearest partner that is not a wall, both in units of that cell's own shortest lag. A single cut-off would have to sit to the right of every mark of the first kind and to the left of every mark of the second, and the two clouds overlap — the extreme cases are 2.236 at 0.0 per cent of the radius and 1.441 at 60.0 per cent. So the interval is empty by a factor of 1.55, while 606 of the 608 cells have a cut-off that works for themselves.

No cut-off makes them one

Two different relations on a head have both been called neighbour: the shortest index lags a count keeps, and a shared Voronoi wall. A cut-off that turns the first into the second exists for almost every cell taken alone, and for no whole head at any size.

tissue · Neighbour definition One cell's six walls, and the two the pair 34 and 55 does not name. The cell of primordium 225, at 50 per cent of the head's radius, with each of its six walls labelled by the index difference across it. Its own counting instrument returns 34 and 55, which names four of them; the two drawn warm are 21 and 21, a family the instrument ranked and discarded. Over the whole head that is 33.80 per cent of the union in dispute, and it is the same fraction in every band.

Two thirds of a cell

The founding claim of this field is that the six sides Euler forces are the spiral families. Measured against the tessellation it names two thirds of a cell's walls exactly, in every band of a head and at every rise of a stem, and the missing third is the same third everywhere.

tissue · Neighbour definition Hops give 55, 34, 89, 21 and walls give 34, 55, 21, 89. The leading families of a 900-point head, ranked twice from the same points. On the left, by the median hop the lag makes divided by the local spacing — the measurement a parastichy count is; on the right, by the share of the 1,732 interior walls the lag carries. The lists hold the same four numbers and four of them change place. The lines between are the permutation, and it is why a family read off a wall count is not a family read off a hop length.

Two rankings, one list

An essay in this collection claimed that the four shortest index hops on a seed head and the four largest shares of its cell walls are the same four numbers in the same order, and called the correspondence exact. Measured again from the same points, the two lists hold the same four families and order them differently, and they order them differently in five of the six bands the head can be read in.

tissue · Neighbour definition A stem is a thread at six rises of the sweep, a ribbon at 23 and a surface at 111. The 140 rises of the sweep, split by how many index families the tessellation carries. On 6 rises above 0.5010 a node has 2 walls and the strip is a thread; on 23 between 0.2187 and 0.4833 it has 4 and is a ribbon; on 111 below 0.2109 it has 6 and is a surface. The comparison means something different in each: two families leave 33.333 per cent in dispute on the surface and 0.00 on the ribbon, and three families leave 0.000 and 33.333.

The third family

On a seed head no threshold makes the counted contacts and the shared cell walls the same relation. On a stem they are the same relation exactly, at every one of a hundred and eleven rises and to three decimal places of nothing, provided the contact cut keeps three families where a count keeps two.

tissue · Neighbour definition The largest empty circle of a 2400-organ 137.508° golden head, ring by ring. Each dot is the largest empty circle in one ring of the head, on the interior's scale. The whole head's is 0.8437 at radius 0.871, at the centre; the rings peak at the radii where the lattice flips from one pair of neighbours to the next, and at the 4 flips beyond radius six the head reads within 0.0020 of the closed form. The dashed line is 1/√2 = 0.7071.

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.

tissue · Gapgrowth How often a cell on a 2,400-organ golden head has a wall the contact cut gets wrong, by its distance from the nearest flip ring. The 1,680 interior cells of a 2,400-organ golden head, grouped by distance from the nearest flip ring in twentieths of the head's wall spacing, 1.92. The bars are the share of each group with at least one wall a three-family contact cut gets wrong; the dots are the share with five or seven sides. 353 cells have such a wall and the farthest is 0.638 spacings from a ring. Of the 1,292 cells 0.66 spacings or more from every ring, none has.

The empty interval is the rings

No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

tissue · Neighbour definition

Named alongside it

The objects these essays reach for when they reach for this one.

Honest limitsRim effectContact familyVoronoi cellsLattice offsetNegative resultRational angleClosed formDisorderEuler's formulaMeasurementParastichy

All concepts