The six are the spirals

The previous essay established that the mean number of sides in a tessellated head is six because Euler’s formula forces it, and that the informative statistic is one step further into the distribution. This essay is about a different thing the mean throws away, and it is not a number at all.

It is which neighbours.

The measurement #

Take a golden-angle head of nine hundred points. Build the Delaunay graph — the contact network of the tessellation, one edge for every pair of cells that share a wall. Cut the rim off. Then label every remaining edge with a single number: the difference between the two nodes’ placement indices.

Node 300 touching node 334 contributes 34. Node 300 touching node 355 contributes 55.

That is the entire method, and what it is not given is worth stating in the site’s usual form: an array of coordinates in placement order, and nothing else. Not the divergence angle, not the model, not the parastichy numbers, not the fact that Fibonacci exists.

The result #

Of 1,903 contacts between interior cells:

Those are the parastichy numbers. The pair a person would count in that band of the head is 34 and 55, and both are in the list; so are 21 and 89, which nobody counts; so, faintly, are 13 and 8.

And the shares add up to the theorem. Every interior cell has 5.72 edges — six, to the accuracy a finite rim-cut sample allows — distributed as about 1.9 from the 34 family, 1.6 from the 55, 1.0 from the 21 and 0.9 from the 89.

So the six sides Euler forces are not six arbitrary neighbours. A cell’s neighbours are its spiral families, and the count “six” is the sum over them.

Why that is not obvious #

It is worth pausing on why this needed measuring rather than deriving.

Euler’s relation is a statement about a planar graph and says nothing about which vertices are joined. Any tessellation of any point set averages six sides. So the theorem constrains the total and leaves the composition entirely open, and the composition is where every fact about the arrangement lives.

There is also a version of the claim that is trivially true and is not this one. Obviously a cell’s nearest neighbours in space are its nearest neighbours along the parastichies — that is close to the definition of a parastichy. What is measured here is that the Delaunay graph, which is a topological object built from circumcircles and not from a distance threshold, picks out exactly those offsets and in a definite proportion. That is a fact about the tessellation, not a restatement of the lattice.

Why the Delaunay graph and not a distance threshold #

The choice of what counts as a contact is the one place this measurement could be made to say whatever the author wanted, so it is worth defending.

The obvious alternative is a distance cut: two nodes are neighbours when they are closer than some multiple of the local spacing. It is simpler, and it is the wrong instrument here for a reason this site has met before in another form. A distance threshold is a free parameter, and the answer moves with it — set it small and only the 34 family survives, set it large and offsets appear that share no wall with anything. A table of shares produced that way is a table about the threshold.

The Delaunay graph has no parameter. Two cells are neighbours when their Voronoi regions share a wall, which is decided by the point set alone. That is the same property the site’s own placement rule was criticised for lacking when reach was a recency window rather than a falloff of distance: a construction with a knob in it states a hypothesis whether or not the author meant to.

There is a second reason and it is about what the measurement is for. The claim is that the six sides Euler forces are shared out among the families, and Euler’s relation is about a tessellation. Counting contacts by a distance rule would produce a number that has no reason to be six and could not be checked against the theorem at all.

The composition changes with the radius, as it must #

One number in the table above is not a property of the head. It is a property of where in the head it is measured.

Measured on the same rule at three hundred points rather than nine hundred, the contacts come out 21 (31%), 34 (25%), 13 (20%), 55 (11%), 8 (7%), 5 (3%) — the same kind of list, shifted one rung down the ladder. The counted pair there is 21 and 34.

That is the site’s oldest finding arriving in a new instrument. Every count is a statement about an annulus; the spiral families of a head are not fixed but change with radius, with computable transitions. The contact graph inherits it exactly, because the tessellation near the middle of a head is a tessellation of a coarser lattice.

So “the six are the spirals” is true band by band, and the shares are properties of a band. Taking a contact histogram over a whole head would mix two or three regimes and produce a list with six or seven families in it, none of them dominant — which is a fair description of the head and a poor description of anything in it.

The control #

The worry with any result on this subject is that Fibonacci numbers have been built in somewhere, and here the output is a list of them, which is exactly what a contaminated measurement would produce.

So run it on a head seeded at the Lucas angle. The contacts come out:

29 (32%), 47 (29%), 76 (20%), 18 (12%), 11 (5%), 7 (2%)

Those are Lucas numbers. Not one of them is a Fibonacci number, the counted pair on that head is 29 and 47, and the same code produced both tables.

Run it on a whorled head at 144°, which is two fifths of a turn exactly, and the contacts are 2, 3 and 5 in nearly equal thirds. Five straight rows, three families, and the arithmetic still holds: 2 + 3 = 5.

What the shares mean #

The proportions are not decoration and they answer a question the counted pair cannot.

A parastichy count reports two numbers as though they were equals. The contact graph says they are not: at this band 34 supplies about a third of all contacts and 55 about a quarter, so the 34 family is the more strongly connected one. That ordering is the same one the position counters use internally — they rank offsets by hop length and take the two shortest — and the contact shares are an independent measurement of the same ranking, from a construction that never computes a hop length.

The two families outside the counted pair are the interesting part. 21 supplies 17% of contacts and 89 supplies 15%, which is not a rounding error; between them they are a third of every wall in the tissue. A description of the head as “34 and 55 spirals” is a description of 58% of its contacts.

Which is not an argument against counting two #

It would be easy to read this as a criticism of the convention, and the essay two fields away establishes that it is the opposite.

The families a head has are closed under addition — every one but the two smallest is the sum of two others — so once two consecutive families are known the rest follow by arithmetic. The 21 and the 89 are not extra information about the divergence angle; 21 + 34 = 55 and 34 + 55 = 89, and a head with families 34 and 55 cannot have anything else.

So the two-number convention is sufficient, and this measurement says what the sufficiency is buying: two counts determine the whole contact network, including the proportions in which the six sides are shared out. That is considerably more than “two spiral counts” sounds like it should determine.

What it says about the tissue laws #

The field’s two standing relations about cellular tissue — Lewis’s law, relating a cell’s area to its side count, and the Aboav–Weaire relation, relating a cell’s side count to its neighbours’ — are both statements about the contact network with the composition averaged out.

This site has already measured both on phyllotactic and random tissue and found that they want opposite kinds of arrangement. What the contact labels add is a reason to expect that: on a golden-angle head the neighbours of a cell are drawn from four families with fixed shares, so “a cell’s neighbours” is a highly structured population, and on a Poisson set it is not a population at all. Two relations averaging over those two situations are averaging over different things.

That is a reading rather than a result, and it is offered as a direction rather than a finding. What would settle it is measuring Aboav’s relation within a family — asking how a cell’s side count relates to its 34-neighbours separately from its 89-neighbours — which the contact labels make possible and which this collection has not done.

The rim, again #

One methodological note, because this measurement is more sensitive to it than the side counts are.

A node near the boundary has its outward neighbours missing. That biases its side count downward, which the previous essay’s rim cut handles. It also biases the composition of its contacts, and it does so unevenly: the highest-order family is the one whose partners are furthest away in placement order, so the outermost ring loses its 89-contacts before it loses its 21-contacts.

Including the rim would therefore not merely add noise; it would systematically under-report the high families and make the head look as though it had fewer of them. The cut is at 86% of the radius, it is the same one the tessellation statistics use, and the shares above are computed on 665 interior nodes out of 900.

What a photograph does not give #

There is a practical obstacle between this measurement and a real seed head, and it is not the one a reader would guess.

Tessellating a photographed capitulum is routine. What the labelling needs, and a photograph does not contain, is the placement order — which primordium was made third and which was made three hundredth. An edge labelled 34 requires knowing that these two organs are thirty-four places apart in a sequence that finished months ago.

On a model head the order is free, because the head was built one point at a time. On a real one it has to be reconstructed, and the usual reconstruction is to trace the generative spiral outward from the centre — which means assuming a divergence angle, or at least a monotone ordering by radius, before the measurement begins.

That is not fatal and it is not free. Ordering by radius is exact for an ideal Vogel-model head and approximate for a real one, and the approximation fails exactly where the pattern is irregular, which is where the measurement would be most interesting. A study using this would have to report how the ordering was obtained and how many organs it was ambiguous for.

The honest comparison: a spiral count needs a photograph and a pencil. This needs a photograph, a tessellation, and a reconstructed ordering, and the third is the part that can go wrong quietly.

What is new here #

The parastichy numbers were already known from three counting instruments. What is new is that they are also the labels on a tessellation’s edges, in stated proportions, and that those proportions account for a theorem’s worth of sides.

That makes four routes to the same integers on this site: the hop lengths on a disc, the hop lengths on a cylinder, the autocorrelation of the divergence sequence, and now the contact graph of the cells. They take four different kinds of input and share no arithmetic.

The premise of this collection is that the spirals are counted rather than admired. Four independent counters agreeing is the strongest form that premise has taken, and the newest of them counts something that is not a spiral at all — it counts walls.

What would make it a claim about plants #

Everything above is a measurement of arrangements this site builds, and it is worth being explicit about what a specimen would have to show for any of it to be about biology.

The prediction is sharp and it is falsifiable. A real capitulum, tessellated and ordered, should give a contact histogram dominated by four offsets, those four should be consecutive members of a sequence closed under addition, and the two largest shares should be the pair a person counts by eye in the same band. The shares should sit near 31/27/17/15 rather than being spread over six or seven offsets.

A head that failed that would be interesting in a specific way. Spread-out shares would mean the tessellation is not of a single lattice — either the band spans a transition, or the pattern is genuinely irregular, and the two are separable by repeating the measurement on a narrower annulus. Offsets that are not closed under addition would mean the ordering was reconstructed wrongly, which is the failure mode the previous section is about.

That is a check with numbers in it, made on data a morphologist already collects, and it costs one tessellation. It is the cheapest prediction in this phase and the only one that does not need a stem measured to a quarter of a degree.

Why the shares are what they are #

The proportions can be read off the geometry, and doing so turns the table from a measurement into something with a reason.

A cell is joined to another when no third point lies inside the circle through them and a common neighbour — which, on a near-regular lattice, means the two are among the closest few. So the share a family takes is set by how short its hop is relative to the others, and the site’s existing hop-length curve is exactly that ranking.

At the band measured, the four shortest offsets are 34, 55, 21 and 89, in that order, and the four largest shares are 34, 55, 21, 89, in that order. The correspondence is exact, and it is the sense in which the contact graph is not an independent fact about the head: it is the hop-length ranking, expressed as a count of walls.

What it adds is the weights. A hop-length curve says which offsets are short and by how much; it does not say what fraction of the tissue’s walls each supplies, and the two are related by the tessellation rather than by anything simpler. That a family a quarter longer than the shortest still carries 27% of the walls, while one half again as long carries 15%, is the sort of number that has to be measured.