Two thirds of a cell

The six sides Euler forces are the spiral families. That sentence is the reason this field exists on this site, and until now it has carried no number saying how much of a cell it accounts for.

It accounts for two thirds. Not approximately, not on average, and not with a tail of exceptions: the two families a count returns name two of every three walls in every band of a head and at every rise of a stem, and the third they miss is the same third everywhere.

That is a better result than the slogan and a worse one, in the two directions a measurement usually goes.

What the claim was, exactly #

The measurement that established it labelled every Delaunay edge of a nine-hundred-point head with the difference between the two nodes’ placement indices, and found the parastichy numbers looking back: 34 at 31.2 per cent of the walls, 55 at 26.1, 21 at 18.1, 89 at 13.2, then 13 and 8 with the remainder.

Those shares sum, over 5.697 walls a cell, to the six that Euler’s relation forces. So the composition of the six is the spiral families, and the claim in that form is exactly right.

What it does not say is which two of them a count returns, and that is where the two thirds comes from.

The arithmetic nobody did #

A count returns two families. A family runs both ways, so two families are four partners. A cell has six walls.

Four against six is two thirds, and the whole of this essay is that the arithmetic is not merely an upper bound — it is what actually happens, to three significant figures, in places where nothing forces it to.

The measurement #

Two neighbour sets on the same points, per cell: the contact set the counting instrument returns at a stated cut, and the adjacency set the Voronoi partition gives. The disagreement is reported as the symmetric difference as a fraction of the union, which is a number nobody can game by making one set bigger.

At the cut a count actually makes — two whole families — the disagreement is 33.80 per cent of the union on the settled nine-hundred-point head.

It is not two thirds by construction. Two families and six walls could overlap in any amount from nothing to four, and a set of four could be a subset of a set of six with a disagreement of exactly a third only if all four are walls. They are.

Which side the error is on #

The 1,230 disputed edges are almost all of one kind. 1,227 of them are walls that no counted family names; the handful left over are counted contacts that share no wall.

That asymmetry is the result. The counting instrument is not seeing things that are not there — it is not seeing things that are. Every one of its four partners is a genuine wall and it stops.

So the error is a shortfall, not a mistake, and the shortfall has a name: it is the third-ranked contact family, which the instrument measured, ranked and discarded.

Cutting one family more, and one fewer #

The whole curve of disagreement against how much of the ranking is kept says the same thing more sharply.

One family leaves 66.76 per cent in dispute. Two leave 33.80. Three leave 7.57. Four leave 24.69, five 39.45, six 49.49.

The minimum is at three and it is not at the cut a count makes. Keeping more is not simply better either — past three the contact set overshoots the tessellation and the disagreement climbs again, which is what makes three a genuine minimum rather than the end of a slope.

The same curve in partners #

Stated in nearest partners rather than whole families the numbers land differently and say the same thing. One partner leaves 83.31 per cent, four leave 33.22, six leave 4.24, eight leave 25.12, ten leave 40.10.

Six partners is the least, which is the mean side count arriving from the other side. But a count does not return six partners; it returns two families, and a family is two partners, so the cut a count makes is four.

The two cuts are not interchangeable and the difference between them is worth a sentence. Keeping the four nearest partners is what a reader imagines a count does. Keeping two whole families is what it does, and on a cell whose two families are unevenly represented the two sets differ.

Band by band, it does not move #

A third on average could be a scatter — a half here, a fifth there. Read band by band from the centre of the head outwards, the two-family disagreement runs 48.7 per cent, then 32.7, 33.3, 33.7, 33.1 and 33.3.

Five bands within a point of each other, out of the middle and out to the rim cut. The inner band is the only one out of line, and it is the band where there is not yet a lattice to count on.

Read on a finer banding of tenths of the radius the values scatter more widely, from 29.9 per cent to 36.9, because each band then holds fewer cells. The two readings say the same thing at different resolutions, and neither has a band anywhere near a half or a fifth.

The one band that is not a third #

The innermost band is at 48.7 per cent and it is worth saying why rather than excluding it.

Inside about a seventh of the radius there are a few dozen organs, the √i radius law has not yet spread them into anything with a rung, and a cell’s own shortest lag is a poor unit because the cell next to it has a different one. The counting instrument still returns two families there; they are simply not the families the partition uses.

That band also holds the two cells with no separating threshold of their own, at 4.7 and 8.2 per cent of the radius. Both defects live in the same place, and the rim cut at 86 per cent does nothing about it because the trouble is at the other end.

And it does not move with size #

Five golden heads, from 300 points to 1,500: 34.83, 34.05, 33.80, 33.88 and 33.61 per cent. The leading wall family changes between 1,200 points and 1,500 — from 34 to 55, as the counts climb with radius — and the shortfall does not notice.

Nor does disorder shift it. Displacing the head by a sixth of a spacing takes the two-family disagreement from 33.80 per cent to 36.31, while the length-threshold disagreement over the same range goes from 1.80 per cent to 40.98.

That contrast is the argument that this number is structural. A quantity set by a tolerance moves twentyfold under noise; a quantity set by counting four things where there are six moves by two and a half points.

A band of walls, drawn #

The per-cell picture and the per-head number can be checked against each other by drawing every wall in a band and colouring each by whether the cell’s own counted pair names it.

The warm ones are the missing third and they are not clustered. They run through the band in the same directions as the named ones, at every radius in it, which is what the tilt statistics say: at the two-family cut the disputed walls average 44.6 degrees from the local radial direction and the agreed ones 44.5.

Direction carries nothing #

That one-tenth of a degree is worth stopping on, because it closes an explanation that would otherwise be the obvious one.

If the missing third were the walls running in some particular direction — the ones most nearly radial, say, or the ones most nearly circumferential — then the shortfall would be a geometric fact about which walls a hop-length ranking can reach, and it would have a picture. It has no picture. The disputed and agreed populations differ in tilt by 0.1 degrees, which is nothing at all.

They differ in length: 1.322 shortest lags against 1.074. So the missing walls are the longer ones, and length is exactly the quantity the contact ranking sorts by. The instrument is doing precisely what it says it does, and what it says it does leaves a third out.

Which families go missing #

By lag, the 1,230 disputed edges are led by 89, with 385 of them, then 21 with 292, 34 with 212, 55 with 141, 13 with 118, and the tail in 8, 5 and 3.

That list is the head’s own family sequence with the counted pair partly removed, and it is closed under addition in the same way the full list is. The largest single contributor is the sum of the counted pair, 34 + 55 = 89, which is exactly the family a count is guaranteed to have and to discard.

So the missing third is not an exotic set. It is the family immediately above the counted pair and the family immediately below it, which is to say the rest of the ladder either side of where the count happened to be read.

On a stem, exactly a third #

The head is a sequence of lattices, so a third holding across it is already a strong statement. The stem is one lattice at a time, and there the number is exact.

Over 111 rises where the tessellation gives every node six walls, the two-family disagreement is exactly 33.333 per cent at every single one, and the three-family disagreement is exactly zero at every single one.

Not 33.3 to a rounding. The same value at 111 different rises, spanning the ladder from a rise of 0.211 down to 0.004 — three rungs of the counted pair and four of anything else.

Which is where the missing third gets its name #

An exact zero at three families over 111 rises is a positive result, and it is the one that turns a third is missing into the third family is missing.

On a stem the contact instrument, allowed to keep three families rather than two, returns precisely the adjacency graph. Nothing left over on either side. The two relations that cannot be reconciled by any threshold on a head are the same relation on a stem, provided the cut is made one family later.

That is the whole account of the two thirds. Six walls, three families, two counted.

The regular lattice says it too #

The sheared triangular control is the case where every quantity is available in closed form, and it agrees.

Its six neighbours are the lags ±1, ±18 and ±17 — three families, six partners, every wall at exactly 1.000000000 shortest lags and every non-wall at or beyond √3. All 320 interior cells have their walls separated from their non-walls, and at the midpoint of that interval the symmetric difference is exactly zero over 1,176 agreed walls.

Three families and six walls, on the tidiest object available. A count on that lattice would return two of the three and would name four of the six.

The two regimes where it is not a third #

The stem has two other regimes and both are informative, because in neither is the answer a third.

At rises above 0.50 a node has two walls rather than six — the pattern is a thread, and the counted pair is not in the tessellation at all, at any of the six rises measured. The two-family disagreement there is 75 per cent.

Between 0.219 and 0.483 a node has four walls: two families, a ribbon. There the two-family disagreement is 0.00 per cent and the three-family disagreement is 33.333. The arithmetic has simply shifted down one, which is the cleanest possible demonstration that the third is a count of families and not a constant of nature.

What a cell with six walls actually holds #

Put the shares back together and the composition of a cell is readable directly. Of 5.697 walls, about 1.9 come from the 34 family, 1.6 from the 55, 1.0 from the 21 and 0.9 from the 89.

The counted pair supplies about 3.5 of them and the uncounted families about 2.2, which is the two thirds in the form a reader can picture: two walls of every three, distributed unevenly between the two counted families rather than two apiece.

That unevenness is why the four-partner cut and the two-family cut differ. A cell whose 34 family supplies two walls and whose 55 family supplies one is a cell where taking four nearest partners and taking two whole families are different operations.

A cut in length does better and is not available #

At the best length threshold anybody could pick, the disagreement on the same head is 1.80 per cent rather than 33.80 — a factor of nineteen better than the cut a count makes.

That is not a recommendation. The threshold that achieves it does not exist as a single value on a head: it is the value that minimises the disagreement rather than one that eliminates it, and it moves with how untidy the specimen is. The comparison is here only to locate the two thirds on a scale.

What it establishes is that the shortfall is a property of counting families rather than of the contact relation. The contact relation, cut by length, gets within two per cent. Cut by the rule a parastichy count uses, it gets within a third.

What this does not say about counting #

It does not say a count is insufficient. Two consecutive families generate all the others by addition, so a reader given the counted pair can write down the missing third without measuring anything.

That is the sense in which the original claim survives intact: two counts do determine the whole contact network, including the proportions in which the six sides are shared. What they do not do is name the network, and the slogan was about naming.

Nor does any of it touch the counting instrument’s own record. The blind disc counter agrees with the cylinder ladder’s prediction of the counted pair in seven of seven bands of this head, which is the strongest agreement between those two routes anybody has recorded.

What it does say about the slogan #

The six sides Euler forces are the spiral families is true. The six sides Euler forces are the counted spiral pair is two thirds true, and the third that is missing is the family above the pair and the family below it.

The distinction matters wherever a wall is used as evidence about a count. A statement of the form these two cells are neighbours, so they are 34 apart is right two times in three, and wrong in a way that has no signature — the missing walls have the same tilt, sit at the same radius, and are only 23 per cent longer.

There is no way to tell, from one wall, which third it is in.

What would refute it #

A head, or a stem, on which the two-family disagreement is not near a third. The reading is one triangulation and one lag scan and it has been made on five golden heads, a Lucas control, seven bands of one head and 140 rises of a stem, and the only values away from a third are the ones where the node count per cell is not six.

The sharper test is the exact one: any rise in the surface regime where the three-family disagreement is not exactly zero. That is 111 opportunities to fail and it has taken none of them, which is what makes the third family a claim rather than a description.

What is left over #

Two things this measurement raises and does not settle.

The three families the graph carries on a stem never differ in share by more than 1.80 percentage points, so reading the two largest is a tie-break rather than a ranking — and the tie-break disagrees with the hop lengths at half the rises. A third that is always missing is one problem; not knowing which two are present is another.

And nothing here reaches a specimen. Labelling a wall by an index difference needs a placement order that a photograph does not carry, which is the standing obstacle between this instrument and a plant. The two thirds is a fact about arrangements this collection builds, and it is stated in a form a morphologist with an ordering could check.