Packing, measured against the interior

A packing number is a distance or an area divided by a scale. In the measurements this corrects, the scale is the square root of a head’s mean cell area, so that a head of nine hundred organs and a head of a hundred and fifty can be put on one axis. The four criteria first measured on a spiral head — closest pair, largest gap, area evenness and six-sidedness — were all computed that way, and three of them depend on the scale.

The mean was taken over every Voronoi cell with a bounded polygon. That sounds like the interior of the head and it is not. It is the interior plus a ring of cells just inside the edge whose polygons reach out to circumcentres far beyond it, and on a small head those few cells outweigh all the rest.

A scale that belonged to the edge #

Counting every bounded cell, the mean area of a golden head is 38.828 at 150 organs, 3.607 at 200 and 22.956 at 600. Those are three heads built by the same rule a few rings apart, and their scales differ by a factor of ten.

Leaving out every cell whose polygon crosses the head’s own radius, the same mean is 3.1600 at 150 organs, 3.1481 at 600 and 3.1425 at 2,000 — between 3.1425 and 3.1634 at all ten sizes. That is π, which is exactly the area the square-root rule gives every organ: organ i sits at radius √i, so the disc out to organ n has area πn and each organ owns π of it.

So one of the two readings is a property of the packing, and the other is a property of where the last ring of organs happened to fall against the convex hull.

Twenty-one cells out of a hundred and thirty-six #

Of the 136 bounded cells on a 150-organ golden head, 21 cross the head’s own radius. The 115 inside it average 3.160 in area. The 21 average 234.154.

A cell at the edge is bounded in the technical sense — its organ is not on the convex hull, so the triangles round it close up — but the triangles it belongs to are slivers along the hull, and a sliver’s circumcentre lies far outside the head. The cell’s polygon is the hull of those circumcentres, and it reaches out to them.

The arithmetic closes. Twenty-one cells at 234.154 are 4,917.2 units of area; 115 cells at 3.160 are 363.4. Together they are 5,280.6 over 136 cells, a mean of 38.83 — the number the criteria were divided by, of which the interior contributes less than a fourteenth.

The defect a repair uncovered #

The scale was not always this far out. Until the triangulation was repaired, its starting super-triangle was too small, and it silently dropped exactly the hull triangles with far-flung circumcentres. Those are the triangles that make an edge cell enormous, so the undersized super-triangle had been shrinking the edge term and hiding it.

Completing the triangulation made every edge cell as large as it really is, and the published packing numbers moved with it — which is how the defect came to be found. The count of sides never noticed, because a cell’s number of sides does not care how far away its corners are.

What the cut is, and what it leaves #

The repaired reading keeps a cell only if every corner of its polygon lies within the head’s own radius. On a 900-organ head that keeps 810 of the 865 cells that are bounded, and at 2,000 organs it keeps 1,855 of 1,965.

It is a cut at the head, not at a number chosen to produce an answer. Its only parameter is the radius the organs themselves reach. And the reading now refuses any cut that leaves the mean cell more than half again as large as the median cell — a check the scale never had, and one the unclipped reading fails by a factor of twelve on the smallest head.

π is a prediction, not a fit #

The cut was not tuned until the scale came out at π. π is what the square-root rule says before any head is built: n organs inside radius √n share πn of area between them. Reading 3.1425 to 3.1634 at ten sizes is therefore an independent check that the cut keeps the interior and nothing else, and it could have failed at any of the ten.

The unclipped reading could not have passed it except by accident, because the area its extra cells add is set by how thin the hull’s triangles happen to be. The other check agrees. On the interior’s scale the mean cell is 1.0008 times the median cell at 150 organs and 0.9999 times it at 900; counting the rim’s cells it is 12.264 times the median at 150 and 1.192 times at 900. A mean that sits on its own median is describing the typical cell.

Area evenness, read both ways #

Area evenness is the spread of the cell areas over their mean. Counting the rim’s cells, the golden head at 900 organs reads 0.9107, twenty-third of 72 angles, and the window runs from 0.0069 to 19.36. A statistic whose neighbouring angles differ by a factor of a thousand is reading where each head’s last ring fell.

On the interior’s scale the golden head reads 0.0102, fourth of 72, and the whole window runs from 0.0074 to 0.1213. The defeat by ninety to one that the earlier reading reported is a factor of 1.4.

The cells a ray evens out #

The evenest readings in the window belong to exact rational angles: 135°, which is three eighths of a turn, at 0.0074, then 144° and 150°. An angle whose organs sit on rays puts each organ at the radial law’s own spacing along its ray, so every cell is a wedge of the same area, however large the empty wedges between the rays have grown.

The unevenest belong to the angles just beside a simple fraction. Half a degree past a third of a turn, 120.5° reads 0.121, more than five times 120° itself; 143.5° and 144.5° read 0.047 and 0.048 against 144°'s 0.009. Those heads have arms that are still shearing apart at nine hundred organs, and an organ caught in the shear owns more room than one on an arm.

So area evenness does rank angles, and what it ranks is nearness to a small denominator, with the exact fraction best and its neighbours worst. The disorder of the tissue dips at the same fractions, which is some evidence that the two statistics are one fact about rays, counted twice.

What area evenness measures instead #

Change the rule that places organs outwards and the reading moves with it, whatever the angle. Put organ i at radius i to the power p rather than at its square root. Then an organ’s area is proportional to i to the power 2p − 1, independent of the divergence, and over organs spread evenly from one to n the spread of that quantity over its mean is

|2p − 1| / √(4p − 1)

— which has no angle in it. At p = 0.4 it predicts 0.2582, and the golden angle, three eighths of a turn and five thirteenths read 0.2534, 0.2522 and 0.2526. At p = 0.6 it predicts 0.1690, and they read 0.1666, 0.1675 and 0.1677. The three never differ by more than 0.0028.

Area evenness, in other words, is an instrument for the radial law. It was never measuring the divergence, except where a near-rational angle’s shear adds a term on top of the law’s.

That exception is the condition under which the account would be withdrawn, and it is large where it applies. At p = ½, 120.5° reads 0.121 where the golden angle reads 0.0102, so an angle within a degree of a small denominator carries a term the radial law does not predict. Any claim that evenness reads the growth law alone has to exclude those angles at the head size it is read at, and a near-rational term found at an angle far from every small denominator would put the law itself in doubt.

What is left at the square root #

At p = ½ the formula predicts no spread at all, and what the heads read instead is about a percent: 0.0102 for the golden angle, 0.0074 and 0.0101 for the two rationals. On the golden head that residual falls with size, from 0.0256 at 150 organs to 0.0068 at 2,000.

Its cause is not derived here. The obvious candidate is the discreteness of the outer rings, which would be a finite-size term rather than a property of the angle, and a fall with size is what that would look like. It is recorded as unexplained rather than explained by the candidate.

Closest pair on the interior’s scale #

The closest pair is the shortest distance between any two organs, over the mean spacing. On the interior’s scale the golden head reads 0.9034, first of 72. The best angle on the half-degree grid is 137.5° at 0.8734, and the worst reads 0.0442.

That is the criterion most readers mean by “evenly spread”, and the earlier reading also put the golden angle first on it at 700 and 900 organs. What changes is that it is no longer alone.

A rational neighbour’s lead expires #

The grid’s best angle is a rational one, and its closest pair does not hold still. 137.5° is 55/144 of a turn. At 400 organs its closest pair is 0.9026, a ten-thousandth behind the golden angle’s 0.9027; at 700 organs it is 0.8748 and at 900 it is 0.8734, while the golden angle’s reads 0.9035 and 0.9034.

The rational angle is not getting worse at spacing in general. Its organs are beginning to line up on its 144 rays, and the first pairs to crowd are the ones that line up first. A rational angle can match the golden angle at one head size and trail it at the next, which is the lesson of the growing gap arriving through the other distance criterion.

Largest gap, read both ways #

The largest gap is the radius of the biggest empty circle among the organs, again over the spacing. Counting the rim’s cells, the smallest in the window belongs to 148° at 0.2497, and the golden head is twenty-fourth at 0.7725. A largest gap a third the size of the golden head’s own, credited to an angle that is 37/90 of a turn — a pattern of ninety rays — is a gap divided by cells that happened to be enormous.

On the interior’s scale the golden head reads 0.8435 and 137.5° reads 0.8434. They differ in the fourth decimal. 137.5° is 55/144 of a turn, a rational angle whose rays have not yet opened a hole larger than the one at its own centre at nine hundred organs.

Two winners, and not three #

Read at 400, 700 and 900 organs, the golden angle is first on closest pair every time on the interior’s scale, and first or level to the fourth decimal on largest gap. Area evenness ranks it fourth, fifth and fourth, behind exact rationals, and six-sidedness ranks it 45th, 40th and 31st, behind the same rationals.

Counting the rim’s cells the same table reads 2nd, 1st and 1st on closest pair and 14th, 26th and 24th on largest gap. The criteria about distance came apart only on the rim’s scale. The published “three criteria, three winners” was two criteria agreeing, read through an instrument that made them disagree.

Six-sidedness is the one criterion the scale never touched, because a count of sides is divided by nothing, and its published verdict stands exactly as it was: at 900 organs three eighths of a turn is 99.2 percent hexagons and the golden head 81.2. It sits with the second moment of the side counts as a reading of a tiling’s topology, and readings of topology have not once singled the golden angle out.

Both are decided in the first handful of organs #

On a 900-organ golden head the closest pair is organs 1 and 4, 0.9034 of a spacing apart at radius 1.363. The largest empty circle is the circumcircle of organs 2, 3 and 5, at radius 0.871. Both are fixed before the head has any lattice to speak of, which is why neither moves: the closest pair reads 0.901 to 0.904 and the largest gap 0.841 to 0.844 at every size from 150 organs to 2,000.

The centre of a Vogel head is also the part of the model least like a plant, so a whole-head packing number is a statement about the most artificial handful of organs in the whole construction.

At the centre, noble angles disagree #

The Lucas angle, 99.502°, is as badly approximable by fractions as the golden angle, and the same rule reaches it on its second branch. Its whole-head closest pair is 0.884 to 0.888, below the golden angle’s. Its whole-head largest gap is 0.824 to 0.828, which is better than the golden angle’s.

So between the two noble angles measured here, the two distance criteria pick opposite winners, and they do so at every size. That is the true form of the disagreement the earlier essay reported: packing criteria disagree about fine detail, and the fine detail is the first five organs.

Removing the centre a ring at a time #

Leave out every cell whose organ lies inside radius two, and the golden head’s closest pair rises from 0.9034 to 0.9347; inside three, to 0.9425; inside ten, to 0.9452. The three noble angles the measurement reads — golden, Lucas, and one more at 132.18° — read 0.9451, 0.9447 and 0.9451 once radius thirteen is removed.

The two near neighbours go the other way. 137.3° and 137.0° hold at 0.6201 and 0.5917, set at radius 13.5 and 14.2, where their twenty-first organs line up. They are irrational, and on a head of about two hundred organs they are badly spaced.

The constant they approach #

Away from its centre a head is locally a lattice. At a rise of t per organ, measured in circumferences, organ offset j is the vector whose sideways component is eⱼ — how far j turns of the divergence miss a whole number of turns — and whose upward component is j·t, and every cell has area t.

The squared length of that vector over the cell’s area is eⱼ²/t + j²·t, which is at least 2·j·|eⱼ|, with equality at t = |eⱼ|/j. The closest pair over a stretch of the head, on the interior’s scale, is therefore √(2 · min j|eⱼ|) over the offsets whose best rise falls inside that stretch. Checked against the six heads with the centre inside radius four removed, it agrees with every one of them to within one percent.

For every irrational angle, infinitely many offsets bring j|eⱼ| down to 1/√5 or below, and for a noble angle none brings it lower in the limit — that is Hurwitz’s theorem. So away from the centre every noble angle’s closest pair tends to √(2/√5) = 0.9457, and a head of 2,400 organs with its centre inside radius thirty removed reads 0.94565. Every other irrational is brought below 1/√8 infinitely often, so no angle that is not noble can hold a closest pair above √(2/√8) = 0.8409 far from the centre.

Packing and approximation are one measurement #

The claim that survives was stated as arithmetic — the golden angle resists rational approximation better than any other number — and packing was described in this collection as a loose version of it. The measurement says otherwise. The closest pair a noble head keeps away from its centre is the arithmetic, read off a set of points: twice Hurwitz’s constant, under a square root.

That turns the relation round. The packing claim is not a looser telling of the approximation claim. Stated as the closest pair away from the centre, it is the same claim with a head attached, and it singles out exactly what Hurwitz’s theorem singles out — the noble angles as a class, of which 137.508° is one.

What the verdict on packing now is #

“137.5° packs seeds most efficiently” is right about distance and wrong about cells. On closest pair and largest gap the golden angle is above every rational angle near it, and far from the centre it is level with every other noble angle. On area evenness and six-sidedness it loses to rational angles whose organs sit on rays.

The earlier verdict, that no criterion singles it out, is withdrawn. So is the conclusion that nothing measured on a finite head separates it from an irrational neighbour: the growing gap separated rational from irrational, and the closest pair separates the noble angles from 137.3° on a head of a couple of hundred organs.

What this does not establish #

It does not single out 137.508° among noble angles. The limit is shared, and on a finite head the differences between noble angles are set at the centre, by their first partial quotients.

That a finite head reaches the lattice’s value is measured, not derived, and the heads sit a few tenths of a percent below it at the radii read. One cut was used, at the head’s own radius, and nothing here shows how the rankings move if the cut is taken deeper inside. The measurement is on Vogel’s model, with equal points, the square-root law and no growth, and it says nothing about what a plant’s objective might be.

What would withdraw it #

A clipped reading of any head whose mean cell area runs more than half again past its median — that is the defect in one number, and the reading now refuses it. A noble angle whose closest pair, with the centre removed, does not approach 0.9457 on a larger head. An irrational angle that is not noble whose closest pair stays above 0.8409 far from the centre. Any one of the three would put the scale, the lattice argument or the theorem in question, and each is a single head to build.

The growth law, read off a head’s cells #

The radial-law result points at the next measurement, and it is not about the angle. If area evenness is |2p − 1|/√(4p − 1) whatever the divergence, then a measured spread of cell areas is a measured radial exponent. A spread of 0.10 means p = 0.455 or p = 0.555, and whether a head’s cells grow or shrink outwards says which of the two.

The test is a head whose placement law is known independently — organ radii measured against their index along a parastichy — with its cell areas measured on the same specimen. If the two exponents disagree by more than a near-rational shear can account for, area evenness is reading something other than the growth law, and this account of it is wrong.