An interior that is nearly neutral

The mean number of sides in a tessellated head is six because Euler’s formula forces it, and a forced mean separates nothing: it is six on a golden head, six on a whorled one and six on a set of random points. The site has already gone one step further into that distribution and found the spread does separate them, and one step sideways and found which neighbours a cell has are its spiral families.

Nobody here has yet asked the plainest question about the same object. The mean is six, so most cells have six sides. What are the ones that do not have six sides doing?

There is a second quantity Euler fixes, and it is the one that makes that question answerable. Give every cell a charge of 6 − n, where n is how many cells it touches, and add the charges up. The total is decided by the patch’s boundary and carries nothing whatever about what is inside it.

The charge of a cell #

A hexagon has charge zero. A five-sided cell carries +1 and a seven-sided cell −1, and a four-sided cell twice what a five carries. The quantity is standard in crystallography under the name topological charge, and it is signed, which is the whole of its usefulness: two exceptions of opposite kind sitting side by side contribute nothing to it while contributing two cells to any count of exceptions.

Sides are counted here as Delaunay neighbours, which is the number of Voronoi vertices around the cell by the other route. The two agree cell by cell, and the essay that labelled every contact by the difference between two placement indices reads the same graph.

What Euler fixes and what it leaves #

For a tessellated patch that is a topological disc, with B cells on its own boundary,

$$\sum_i (6 - n_i) = 6 + 2B$$

The derivation is four lines. Euler gives V − E + T = 1 for a disc; a three-valent dual gives 3T = 2E − B; eliminating T gives E = 3V − B − 3; and the charge sum, which is 6V − 2E, comes out at 2B + 6.

The right-hand side contains one number, and that number is a property of the rim. It says nothing about how many cells inside are exceptions, nothing about what kind they are, and nothing about where they sit. Those three are left free, and everything below is about what a head does with the freedom.

Confirmed rather than assumed #

An identity that follows from Euler’s formula does not need checking, in the sense that the arithmetic is not in doubt. What needs checking is whether the neighbour graph this site builds is the object the identity is about.

So it is measured rather than assumed, on thirty patches: six arrangements at one size, the golden head at six sizes, at six rim cuts, and at twelve amounts of displacement. Each patch’s charge is summed over every cell from the adjacency lists, and 6 + 2B is computed from nothing but a count of boundary cells.

Thirty patches and one exact arithmetic #

They agree on all thirty, as integers, with no tolerance anywhere. The golden head of 2,400 organs cut at 0.86 of its radius comes out at 294, which is 6 + 2·144. The Lucas head at 404 = 6 + 2·199. The whorled head at 16 = 6 + 2·5, which is a boundary of five cells because a whorled arrangement stacks its organs on rays and the hull of a set of rays is nearly a polygon. The Poisson set at 318 = 6 + 2·156.

And V − E + T = 1 on all thirty, which is the check that each patch is a disc rather than something with a hole in it.

Why the range matters #

Thirty agreements at one value would be one agreement repeated. The forced total runs from 16 to 404 across these patches, so the identity is tested at many different numbers, and a graph that satisfied it by construction rather than by being right would have to satisfy it at all of them.

That is the same design the site uses when it asks whether a counter reads a lattice or a folklore: give it inputs whose answers differ, and see whether the answers track. A check that can only come out one way is not a check.

Why this is an instrument and not a result #

A neighbour graph that has lost one wall misses the identity by exactly two. That is the sensitivity, and it is why this arithmetic is the first thing run rather than the finding: any statement below about how many exceptions a head holds is a statement about the adjacency lists, and the adjacency lists have to be shown to close up first.

The refutation is built in. Removing a single edge from the adjacency of a 900-organ head makes the identity throw, which is what says the check has teeth rather than being satisfied by anything that parses.

The interior of a golden head #

With the instrument checked, the question can be put to the interior — the cells strictly inside the patch, none of which touches the cut.

A golden head of 2,400 organs, rim cut at 0.86, has 1,631 interior cells. Of those, 135 have five sides and 129 have seven. Nothing else. The remaining 1,367 are hexagons, which is 83.8 per cent of the interior, and the number worth carrying is not that share but the one beside it.

Two hundred and sixty-four exceptions, charge six #

Two hundred and sixty-four cells are not hexagons, and the charge they carry between them is +6.

That is the finding in the title. The interior is not clean and it is not neutral by being empty: it is stuffed with exceptions which very nearly cancel. Since every exception is a five or a seven, the interior charge here is just the excess of fives over sevens, and the excess is six out of two hundred and sixty-four.

Euler does not require this. The theorem fixes the total over the whole patch at 294 and says nothing at all about how that 294 is distributed between the boundary cells and the rest.

Every exception is a five or a seven #

Not one four-sided cell and not one eight-sided cell appears in the interior of the golden head, the Lucas head, the head at a rational 137.5°, or the head at 137.0°. The exception set is exactly {5, 7} on all four.

That is a stronger statement than a low variance, and it is the sort of statement a summary throws away. A distribution reported as mean six, some spread is consistent with a tissue holding fours and eights in balanced numbers; this one holds none, and the reason a second moment was the right next statistic is that it is sensitive to exactly this and still does not say it.

Where the charge actually is #

If the interior of this head carries +6 and the whole patch carries +294, then the 144 cells on the boundary carry +288 between them. The charge lives on the rim.

Which is what the identity said would happen, read the other way round. A disc has to close up, the closing is done at the edge, and the interior is left to do whatever the arrangement makes it do. A reader who takes “total topological charge” as a property of a tissue is taking a property of where somebody drew the cut.

What a set with no rule in it does #

The Poisson control is the same disc filled with 2,400 points at random. It has 1,664 interior cells and 1,214 of them are not hexagons — 73 per cent, against 16 per cent on the golden head at the same cut.

Its interior charge is +45. That is the one arrangement measured here that leaves a real residue, and it is worth noticing that +45 out of 1,214 exceptions is still a small number: a random tiling is nearly neutral too, just much less tidily.

Degrees from three to eleven #

The Poisson set holds 405 cells outside {5, 6, 7}, with degrees running from three to eleven. The golden head holds none, at any size measured.

So the two arrangements differ in two ways that a mean cannot see and a variance conflates: how many exceptions there are, and what kinds are available. The ordered head has a great many exceptions of exactly two kinds; the random set has more exceptions of nine kinds. This is the same shape as the two tissue laws that want opposite material — the ordered tiling and the random one are not the same object with different amounts of noise on it.

The head with nothing to balance #

The whorled head at 144° is the opposite extreme. Its interior holds 1,770 cells and four exceptions: two fives, no sevens, and two cells outside {5, 6, 7}. Its interior is 99.8 per cent hexagons and its interior charge is +6.

That is a head in which the question has almost no content. There is nothing to cancel, so cancellation says nothing, and any statement about how exceptions balance is being made about four cells. It is included because a control that produces almost none of the thing being measured is the right control for a claim that the thing is balanced.

Three more angles, and one of them is rational #

The Lucas head at 99.502° gives 1,576 interior cells, 113 fives, 117 sevens and an interior charge of −4. The head at 137.0° gives 1,641 cells, 112 fives, 106 sevens and +6.

And the head at a rational 137.5° — eight thousandths of a degree from the golden angle — gives 1,631 interior cells, 135 fives, 129 sevens and +6. Not a similar table. The identical one, cell for cell.

It does not single out the golden angle #

That identity is the finding, not a footnote to it. A criterion on which the golden angle and a nearby rational return the same numbers is a criterion that does not distinguish them, and this is now the eighth such criterion the site has put to the angle.

The others are on record: packing measured four ways gives three winners and none of them the golden angle, the disorder sweep is a staircase whose tread runs from 137.47° to 137.54°, and the most badly approximable angle is not the most disordered. The one claim that has survived is the growing gap, which is asymptotic and about rationals rather than about this angle.

With head size #

The same head at six sizes, cut the same way. At 300 organs the interior holds 167 cells with 46 fives and 40 sevens; at 900, 576 cells with 80 and 74; at 1,500, 965 cells with 135 and 129; at 2,400, 1,631 cells with 135 and 129; at 4,000, 2,725 cells with 224 and 218.

Five of the six come out at an interior charge of exactly +6, which is not forced and is therefore worth stating as a measurement rather than as a rule.

The head that comes out at minus eighteen #

The sixth is 600 organs: 355 interior cells, 56 fives and 74 sevens, interior charge −18. Eighteen more sevens than fives, in a head where every other size measured runs a surplus of six the other way.

Nothing here explains it and it is reported rather than explained. What it does is settle a question the other five rows would have left open: whether +6 is arithmetic or coincidence. It is coincidence — or rather, it is a property of which exceptions a particular head size happens to have room for, and one head in six has room for a different answer.

Every extra cell is a hexagon #

The most informative pair of rows is 1,500 and 2,400. The larger head holds 69 per cent more interior cells and holds the same 135 fives and the same 129 sevens.

Every one of the 666 extra cells is a hexagon. Growing the head between those two sizes added no exceptions at all, which means the exceptions are not distributed through the tissue in proportion to its area — they are somewhere specific, and there is a stretch of head between the two sizes with none of them in it.

Going further out, 4,000 organs adds exactly 89 fives and 89 sevens to the 2,400 figures. Equal numbers, and 89 of each. That the count is a Fibonacci number is the first thing here to point at the ladder that runs up a head, and where those 178 cells sit is the argument the next essay makes.

What a defect fraction is a statement about #

Cut the same golden head at 0.70 of its radius rather than 0.86 and the interior holds 1,032 cells instead of 1,631. The exception counts do not move: 135 and 129 at every cut from 0.70 to 0.96, with the same charge of +6.

The share moves by a factor of two, from 25.6 per cent down to 12.8 per cent. So a published defect fraction is a statement about where somebody put the rim, and a defect count is a statement about the head. It is the same trap the rim set for the contact-family shares, arriving on a different statistic.

The cut at the head’s own edge #

At a rim of 1.00 the numbers change completely: 2,366 interior cells, 224 fives and 130 sevens, interior charge +96. Eighty-eight fives arrive with no sevens beside them, where at every cut inside the hull the pairing is total.

That row is not a measurement of tissue, and it had two causes rather than one. A cell near the head’s boundary has its outward neighbours missing and so loses sides it would have had — which is the cut, and is unavoidable. And the triangulation this site builds was incomplete there — which was a bug, and is not.

A defect in the neighbour finder, named and repaired #

delaunay() built its Bowyer–Watson super-triangle at four times the point set’s own radius, which is too small. An enclosing triangle’s vertices take part in every circumcircle test, so one sitting close to the data is a real competitor in it, and a triangle against the convex hull could be rejected in favour of a super-vertex that a distant one would never have beaten. The finished triangulation was then short 13 triangles on a 600-organ head and 21 on a 2,400-organ one, all of them against the hull, and the symptom was that the triangle count failed its own relation to the point count: thirteen points strictly inside the hull were left sitting on the triangulation’s boundary, and the cells just inside them lost sides.

Nothing measured above was ever touched, because every statistic here cuts the rim at 0.86 and the damage was beyond 0.95. At a factor of one hundred the triangulation is complete on every head and every random disc tested — the count is exactly the 2n − 2 − h Euler’s formula requires, and no interior point is stranded on the boundary — which is the setting the site now builds at, and the 21 cells it recovers on this head are the difference between the 2,345 that row used to report and the 2,366 above.

Not one of the twenty-one is a five or a seven, which is why those counts and the charge are the numbers they were: the missing triangles cost the hull’s neighbours sides without moving anybody across the boundary between six and not-six. So the repair removes a cause and does not rescue the row. At the head’s own edge the cut alone still strands eighty-eight fives, and saying which of the two causes was a fact about tissue and which was a fact about a super-triangle is part of the result rather than an aside.

What the instrument refuses #

Four refusals, each provoked on purpose. A neighbour graph with one edge removed makes the charge identity throw. A patch that is not a topological disc, so that V − E + T ≠ 1, throws. A boundary pinched at a vertex, where the count of boundary edges and the count of boundary cells disagree, throws. And an arrangement the file does not build — asking it for a “hexagonal” divergence angle — is refused by name rather than answered from the nearest thing it has.

An assertion that has never rejected anything proves nothing, which is why each of these is driven rather than described.

The rhombus and the disc #

The fifth control is the most instructive. A 24 × 24 rhombic patch of the perfect triangular lattice has 484 interior cells and not one five and not one seven. Every measurement of how exceptions balance returns nothing at all on it, rather than returning a share of an empty set.

Cut the same lattice into a disc instead and eight pentagons appear one ring in. Nothing about the lattice changed; the cut did. That is the clearest statement available of what this whole essay is about — the charge is a boundary quantity, and the shape of the boundary decides how much of it there is and where it has to sit.

What this rules out #

It rules out reading the six-sided mean as the whole topological story. Euler fixes the mean and fixes the total charge exactly at 6 + 2B, and fixes neither how many cells are exceptions nor what kind they are nor where they are. On the golden head the number is 264 and it is not a number the theorem knows.

It rules out reading a defect fraction as a property of a head, since the same head reports 25.6 per cent or 12.8 per cent according to a cut while its counts do not move by one cell.

And it rules out treating “an ordered tissue has few defects” as the difference between an ordered head and a random one. The golden head has 264 exceptions in 1,631 cells. That is not few. What it is, is balanced.

What it does not support #

It does not support any claim about the golden angle, for the reason given: a rational angle eight thousandths of a degree away returns an identical table.

It does not support a claim about real tissue. Every number here is measured on Vogel’s model, where organ i sits at i times the divergence angle and a radius proportional to the square root of i, and what that model has and has not established is a standing caveat on the whole field. A photographed capitulum has scatter, and how much of this survives scatter is a separate measurement.

And it does not support the reading that the interior must be nearly neutral. Five sizes out of six came out at +6 and the sixth came out at −18, which is what a measurement looks like when the quantity is free.

What the counts are for #

The count of exceptions and their kinds are now known: 264, all fives and sevens, balanced to six, unchanged across a wide band of rim cuts, and unchanged across a size step that adds 666 cells.

Every one of those facts points the same way and none of them says it. A tissue whose exception count is flat against area, whose exceptions arrive 89 at a time in equal numbers of both signs, and which has a stretch of radius holding none, is a tissue in which the exceptions are somewhere rather than everywhere. Where they are is a question about position that a table of counts cannot answer, and it has an answer.