Two rankings, one list

Two instruments on this site produce an ordered list of the same integers from the same points. One ranks index offsets by how short a hop they make; the other ranks them by how many cell walls they carry. The essay that labelled a tessellation’s edges by index offset reported that the two lists come out the same, in the same order, and called the correspondence exact.

It is not exact. It is not even close enough to be a rounding.

The claim, as it was made #

The sentence reads that 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. Two rankings, one list.

It is a good claim to have made, because it says something that could be false and says it in a form a second measurement can check. The second measurement has now been made on the same nine-hundred-point head, with the same rim cut, from the same placement rule, and the two halves of the sentence do not agree with each other.

The measurement #

The head is nine hundred points at the golden angle, cut at 86 per cent of its radius, leaving 608 interior cells and 1,732 interior walls. Every wall carries the difference between the two nodes’ placement indices, and every index offset carries a median hop length, divided by the local spacing so that a number near 1 means a hop about as long as the distance to the nearest point.

Both lists come out of that one reading. Nothing is fitted and nothing is compared against a model; the two orderings are two sorts of two columns of the same table.

The rim cut matters more to the hop column than to the wall column, and it is the same cut in both. A cell near the boundary loses its outward partners, which shortens nothing but removes the longest family first; leaving those cells in would bias the hop medians downward for the high offsets and the wall shares downward for the same offsets, in different amounts. Cutting once, before either column is computed, is what makes the two sorts comparable at all.

The two lists #

By hop length, shortest first: 55 at 1.037 local spacings, 34 at 1.130, 89 at 1.414, 21 at 1.527.

By wall count, most first: 34 at 31.2 per cent of the walls, 55 at 26.1, 21 at 18.1, 89 at 13.2.

The permutation is two transpositions #

The published list is the measured hop list with both of its adjacent pairs swapped. Take 55, 34, 89, 21, exchange the first two and exchange the last two, and the result is 34, 55, 21, 89 — which is precisely the wall list.

That is a strong hint about how the sentence came to be written, and it is worth taking slowly, because a permutation of four things is not evidence of anything on its own.

What is right in it #

The wall half is right, exactly, to the digit. The share ranking on this head is 34, then 55, then 21, then 89, and the shares are 31.2, 26.1, 18.1 and 13.2 per cent. The published table reported 31, 27, 17 and 15 on a slightly different interior set and the ordering it gives is the ordering measured here.

So the essay’s central finding survives untouched. A cell’s walls really are carried by a short list of index families in stated proportions, the list really is the parastichy numbers, and the six sides Euler’s relation forces really are shared out among them.

What is wrong in it #

The hop half. The shortest hop on this head belongs to 55, not to 34, and the third shortest belongs to 89, not to 21.

Neither gap is small enough to be noise. The 34 family’s median hop is 9.0 per cent longer than the 55 family’s, and the 21 family’s is 8.0 per cent longer than the 89 family’s. A measurement whose two candidate orderings differ by 9 per cent of the quantity being ordered is a measurement that has decided.

The family with the most walls is not the family with the shortest hop #

That is the whole of the correction, stated as a sentence about the tissue rather than about a claim.

The 34 family carries a third of the walls in the interior and makes the second shortest hop. The 89 family makes the third shortest hop and carries the fewest walls of the four. The map from hop length to wall share is monotone nowhere: it is not merely noisy, it reverses twice.

The top two survive, which is why nothing noticed #

Both instruments name 34 and 55 as their two largest on the whole head. A reader checking the claim against the number that matters — which pair a person would count in that band — would have found the two lists agreeing, and would have stopped.

That agreement is real and it is also the narrowest possible form of the claim. Two rankings that agree on the identity of a set of two, and disagree on the order within it and on everything below it, are not one list.

Below fourth place they do not even hold the same families. The hop ranking continues 110 at 2.048 spacings and 144 at 2.152; the wall ranking continues 13 at 6.9 per cent and 8 at 2.6. So 110 and 144 are short enough to appear on one list and carry too few walls to appear on the other, while 8 and 13 carry walls in the inner bands and have long hops when the median is taken across the whole interior. The agreement on membership stops at four, and it stops there because the hop median mixes bands that the wall share also mixes, in opposite directions.

Band by band they do not survive #

The head is read in six bands between 0.2 and 0.8 of the radius. One of the six orders the four families alike. Four of the six name the same leading pair, which leaves two that do not.

The band that is a whole rung out #

Between 0.4 and 0.5 of the radius the hop lengths rank 34, 21, 55, 89 and the wall counts rank 55, 34, 21. Reading the two largest off each gives 34 and 21 against 55 and 34.

Those are not two orderings of one pair. They are two consecutive rungs of the ladder — the pair one rung down and the pair one rung up — so an observer who trusted the wall shares in that band and an observer who trusted the hop lengths would report counts a whole transition apart from the same annulus of the same head.

The band that is not a rung at all #

Between 0.5 and 0.6 the hop lengths give 34, 55, 21, 89 and the wall counts give 21, 55, 34. The two largest by hop are 34 and 55; the two largest by wall are 21 and 55.

That pair is not a rung of anything. 21 and 55 are two apart in the sequence, with 34 between them, so a wall count read as a parastichy count in that band would return a pair no additive family can generate — and the check that would catch it is arithmetic rather than measurement.

The one band that agrees #

Between 0.6 and 0.7 both instruments give 55, 34, 89, 21, in that order, on all four. It is the only band where the correspondence the published sentence claims actually holds.

Why the middle of the head is the worst place to check #

The two bands that disagree about the pair are adjacent, and they sit either side of a transition. That is not a coincidence and it is the useful part of the negative result: where the head is between two rungs, the wall shares of three families are close together and the hop lengths of the same three are close together, and two close orderings decided independently have no reason to come out alike.

An instrument comparison run only in the middle of a head is therefore run in the one place the two instruments are least determined. An instrument comparison run only on the whole head averages that away.

How a sorted list becomes a ranking #

The likeliest account of the original sentence is not a wrong measurement. It is a list that arrived in a fixed order and was read as though the order meant something.

A counter that finds several short offsets has to return them somehow, and returning them in numerical order is the obvious thing to do — it makes two results comparable and it makes a change visible at a glance. A list sorted for storage looks exactly like a list sorted by size, carries no marker saying which it is, and can be read as a ranking for free. That reading is invisible afterwards, because the numbers are all correct.

Which is a general hazard and not an accident here #

This collection has taken an ordering off an instrument before and found it belonged to the instrument. The shape recurs because an ordering is the cheapest thing a table appears to offer and the most expensive thing to check: verifying that a list is sorted by size means recomputing the sizes, which is the whole measurement again.

The defence is a habit rather than a gate. A claim of the form these two rankings are the same ranking has to name where each ranking came from and give the values it was sorted on, so that a reader can see two columns of numbers rather than two rows of integers. The lists above are printed with their hop lengths and their shares attached for exactly that reason.

The head size moves the list as well #

The wall ranking is not fixed even for one instrument. Its leading family is 34 on a head of 1,200 points and 55 on a head of 1,500, and its third and fourth members change places between 900 points and 1,200 — 34, 55, 21, 89 becomes 34, 55, 89, 21 with no change to the rule that built the head.

So neither list is a property of the pattern #

Both are properties of a pattern and a window. That is the site’s oldest finding about counting arriving in a second instrument: a count is a statement about an annulus, and a wall share is a statement about the same annulus by a construction that never computes a hop.

A correspondence between two such lists therefore has to be claimed at a stated radius, on a stated head, or not claimed at all. The published sentence names a band in its surrounding prose and its number is a whole-head number, which is the third distinct way the claim is looser than it reads.

What the correspondence would have bought #

Had it held, it would have been worth a good deal. It would have meant the tessellation adds no information to the hop lengths — that the contact graph is the hop-length ranking expressed as a count of walls, and that a wall share could be predicted from a hop length by a monotone function nobody would need to measure.

That is the reading the published essay draws, in the sentence immediately after the claim, and it is the one part of that essay this correction removes.

What the two lists do buy, once they are two #

An independent measurement rather than a restatement. The wall shares are now evidence about the head that the hop lengths do not contain, because the two disagree in a way that is not noise and is not the same in every band.

The disagreement is also the reason the two definitions of neighbour on this site are two definitions. No cut-off on distance makes them one relation, and the ordering they induce on the families is a second place where the difference shows, narrower than the cut-off question and much cheaper to check.

The shortfall does not depend on which list is right #

Neither ranking rescues the two-number convention as a description of the walls. Whichever two families are named, the pair accounts for about two thirds of a cell’s walls and the remaining third belongs to families the count discarded — the same third in every band of the head.

Three families, not two #

Allowing the contact cut a third whole family takes the disagreement on the nine-hundred-point head from 33.80 per cent to 7.57 per cent, and a fourth takes it back up to 24.69. The minimum is inside the range rather than at its end, so keeping more is not simply better, and the number of families that best reproduces the tessellation is three.

The shape is the same on every head the comparison was measured on, which is what makes three a property of the tessellation rather than of one size. What changes with the head is how deep the minimum goes, and it never reaches zero on a disc.

On a stem the tie is exact #

The ordering question has an extreme form on a cylinder, where the counted pair is the same the whole way up and nothing can be blamed on the band. Across 111 rises the contact graph carries three families whose shares never differ by more than 1.80 percentage points, and reading its two largest names a different pair from the hop lengths at 55 of those 111 rises.

Which makes the head’s agreement a coincidence of margin #

Half of a set of rises is what an ordering decided by a margin of nothing looks like. On a head the margins are wider — 31.2 against 26.1 per cent for the leading pair — so the wall ranking is stable there in a way it is not on a stem, and the top-two agreement on the whole head is a fact about the size of that margin rather than about the two instruments being one.

What the missing third family turns out to be is the positive result this correction sits under, and it is on the stem rather than on the head.

What this does not overturn #

Nothing in the tissue field’s measurements. Wall counts, mean side number, the second moment of the side distribution and the two standing relations about cellular tissue are all statements about the tessellation, computed from it, and none of them uses a hop length at any point.

Nor does it touch the counting instruments. A counter that never sees a position and a counter that walks the hop lengths both return 34 and 55 on this head, and they are right. The correction is to a sentence about the relation between two measurements, not to either measurement.

What would refute the correction #

A hop ranking computed some other defensible way that comes out 34, 55, 21, 89. The ranking here is by median hop over the interior after the rim cut, divided by the local spacing; a mean rather than a median, a different rim, or a hop measured in absolute units rather than in spacings are all reasonable and none has been tried.

The claim being made is therefore narrow: on this head, under this rim cut, by median hop in local spacings, the order is 55, 34, 89, 21 and not the published one. Anyone who can produce the published order from the same points by a stated rule has found something worth knowing, and the margins of 9 and 8 per cent say how much the rule would have to change to do it.

What a reader should carry away #

That the two lists share their members and not their order, that they share the identity of their leading pair on the whole head and lose it in a third of its bands, and that the sentence claiming otherwise is now marked where it stands.

The essay it appeared in is otherwise intact, and its argument — that a cell’s neighbours are its spiral families and that the six sides are the sum over them — is what the wall shares measured here confirm. A collection that reports its own numbers has to be able to correct one of them without withdrawing the essay it sat in, and this is what that looks like.