A basin with no upper edge

A basin is the stretch of starting angle from which a stem ends up at one destination, and the widest one the settling table holds was reported as fourteen consecutive angles spanning 47.5 degrees. That figure came with a bracket attached rather than an error bar: the true edges lie somewhere in the gaps beyond the outermost agreeing angles, so the width is at least 47.5 and at most about 57.

A bracket like that closes by sampling near an edge, not by sampling everywhere. Three basins were cut at a quarter of a degree — one step of the 1,536-sample azimuth grid the organs are placed on — ten degrees at a time across each of their two sides. Seven windows, 41 runs a window, 574 stems grown.

All six boundaries are located, each to an eighth of a degree. The bracket closes at 55.625, near the top of its own range. And the basin it belongs to turns out not to have a width in the sense a width is subtracted, because it does not have an upper edge.

Why a quarter of a degree and not less #

Because a finer grid asks the placement rule a question it cannot answer. Each organ is put at the minimum of a sum over its neighbours across 1,536 candidate azimuths, so the azimuth grid is a quarter of a degree, and two starting angles closer together than that are two the rule cannot tell apart.

That makes the sampling step here a property of the machinery rather than a budget. It is the same distinction a run length that could not be spent turned on: a limit belonging to the rule stays where it is however much processor time is thrown at it, and a limit belonging to the run does not.

So the six boundaries below are located to plus or minus an eighth of a degree, and a width built from two of them carries a quarter. Neither number can be improved by sampling.

The three basins #

The widest basin sits at a rise of 0.030 and a falloff exponent of 2, reaches 139.3 degrees, and is the one the forty-angle table bracketed over 128.75 to 176.25 degrees — fourteen angles, 47.5 degrees of bracket. It is confirmed as the widest run of consecutive starting angles that table holds.

The same-rise basin is the same rise at an exponent of 3. It reaches 139.1 degrees, which is the same destination to a fifth of a degree, and the table bracketed it over 128.75 to 165 across eleven angles. One parameter of the placement rule differs, and nothing else does.

The narrow basin sits at a rise of 0.020 and an exponent of 3, reaches 151.0 degrees, and was bracketed over 146.875 to 161.25 across five angles — 14.375 degrees. It is the cheapest basin in the table with two sides worth cutting.

Six boundaries and five names #

The question the cutting was designed to answer offered two possibilities: a basin’s border is either a change of destination or a fade into angles that reach nothing. Six boundaries needed five names, and only three of the six are edges in the sense the question meant.

They are a wall, where the destination changes and nothing stops settling; a fade, where the angles beyond reach nothing at all, of which there are two; a fringe, where settling and non-settling angles alternate outside the crossing; a sliver, where a wedge of a neighbouring destination is driven into the basin; and a puncture, which is a hole inside a basin rather than a side of one.

What separates a wall from a fade, and why the obvious instrument for telling them apart does not work, is a question about the rule rather than about the basin. This essay locates the six and prices them.

The one clean wall #

It is the narrow basin’s lower boundary, at 144.875 degrees. Every one of the 41 sampled angles in that window settles somewhere — there is no non-settling angle anywhere in ten degrees — and three destinations abut with nothing between them.

136.9 degrees holds 140.25 to 144, sixteen angles over 3.75 degrees. 101.8 degrees holds 144.25 to 144.75, three angles over 0.75. And 151.0 degrees holds 145 to 150.25, twenty-two angles over 5.25 degrees.

What a wall is made of #

Three destinations meeting in ten degrees, with the middle one three sampled angles wide. That 0.75-degree stretch reaching 101.8 degrees is a basin in its own right, and it is a quarter the width of the narrowest thing the forty-angle table could have seen.

So the wall is not a boundary between two large regions. It is a boundary at which a third, small region happens to sit, and the only reason it is called a wall rather than a pair of walls is that the narrow basin’s own side of it is where the cut was aimed.

That matters for reading the earlier tables. A destination reached from a single starting angle was recorded there as a basin the sampling could not measure rather than a narrow one, and this is the first direct sight of what one of them actually is: three quarters of a degree of starting angle, wedged between two much larger stretches.

The fade above it #

The same basin’s upper boundary, at 161.625 degrees, is the opposite kind of thing. Twenty-seven consecutive angles from 161.75 to 168.25 reach nothing at all. Not a different destination, not an intermittent one — nothing, for six and a half degrees.

One basin with one of each #

The narrow basin is walled below and fades above. That is the single most economical thing in this reading, because it disposes of the assumption behind the question.

An edge was being treated as a property of a basin — the sort of thing a basin has, like a width or a destination. It is not. What a boundary is is a fact about the placement rule at a place, and one region can be bounded by two different kinds of thing without anything about the region changing between its two sides.

The only quotable width #

The narrow basin runs from 144.875 to 161.625 degrees, so its width is 16.75 degrees plus or minus a quarter, against the 14.375 the table bracketed. It is the only one of the three basins with a width that can be quoted at all, and it is the narrowest.

The other two are refused, and the refusal is machinery rather than judgement: a basin with a hole above it has no upper edge to subtract from, and a basin with a wedge cut into it below has a lower edge whose meaning depends on which side of the wedge is being counted.

That is an uncomfortable result to report and it is the honest one. Two of the three widest basins in this table do not have widths, and the reason is not that the sampling was too coarse.

The fringe below the widest basin #

The widest basin’s lower boundary sits at 124.375 degrees, standing on twenty-nine consecutive angles that reach 139.3. Beyond it the picture does not stop; it frays.

Seven of the twelve sampled angles below the crossing still reach 139.3 degrees, in three separate islands, the furthest 2.875 degrees outside the located edge. Downwards from the crossing the islands are 123.25 to 123.5, then 122.75 alone, then 121 to 122.25 — two angles, one, and six. Between them sit 123.75 to 124.25, 123 and 122.5, reaching nothing.

Where a fringe leaves a boundary #

The crossing is still located, because the rule for locating one is the first angle outside a run of consecutive agreeing angles, and that angle is unambiguous. What the fringe removes is the meaning the located number was going to carry.

A boundary is worth locating when it separates a region from what is not that region. Here it separates a run of twenty-nine agreeing angles from a stretch in which the same destination recurs six times more across six degrees, and calling the first part the basin and the second part outside it is a decision about the run length of agreement rather than about the rule.

The same shape has come up before under a different name. Islands of one answer inside runs of another turned out to be speckle rather than a period once every step was swept rather than sampled, and the resolution there was to stop looking for structure in the alternation. The resolution here is the same: three islands over six degrees is a frayed border, not six boundaries.

The hole at a hundred and eighty #

The widest basin’s upper side is the finding this essay is named for. Every quarter degree from 173.25 to 179.5 reaches 139.3 degrees — twenty-six consecutive samples. Then 179.75, 180 and 180.25 reach nothing. Then 180.5 to 183.25 reaches 139.3 again, twelve consecutive samples.

The hole is 0.75 degrees wide and centred on 180.000.

Why 180 degrees is not an ordinary angle #

Because it is the one starting angle at which the placement rule has no handedness. A stem started at 180 plus some amount and a stem started at 180 minus the same amount are the same placement problem reflected, so the range of starting angles does not continue past 180 in any meaningful sense — it folds back on itself.

That is a statement about the rule and it is testable, which is what makes it worth more than a remark. If the fold is real, every sampled angle above 180 reproduces its mirror below, run for run, and not merely in the destination it reaches.

Thirteen pairs, and all thirteen identical #

Thirteen of thirteen sampled angles above 180 degrees reproduce their mirror below it exactly: the same settling, the same organ, the same divergence to a thousandth of a degree. 180.5 and 179.5 both settle at organ 51 on 139.297 degrees.

So the hole is inside the basin #

Not a side of it. The basin’s angles run up to 180, the hole sits three quarters of a degree across the fold, and what appears above 180 is the basin’s own interior read backwards. A window that looked as though it held a boundary held a reflection.

That makes the hole a degenerate point rather than an edge, and it is the second reason the widest basin has no width. The first is that there is nothing above the hole that is not already below it.

The right statement is therefore an extent rather than a width: from the located lower boundary at 124.375 degrees to the reflection point, 55.625 degrees plus or minus an eighth. One end is a measurement and the other is a symmetry of the rule.

What the old bracket was doing #

It was bracketing something that does not exist, and it was doing it well. At least 47.5 and at most about 57 was an honest statement about where two edges could be, given angles three to five degrees apart, and 55.625 sits inside it near the top.

So the number is confirmed and the object is not. That is a shape worth naming, because it is not the usual way a bound fails: the bound was right, the arithmetic that produced it was right, and the quantity it bounded turned out to be the wrong quantity to want.

The reason the old table could not have known is that it had no angle beyond 176.25 in that basin and nothing at 180 at all. The fold is invisible to any sampling that stops short of it.

What the table understated #

By 4.375 degrees below — 128.75 against a boundary at 124.375 — and by 3.75 above, 176.25 against a crossing at 179.625. Both understatements are the size the bracket predicted, which is one to two of the table’s own spacings.

Between the two located boundaries there are 55.25 degrees of contiguous settling angle. That is a different number from the 55.625 extent above and it should be, because one runs to the last angle before the hole and the other runs to the fold.

The same understatement holds on the narrow basin: 16.75 located against 14.375 bracketed. In every case the finer sweep found the basin larger, which is what a lower bound is supposed to do and is the property that made the earlier widths worth quoting even before anything was located.

The middle nobody has swept #

Of the widest basin’s extent, 41.75 degrees — from 131.5 to 173.25 — are sampled only at the forty-angle table’s own three-to-five-degree spacing. The quarter-degree windows sit at the ends, because that is where the edges were expected to be.

So the middle is a lower bound of a different kind: fourteen agreeing angles at the table’s spacing, with everything between them unvisited. Nothing here says it is unbroken.

There is direct evidence that it might not be, and it comes from the second basin.

The sliver #

The same-rise basin — same rise, same destination to a fifth of a degree, one exponent higher — has a 0.75-degree wedge at 124.25 to 124.75 in which all three sampled angles settle on 101.4 degrees, with 139.1 on both sides of it. Above the wedge the basin runs unbroken from 125 to 131.5 across twenty-seven samples; below it, from 121.5 to 124 across eleven.

That is a whole neighbouring destination cut three quarters of a degree into a basin, and it is the size of thing that would be missed by three-degree sampling four times out of five.

So the widest basin’s unswept middle could hold one and nothing in this reading would know. What a quarter of a degree cannot see is the limit that bounds this whole result, and the sliver is the demonstration that the limit is not theoretical.

The fade with a speckle in it #

The same-rise basin’s upper boundary, at 167.625 degrees, is a fade that is not quite clean. 167.75 reaches nothing, 168.0 reaches 139.1 degrees, and then sixteen consecutive angles from 168.25 to 172 reach nothing at all.

One angle out of eighteen, the second sampled angle past the crossing, returns to the basin’s own destination. Whether that is a real detached sliver of basin or a single run that happened to converge is not decidable from one sample, and calling it a fade with a speckle is the most that can be said.

It is worth recording rather than smoothing away, because a second such speckle on a later basin would make it two, and nobody goes back to look at a boundary already named.

What the run length did not buy #

Every window was swept twice, at 1,200 organs a run and at 3,200. 287 pairs of runs, and the comparison is the shortest paragraph here: none changed whether the stem settled, none changed the organ it settled at, none of the six boundaries moved by a digit, and none of the five kinds changed.

That statement had been made before, over seventy-two pairs of runs three to five degrees apart, where a difference had least reason to appear. Making it a quarter of a degree either side of a boundary is making it where a budget would show — a run that is about to fail to settle is exactly the run a longer budget might rescue.

None was rescued. So the wall in the settling share holds at this resolution too, and the sweep can be read as being about the rule rather than about how long anything was allowed to run.

What settled, and where it went #

434 of the 574 stems settled, which is 75.6 per cent — much higher than the thirty per cent the whole settling table gives, and it should be, because six of the seven windows were aimed at the inside of a basin.

Across every window the runs that settled reached seven distinct settled values, read to a tenth of a degree: 101.4, 101.5, 101.8, 136.9, 139.1, 139.3 and 151.0 degrees. Seven values is not seven destinations, because two settled values half a degree apart or less are one destination on the tolerance this collection has always used — and two groups here sit inside it, the three near 101 and the pair at 139.1 and 139.3.

So the seven are a reading of what these windows reached rather than a census. A sweep aimed at the edges of known basins finds the neighbours of known destinations, which is the least surprising thing here and is worth stating so that nobody reads the list as an addition to the table’s own.

What this does not settle #

Nothing here bounds a basin narrower than half a degree, because two angles a quarter of a degree apart are the closest pair the rule distinguishes and a basin needs two angles to show at all. The 0.75-degree wedge is at the floor of what this design can resolve, and something narrower would appear as a single angle or as nothing.

Nothing here says the three basins are typical. They were chosen as the widest, its neighbour at one changed parameter, and the cheapest with two sides — which is a design for locating edges, not a sample of the table.

And nothing here revisits the count of basins, which grows with the sampling and will go on doing so until the spacing is finer than a typical basin. Six boundaries located is six boundaries located.

What it cost #

About 939 seconds of processor time on its own, and 4,031 seconds of process time across five shards for 875 seconds of wall clock, on a machine at a load average between 60 and 84.

The gap between the first number and the third is the whole argument for sharding a sweep of independent runs, and the gap between the second and the third is what contention costs. Neither number is a property of the basins.

Against that, the forty-angle table it refines took about twenty-two minutes for 1,280 runs. A targeted sweep of 574 stems located six boundaries that 1,280 could only bracket, which is the usual return on sampling where the answer is rather than everywhere.

What is claimed #

That the widest basin the settling table holds runs from a located lower boundary at 124.375 degrees to the reflection point at 180, an extent of 55.625 degrees plus or minus an eighth — so the bracket of at least 47.5 and at most about 57 closes near the top of its own range, and closes on an extent rather than on a width.

That the basin’s upper side is a hole three quarters of a degree wide centred on 180.000, with its own destination on both sides of it and all thirteen sampled angles above the fold reproducing their mirrors run for run, so there is no upper edge to measure.

That six boundaries on three basins needed five names, only three of them are edges, and the only quotable width among the three is the narrowest basin’s 16.75 degrees plus or minus a quarter against the 14.375 the table bracketed.

And that none of it depends on the run length: 287 pairs at 1,200 and 3,200 organs agree on every settling, every organ, every located boundary and every name.