The pass's depth belongs to the projection, and it is not a constant

The first break is mostly its denominator took the threshold at which the near-optimal aspect set comes apart and did one multiplication to it. The threshold is a ratio — the depth of a pass over the best score the region admits — and separating the two showed that as a region grows, the best score rises faster than the pass’s own depth falls, so about two thirds of the threshold’s fall is the denominator’s.

That leaves the numerator as a quantity in its own right, and the obvious next question about it is what it depends on. The separation was made on one projection at one latitude, varying only the region’s size. Here the same pass — the one that actually fails first — is measured on three projections at four latitudes, four region sizes each.

Why the value matters once the behaviour is known #

A reader might reasonably ask why a number’s value matters once its behaviour is known, and the answer is that this one was being used as an explanation.

The threshold is what four measurements have been fitting, and every account of it has had the form the threshold falls because one part rises and the other does not. That account is only as good as its parts are identifiable. If the pass depth were fixed — a number one could look up per projection, or derive — the threshold would reduce to a known numerator over a computable denominator. If it depends on everything, it is a second unknown.

The measurement says it is neither: it depends mainly on one of the three variables, and weakly on the other two.

The projection sets the level #

Hold the projection fixed and move the region’s centre from 8° to 52° north, and the pass depth barely moves on Robinson — from 2.06 to 1.80 × 10⁻², a factor of 1.14. It moves more on the other two, by 1.39 on the Winkel tripel and 1.95 on Mollweide, whose depth at 52° is nearly twice its depth at 38°. But the three projections’ bands hardly overlap. Robinson’s is the lowest at every latitude, Mollweide’s the highest at every latitude, and on average they sit a factor of 2.9 apart.

So the title’s claim is the part of this measurement that is firm. The pass depth’s level is a property of the projection: Robinson’s near-optimal set comes apart over lower ridges than Mollweide’s, at every latitude tried, by roughly a factor of two and a half. The latitude moves the depth around inside that band, and on Robinson it hardly moves it at all.

And it is not a constant of anything #

The first version of this measurement found the pass depth holding to between one and twenty per cent across region sizes within each case, and called it a constant of the projection and the latitude. That reading came from a grid that counted each pole-row map twelve times; the pass was a map meeting itself sets out how that made the “pass” at small regions the cost of one grid step, which hardly changes as a small region grows. Corrected, the depth moves with the region’s size in every case.

On Robinson at 38°, the case every earlier measurement used, the depth falls from 0.0260 at a 6° region to 0.0147 at 22°. The threshold it produces falls by 4.8 over the same range, because the best score is rising at the same time, and the pass’s share of that fall is the smaller one — which is the earlier separation’s result, and it survives. What does not survive is the idea that the pass holds still.

Case by case the pattern is the same. Only Mollweide at 52° holds within fifteen per cent, and ten of the twelve cases end lower at 18° than they began at 6°. The fall is slow — a factor of 1.2 to 1.9 over a threefold growth of the region, where the best score rises by about 2.5 — and it is steady rather than a step. The pass depth is a function of the region’s size, one that falls gently, not a number the size leaves alone.

The twelve cases, at the smallest and largest region, show both halves of that at once:

Read down the last column and the three projections separate cleanly. Read across any row and the depth falls from the small region to the large one — in ten rows of twelve. The two exceptions are both Winkel tripel, at 8° and 23°, where the largest region’s pass stands above the smallest’s — at 8° by half again. That projection’s rows are the least orderly of the three: its passes jump between neighbouring region sizes, 0.0366 then 0.0517 then 0.0260 at 23°, in a way Robinson’s and Mollweide’s rarely do, and it is the one whose four-latitude fit, below, scatters. A pass that jumps between neighbouring sizes is a sign that the deciding pair of basins changes from one size to the next, which the earlier merge measurement found is what the first pass does.

What a projection’s level would have to be about #

It is worth being clear about what kind of property could give a projection its own band, because the band is the one firm result here and it constrains the explanation.

A pass in this landscape is a ridge between two good placements of the projection over the region — the least-bad arrangement that has to be crossed to turn one good placement into the other. Crossing it means rotating the projection’s pattern across the region, and the height of the ridge is how bad the region’s distortion becomes somewhere along that rotation. So the level is a statement about how much worse the projection gets between its good arrangements, and that is a property of the pattern’s relief rather than of its lowest point.

Robinson was designed by eye to spread its distortion thinly: no part of the map is very good and no part very bad, and its scale field is gentle everywhere except near the poles. A region rotated across such a field meets no deep trough and no high wall, and the ridges between good placements are low. Mollweide is equal-area, and pays for it in angular deformation that grows steadily towards the edges of its ellipse; a region rotated towards those edges meets a steep wall, and the ridges between good placements are correspondingly higher. The Winkel tripel, the average of two projections, sits between them.

That is a plausible account and it has not been measured. It predicts that a projection’s level should be computable from the projection’s scale field alone — some measure of its relief — without any region or any search. The account is testable on the three projections here and would be worth more if tested on a dozen.

What the level means for a search #

The level has a practical reading, and it does not depend on why the level is what it is.

The threshold at which the near-optimal set first comes apart is the pass depth over the best score. Mollweide’s pass depth is about two and a half times Robinson’s and its best score only about a fifth higher, so at any stated tolerance Mollweide’s near-optimal set is more broken up than Robinson’s: its good placements are separated by higher ridges. A search that walks downhill from a random start is more likely, on Mollweide, to end in a good placement that is not the best one, because the ridge it would have to cross to reach the best is higher relative to how much better the best is.

So for a small region, the choice among these three projections decides how hard the placement is to find much more than how good it can be. Robinson’s good placements are easy to move between and nearly equally good; Mollweide’s are nearly equally good and harder to move between. A search on Mollweide wants more starting points for the same confidence that it has found the best, and a report of a Mollweide placement is more likely to be one of several — which is report the map, not the parameters with a projection-dependent weight on it.

Which half moves depends on what is varied #

This is the part of the earlier version that survives whole, and it is a warning about a class of measurement rather than a fact about passes.

A threshold here is a ratio of two quantities. Sweep the region’s size and both move, but the denominator moves more: the best score rises by about two and a half across 6° to 18° while the pass falls by about one and a half, so the ratio’s behaviour is mostly the denominator’s. Sweep the projection or the latitude and the denominator holds nearly still — the best score a region admits varies by only half again across all twelve cases — while the numerator moves by a factor of 4.4, so the ratio’s behaviour is mostly the numerator’s.

Both readings are correct and they point in opposite directions. Which one a measurement arrives at is decided by which variable it happened to sweep, and neither is visible in the ratio itself. Every earlier measurement on this ground — where the valley breaks in two, the height of the pass between two basins, the basins have widths as well as depths, the threshold is not a percolation — varied the region and held the projection, so every one was looking mostly at the denominator.

The best score a region admits hardly cares what projection it is #

The denominator’s steadiness deserves a sentence of its own, because it is unexpected and it is what makes the reversal sharp.

The best score is what the region can be served to at all — the value at the bottom of the deepest basin, over every placement of the projection. One might expect that to be strongly projection-dependent. Over these twelve cases it varies by a factor of 1.5, from 1.94 × 10⁻² to 2.93 × 10⁻². Whatever else a projection’s identity decides, it barely decides how well a small region can be served once the placement is free to be chosen — which is a restatement, from an unexpected direction, of what every projection minimises something says about compromise maps being more alike than their names suggest.

The pass depth, over the same cases, varies by 4.4, and most of that is the projection. The identity of the projection shows up in the structure of the landscape rather than in its floor: how high the ridges are between good placements, not how good the best one is. For somebody choosing between these three for a small region, the choice is not about how well the region can be served; it is about how isolated the good placements are from one another, which is what the shape of the valley is about.

What the region sees does not decide it #

Where the valley breaks in two found three projections lying close to one curve when their fracture thresholds were plotted against how much of their own variation the region sees. The obvious test was whether the same quantity explains the pass depth, and it does not. Pooled across the twelve cases the fit explains nothing — $R^2$ of 0.00 — because the three projections sit at three different levels and the variable does not know which projection it is looking at. Within a projection it is no better as a law: Robinson’s depth is flat against it, Mollweide’s rises with an exponent of 0.56, and the Winkel tripel’s scatters.

The first version of this measurement found the opposite — a tight power law within each projection — and the next two measurements on this ground were built on those curves. The curves were the grid step again. The corrected statement is narrower and more useful: the threshold collapses against what the region sees because its denominator does; its numerator is set by the projection.

What this leaves of the account #

Three things are now settled about the threshold, and they came from different measurements.

The pass that decides it is the last good placement to join the best placement’s basin, standing 1.02 to 2.09 times above the pass between the two deepest basins, and falling at nearly the same rate.

Over region sizes the threshold’s fall is mostly the denominator’s: the best score rises faster than the pass falls.

Over projections and latitudes it is mostly the numerator’s: the pass depth moves by 4.4, most of it between projections, and the denominator by 1.5.

What is not settled is what sets a projection’s level. That is a question about the projection alone, and it is where the explanation will have to come from.

What each number was checked against #

Across rows the depth must not be one number, or there is a universal constant after all. The extremes are 1.80 × 10⁻² and 7.86 × 10⁻².

The projection must account for most of the spread, or the title is an over-reading. Within each projection the four latitudes span factors of 1.14, 1.39 and 1.95; between projections the bands sit 2.9 apart on average, and Robinson’s highest value is below the Winkel tripel’s lowest.

The denominator must be the steadier of the two across rows, which is the whole of the reversal: 1.5 against 4.4.

The pass is the corrected one throughout. Every depth here is the highest merge in the filtration, with components of fewer than three cells excluded and both of the grid’s ways of holding one map twice joined.

And the spread within a row is a spread over sizes, not a standard error. The four values averaged are four different regions, nested about one centre, so the whiskers are a real range rather than an estimate of sampling noise.

What twelve cases do not establish #

Three projections and four latitudes. Robinson, the Winkel tripel and Mollweide are one compromise projection, one arithmetic mean of two projections and one equal-area pseudocylindrical. A cylindrical is not included, and a projection whose level sat between two of these would test whether “level” is one number a projection has or a range.

The latitudes are northern and the regions square. Every case is a synthetic region of unit aspect ratio centred on a meridian, and square regions carry a mirror twin for every placement that a real region does not. A region long in longitude would sample a different part of each projection’s pattern.

Small regions only. The depths are measured from 6° to 18° of span, which is where the earlier separation was made.

And a band is a range, not a law. Twelve cases give three bands; they do not give a function of the projection. What property of a projection’s scale field sets its band is not measured here.

Still open: what sets a projection’s level #

The pass depth sits in a band that belongs to the projection — Robinson’s around 0.02, the Winkel tripel’s around 0.03, Mollweide’s around 0.05 — and the question is what property of a projection decides it.

Two candidates are cheap to test. The level might track how smooth the projection’s scale field is — a gentle field giving low ridges between good placements, a sharp one high ridges — which would make it computable from the projection alone, with no region in it. Or it might track how much of the field lies in longitude rather than latitude, since the ridges between placements are crossed by turning the projection, and turning moves the longitude pattern. Mollweide’s level is highest and its scale field varies most in longitude of the three; that is one ordering, and three projections cannot tell it from coincidence.