The height of the pass between two basins

Where the valley breaks in two measured the threshold at which the set of near-optimal aspects for a region stops being one piece, and found it falling faster than the argument behind it predicts: as the −1.57 power of the fraction of its own pattern the region sees, for Robinson, and −1.36 pooled over three projections, against a prediction of −1.

That essay recorded a shortfall and named the suspect. A level set says whether two basins are joined at a height and not how high the ground between them is, and the exponent is a statement about that height — so measuring the height directly ought to recover the prediction.

It moves the answer most of the way there, and not all of it.

What a pass is, and how to find it #

Between two basins of a landscape — a landscape in parameter space rather than on the ground, though the ground version behaves the same way — there is a lowest crossing, and every path from one to the other must rise at least to it. That is the minimax path: the path whose highest point is as low as possible, and its height is the pass.

Finding it does not need a search over paths. Sort every cell of the surface by score and add them one at a time, joining each to any neighbour already added; the height at which the two basins’ components merge is the pass, exactly. One sweep, one union-find structure, no tolerance and no iteration.

That is a better instrument than a level set for this purpose. A level set at threshold $t$ answers are these two connected at $t$, and finding the crossing point means bisecting on that yes-or-no answer. The union-find sweep returns the crossing directly and returns the two basin floors at the same time.

It has one requirement the level set shares and that an earlier version of this measurement did not meet: two cells that are the same map must be joined before anything is counted. The aspect grid holds a map twice in two ways. A pole and its antipode with the turn reversed draw the same picture upside down, and on the two pole rows, where longitude and turn are one rotation, each map appears twelve times. Left apart, those copies count as separate basins, and the “pass” between two copies of one map is the score of whatever lies between them. The first version of this measurement joined neither; its second basin was the best map’s own copy, and at small regions its pass was the climb from one copy to another. The pass was a map meeting itself sets out what that did to the measurements that followed. Every number below joins both.

What the copies had done to this sweep #

The correction is worth seeing point by point, because it did not move every point and the ones it moved say what was wrong. Measured with the copies left apart, the six passes were 1.869, 1.069, 0.566, 0.727, 0.198 and 0.127 above the best score, for regions from 6° to 40°. With them joined they are 1.117, 0.709, 0.566, 0.315, 0.198 and 0.126.

Three points are untouched: at 16°, 30° and 40° the sweep had found a climb between two genuinely different placements all along. The other three moved, and all three moved down, because a climb from one copy of a map to another has to go through whatever lies between the copies on the grid, and on these regions that was higher ground than the real pass to a different placement. The largest change is at 22°, where the published pass of 0.727 was more than twice the real one. That single point, sitting far above the line through the others, is what bent the published fit to −1.60 at $R^2$ 0.913; with it corrected, the six points fall on a line at 0.986.

A published exponent carried by one point is worth the sentence, because the fit gave no warning. An $R^2$ of 0.913 over six points looks like a result. It was a result about three points, two copies and one copy far from the rest.

The pass is not the first break #

The first check is whether the pass between the two deepest basins is the threshold at which the set first comes apart, and it is not. At a 20° region the pass stands 0.349 above the best score and the set first breaks at 0.497, thirty per cent higher. Across the sweep:

The pass is never above the first break and only once meets it. That is exactly what it should be, and it says what the level set was measuring. The near-optimal set comes apart at the highest merge in the whole landscape, wherever it is, and the two deepest basins are only one pair among several. When the set breaks first, some other pair — shallower basins further from the best — is separating, and the two deepest are still joined. At 16° the two happen to be the same event; elsewhere they are different events a factor of up to 1.9 apart. The pass that fails first is not the one that was measured is the measurement that tracks every merge rather than one, and finds out which pair it is.

The exponent moves towards the prediction #

The pass height falls as the −1.34 power of the fraction seen, with $R^2 = 0.986$ over six regions.

The level-set measurement on the same projection was −1.57, and the prediction is −1. Measuring the quantity the shortfall named, with a sharper instrument, has moved the measurement from 0.57 away from the prediction to 0.34 away: most of the way, and by a clear margin short of it.

So the shortfall’s hypothesis is half right. Part of the level set’s steepness was a property of what it measured rather than of the landscape — it reads the highest merge, and the highest merge belongs to a pair of basins whose separation grows faster than the deepest pair’s. The rest is in the pass itself: the ground between the two best placements rises faster than the fraction seen, and no choice of instrument makes it rise as the fraction does.

The second basin is usually the first one reflected #

The second half of the shortfall’s plan was a decomposition. A pass sits above the second basin’s floor, and the second floor might rise as the region grows, so the measured exponent could be a sum of two — how the barrier scales, and how the second basin’s depth scales — and fitting both would say which carried the discrepancy.

At five sizes out of six there is nothing to split, because the second basin is not a worse map. It is the best map’s mirror twin: a different placement of the region on the projection, in which each point of the region sees exactly the distortion that its reflection across the region’s central meridian saw in the best placement. The regions here are squares, and a square is its own mirror image, so the twin scores alike to the last digit. The projection’s own symmetry about its central meridian is what makes such a placement exist at all.

Why a twin should exist is worth a sentence. Every projection in this comparison is symmetric about its own central meridian: a point a degree east of it and the point a degree west are drawn with the same distortion. A placement of the region is a rotation of the sphere onto the projection, and composing that rotation with the reflection across the region’s own central meridian, and then with the reflection across the projection’s, gives a second rotation — two reflections make a rotation — that lays the region down with east and west exchanged. Each point then sits where its mirror image sat, in a part of the projection distorted exactly as the part its mirror image occupied. For a region that is its own mirror image, nothing distinguishes the two.

That is a fact about the test regions, not about aspect searches in general. A real country is not symmetric about a meridian, and it has no mirror twin; its second basin is a genuinely different map with a floor of its own. The one size here where the square’s twin is not the second basin — 40°, where a different placement 2.1 per cent worse comes second — is a glimpse of what the decomposition would look like on a real region: a barrier of 0.102 above a second floor of 0.021, so that the barrier is most of the pass and the second basin’s depth a fifth of it.

Three predictors that cannot be told apart #

Three candidate predictors were tried: the fraction of its own pattern the region sees, which is the argument’s variable; the raw scale spread over the region; and the region’s own angular span. They give exponents of 1.34, 1.22 and 1.14, with coefficients of determination of 0.986, 0.988 and 0.958.

None of the three is distinguishable from the others. Six points on a log-log plot, over a factor of five in the independent variable, cannot separate three quantities that are all monotone functions of the region’s size, and the fits are tight enough that the choice among them is decided by which variable the reader finds natural, not by the data.

What the argument actually predicted #

The prediction of −1 deserves restating, because knowing what was assumed says where it could fail.

The argument is that two aspect placements — and the aspect has three numbers rather than one — score similarly until one of them moves part of the region into a zone where the projection behaves differently. How much of the projection’s own variation a region sees is therefore the natural variable: a region that sees none cannot tell the placements apart and its near-optimal set stays connected to a high threshold; a region that sees most of it fractures early.

Making that a quantitative prediction takes one more step, and it is the step with the assumption in it. If the score surface’s shape is fixed and only its amplitude scales with the fraction seen, then the pass height scales inversely with that fraction — exponent −1.

The measurement says −1.34, so the amplitude is not the only thing that scales. The shape changes too: as the region grows, the basins narrow relative to the parameter space and the ground between them rises faster than a pure amplitude scaling would give. That is a statement about the landscape’s geometry rather than its depth, and it would need the widths of the basins to settle.

Six points, and why not more #

The limitation is computational and it is worth stating so that anybody extending this knows what it would cost.

Each point requires a full aspect score grid — a three-dimensional sweep over the two angles that place the pole and the turn about it, with a region-wide distortion integral at every node. The grids here are 24 × 13 × 12, which is 3,744 evaluations of a regional criterion, and each of those is a sum over sample points. Six regions is about a minute. Extending the range by a factor of five in either direction means finer grids as well as more of them, because a larger region’s near-optimal set is smaller in parameter space and needs more resolution to resolve its basins.

That is not a reason the exponent is wrong. It is the reason the honest statement is that three predictors are indistinguishable at this sample size, rather than a choice between them.

Reading the sweep as a practical statement #

Underneath the exponent there is a plain result a user of an aspect search can act on.

Small regions have one basin and large regions have several, which is how many sheets an atlas needs arriving as a statement about parameter space. At six degrees across, the near-optimal set of aspects for a Robinson map of a square region is a single connected sheet up to a threshold of 2.1 — almost any aspect within three times the optimum’s cost is reachable from any other by a continuous change, and the best placement and its twin are joined by a pass of 1.1.

At forty degrees across the two best basins are joined at 0.126. Two placements differing by thirteen per cent in score can be genuinely different maps with no continuous path between them that stays inside that margin.

The practical form is a warning about optimisation. On a small region, a search that stops anywhere near the optimum has found something close to the answer, or its mirror. On a large one it has found one of several, and reporting its parameters says which basin the search happened to land in rather than what the region wants.

What a half-confirmed hypothesis is worth #

The shortfall asked a specific question: is the level set’s steepness an artefact of reading heights through a yes-or-no test? The answer is that half of it is, and the half that is not has a location.

The instrument is better. The union-find sweep is exact where the bisection was approximate, and it returns the basin floors as well, which is how the twin was found.

The discrepancy is split. Of the 0.57 by which the level set misses the prediction, 0.23 belongs to the level set reading a different, faster-separating pair of basins, and 0.34 belongs to the pass between the best two. The first part is the next measurement’s subject and the second is the basin widths’.

And the test regions carry a symmetry a real region does not. Every square region in the sweep has a mirror twin that scores alike, which is harmless for a pass between the two deepest basins — the pass between a map and its twin is still a real climb between two placements — and which would be wrong to generalise: report the map, not the parameters is the advice for real regions, where the second basin is a different map.

Where the model stops #

Everything here is one projection and one region shape at one central latitude. The exponent is fitted over six regions spanning a factor of five, and the fit, at $R^2$ 0.986, establishes the value better than the sample size would suggest and still over too short a range to rule out curvature.

What would settle whether −1.34 is a property of the landscape or of the range is an argument rather than more points: deriving the barrier’s scaling from the assumption that produced −1, and finding which part of it fails.

Where the argument goes next #

The basin widths. The exponent’s remaining discrepancy lives in how the shape of the score surface changes with the region, and shape means widths as well as heights. Measuring a basin’s width in a three-dimensional parameter space means measuring the volume of its sublevel set, which the union-find sweep computes on its way past and which nothing here reads out.

The first break. The level set reads the highest merge in the landscape, and the pass here is not it. Which pair of basins separates first, and how its height scales, is a measurement of every merge rather than one.

Who found it, and when #

The minimax path is a standard object under several names — the bottleneck shortest path, the minimum spanning tree’s own structure, the merge tree of a scalar field — and the union-find sweep that finds it is the same algorithm Kruskal’s minimum spanning tree uses. It is imported here rather than invented.

What is not standard is applying it to a projection’s aspect landscape, and the reason is that nobody usually looks at that landscape at all. An aspect search returns a set of parameters, and fitting the aspect to the region is what it is for; the surface it searched over is discarded, and every structural fact about it — the number of basins, their twins, the height between them — is discarded with it.