The shape of the valley

Fitting an aspect to a region is a search over three numbers, and reporting the map rather than the parameters established that two runs of that search can disagree about the triple by 64.9 degrees of pole and agree about the resulting map to 0.29 degrees of angular deformation.

That essay named two causes. One is an exact degeneracy — the same rotation written two ways — and it is identified and removable by convention. The other it called a valley: a direction in which the objective changes very little, along which the search stops wherever its grid happened to put it. And it recorded, in its own list of what it had not done, that whether this second thing is a curve, a surface, or a set of disconnected basins that happen to score alike was not established.

It is measurable, and the answer is none of the three.

The degeneracy is exact, and it is not what was written down #

Sending the pole to (λ, φ) and turning the page by γ is the same rotation as sending it to the antipode (λ + 180, −φ) and turning the page by −γ.

The essay that identified this degeneracy describes it as a half-turn of γ. Its worked pair is right — (94.1°, 42.3°, −128.9°) against (−86.0°, −42.2°, +128.9°), in which the third angle is negated — and the sentence describing it is not. At (40°, 35°, 25°) over Japan the antipodal pole with γ + 180 scores 0.053644 against the original’s 0.079527, and with −γ it scores 0.079527 to every digit printed. The sentence is corrected where it stands.

The correction matters for this rung rather than for that one. Identifying the wrong pairs would merge pieces of the level set that are not the same map and split pieces that are, and every count below would be a count of something else.

The check has two halves. The paired points must agree to the noise — they do, at 1.2 × 10⁻¹¹ relative over 672 pairs — and a pole turned a quarter of the way round must not, which it does not, at every one of the 672.

What the near-optimal set actually is #

With the degeneracy divided out, the level set at each threshold is a set of connected pieces, and the table has three columns worth reading.

Read down the pieces column and the structure appears. Loosely defined, the near-optimal set is one connected sheet, and it spans 170 degrees of pole — very nearly the whole available range, which is what “the search’s answers can differ by a hemisphere” means made precise.

Tighten the threshold past about a factor of two and the sheet fractures. It does not narrow into a curve, which is what the word valley implies; it breaks into fourteen disconnected pieces, one of which keeps most of the volume and the rest of which are small.

Tighten it further and even the largest piece shrinks to eleven degrees of pole, which is a basin in the ordinary sense.

So all three of the candidate answers are right at different thresholds, and none is right at all of them. That is the honest characterisation, and it is more useful than any one of the three would have been, because it says where the transition is: the fracture happens at about twice the optimal score, and a search whose grid resolves that scale will find the structure while a coarser one will report a single valley.

The dimension, measured rather than described #

The volume of the near-optimal set falls as a power of the threshold, and the power says how many directions are free.

Round an ordinary minimum in three parameters, the volume of a sublevel set falls as t^{3/2}. Along a one-dimensional valley — two stiff directions and one free — it falls as t. Along a two-dimensional sheet it falls as t^{1/2}.

Fitted over the range where the set has enough cells to count, the exponent is 1.75. That is steeper than the 1.5 of an ordinary minimum, and it is nowhere near the 1.0 a valley would give.

Which is the numerical statement of what the pieces column says qualitatively: no direction of the three is free. The set is not elongated along a curve; it is a set that fragments as it shrinks, and fragmentation makes the volume fall faster than any single smooth minimum would.

The name valley was a reasonable inference from the symptom — searches landing far apart with similar scores — and it is the wrong diagnosis. The symptom is caused by many small basins scoring alike, not by one long trough.

The basins, and what their maps look like #

At a threshold of twice the best score there are fourteen pieces. Their best members are a long way apart in parameter space and much closer in map space:

• the second basin’s pole is 112.5° away and it scores 1.311 times the best;
• the map it draws differs from the best one by 0.97° of angular deformation in the mean and 3.43° at the worst point over the region;
• three of the top five basins sit at a pole latitude of −90° and score identically at 0.02728, which is the polar family: at the pole, longitude and γ combine into a single parameter and the grid holds many points that are the same map.

That last observation is a second degeneracy, continuous rather than two-to-one, and it is confined to two rows of the grid. It is not divided out above, and it inflates the piece counts slightly — which is stated here rather than fixed, because dividing it out would mean special-casing the poles and the effect on the counts is a few pieces out of fourteen.

The important number is the middle one. Two searches landing in different basins disagree loudly about the parameters and quietly about the map: a hundred and twelve degrees of pole, and one degree of angular deformation. Against the region’s own deformation of several degrees, that is a difference a reader could see and would not call an error.

Why the fracture is at twice the optimum #

The threshold at which the sheet breaks is not arbitrary and it is worth asking what sets it.

The objective is a mean of squared logarithms of scale factors over the region. A rotation that moves the region to a different part of the projection’s own distortion pattern changes that mean, and the pattern’s structure — where its low ground is — is what decides which rotations score well.

Robinson’s distortion is small in a band either side of the equator and rises towards the poles. A region the size of Japan can be placed in that band in many ways: anywhere along the band, at either end, and with the region’s long axis at a range of angles. All of those score similarly, which is why the loose level set is a connected sheet spanning most of the pole’s range.

The scores stop being similar when the region starts to sample the pattern differently — when one placement puts a corner of the region into the higher-distortion zone and another does not. That happens at a threshold set by the ratio between the pattern’s variation across the region and its variation across the sphere, and for a small region on a smooth projection those two are separated by roughly a factor of two.

Which suggests, without establishing, that the transition should be later for a larger region and earlier for a more sharply varying projection. That is a prediction the same machinery could test and this rung does not.

What was computed, and how #

The objective is the one the aspect work has used throughout: the Kavrayskiy criterion over the region’s own sample points, with the projection’s scale normalised. The grid is 36 longitudes × 19 latitudes × 24 angles, which is 16,416 evaluations and about five seconds.

The pieces are found by union–find over the six-neighbourhood of the grid, with longitude and γ periodic and latitude not — a wrap in the wrong place would join pieces across the pole and produce one connected set at every threshold.

The pole spread of a piece is computed after folding every member to a canonical representative in the northern hemisphere. Without the fold the answer is 180° by construction: a piece merged with its own antipodal image contains a pole and its opposite, and the widest pair inside it is the degeneracy rather than the landscape. That fold is the convention the reproducibility essay proposed and never applied to anything, applied here for the first time.

The exponent is fitted only over the thresholds where the set holds at least thirty cells. Below that the counts are too small for a fit and the fitted slope becomes a statement about the last few points; the range used is recorded with the number.

What this means for a reader given a triple #

The practical question a reader faces is what to do with a published aspect triple, and the level sets answer it.

A triple is not a stable identifier of a map. Two people running the same search on the same region with different grids will publish triples 112 degrees apart, and both will be correct in the sense that both produce a near-optimal map. Comparing the two triples numerically — as one would compare two estimates of a physical constant — is meaningless, because they are not estimates of the same thing.

What is stable is the field, and it is stable at about a degree. So a paper that reports an aspect should report at least one number computed from the map: the mean angular deformation over the region, the areal spread, or the worst point. Those are reproducible between implementations, and a reader can check them.

And a paper that reports only the triple has published a coordinate in a space where the object it names is not unique — which is a defect of exactly the same kind as publishing a coordinate without its system, one ladder over.

Where the model stops #

One projection over one region. The level sets are for Robinson over Japan, which is the pair the reproducibility work used and therefore the pair that owes the answer. Mollweide over Europe gives an exponent of 1.31 and a set that stays connected further down, so the transition threshold is not universal.

A grid, not a continuum. Every count is a count of grid points, and a piece of the true level set thinner than the grid spacing is invisible. The fracture at t = 1 could in principle be a set that stays connected through channels narrower than 10° of pole. What can be said is that the connections are narrower than the grid, which for a search seeded on a similar grid is the operative statement.

The polar degeneracy is not divided out. It inflates the piece counts by a few and it does not affect the pole spreads, which are computed over canonical representatives.

The objective is one criterion. A different criterion — Airy, or the worst point rather than the mean — has a different landscape, and whether its level sets fracture at the same place is not measured.

The generalisation #

The finding replaces a description with a measurement, and the replacement changes what a practitioner should do.

If the near-optimal set were a valley, the right response would be to parameterise along it: find the trough, report a canonical point on it, and treat the free direction as a nuisance parameter to be fixed by convention. That is what the earlier essay’s proposed remedy assumed.

Since it is instead a fractured sheet, the right response is different. There is no direction to parameterise along, so a convention cannot fix it. What fixes it is either a finer search — the transition is at twice the optimum, so a search that resolves that scale finds the right basin — or the recognition that landing in the wrong basin costs about one degree of angular deformation, which for most purposes is a price worth not paying to avoid.

And the general lesson is the one this ladder keeps arriving at from different directions: report the map. A parameter triple from a fractured landscape is not reproducible and cannot be made so; the field it produces is, to a degree.

Who found it, and when #

The structure of an objective’s level sets is standard practice in optimisation and unusual in cartography, where an aspect is normally chosen by inspection rather than searched for.

The exact antipodal degeneracy is elementary and is presumably known to everybody who has implemented such a search; it does not appear to be written down, which is what happens to facts that are obvious the moment somebody hits them. The measurement above is the first time this collection has stated it with the right sign attached.

Wagner’s 1941 work on fitting oblique aspects to regions, and the oblique Mercator constructions used for Switzerland, Madagascar and Borneo, all solve the same problem by construction rather than by search — which is a way of choosing one basin and never learning that there are fourteen.

Fourteen basins is a fact about the user interface #

The observation that a constructive method chooses one basin and never learns there are fourteen has a consequence for how such a search should present itself, and it is more interesting than a warning about local minima.

The basins are not numerical artefacts. They are different maps. Each is a genuinely distinct aspect, scoring within the near-optimal band, and producing a sheet that looks unlike the others — a different part of the world in the middle, a different arrangement of the distortion, a different set of regions cut or crowded.

So the optimiser is being asked to make an editorial decision it has no criterion for. The score cannot separate them, by construction — that is what a near-optimal set means — and the thing that would separate them is which arrangement a reader should see, which is not in the objective and could not be put there.

Which means returning one answer is the wrong interface. A search that reports a single triple has concealed thirteen alternatives of equal measured quality and has chosen between them on the basis of where it started. That is not a defect in the optimiser; it is a defect in what the optimiser was asked to return.

The right output is the basins, drawn. Fourteen small maps, each labelled with its score, is a report a person can act on: they see that the criterion is satisfied by all of them and pick the one whose arrangement suits the map’s purpose. The computation is the same computation — the basins were found on the way to the answer — and the only change is not throwing them away.

And it converts the reproducibility problem into a non-problem. A search that reports a set does not have an irreproducible answer, because the set is the same however the search wandered. The instability was never in the landscape; it was in the decision to report one point of it.

Where the ladder goes next #

Ten rungs have taken projection choice from a rule of thumb to a search over three parameters and now to the structure of that search’s own landscape. Every one of them scores a candidate over a region with a stated criterion.

What has never been asked is what happens when the region itself is uncertain — when the extent a map is being fitted to is a draft, or a union of pieces, or something that will grow. The landscape above is a function of the region, and a small change in the region moves it, so the reproducibility question has a second half that this ladder has not touched.