The pass was a map meeting itself

The sinusoidal is the edge of its family, not outside it ended by asking for a count. Its measurement — how deep the pass is that first splits a region’s set of near-optimal aspects, and how that depth grows with how much of the projection’s variation the region sees — held a tight power law along most of a family of equal-area maps and came apart over its last twentieth. The question it left was whether the aspect search’s landscape changes character there: many shallow basins where there had been a few deep ones.

Counting the basins answers a different question first, and the answer takes the earlier measurements apart. Listed one by one with the place each basin’s floor sits, the census kept returning groups of basins with identical floors — the same score to eleven decimals — at different places in the search. They were not symmetric partners of the kind the shape of the valley had already found and identified. They were all on one row of the grid, and they were all one map.

Two angles are one rotation at the pole #

The search describes an aspect by three angles, as the aspect has three numbers, not one set out: the longitude and latitude at which the projection’s own pole is placed, and a turn about that pole. Each is sampled on a regular grid — 24 longitudes, 13 latitudes from pole to pole, 12 turns — and every cell is scored.

Most of the time three angles are three independent freedoms. At a pole they are not. Put the projection’s pole on the Earth’s pole and the first angle, which moves the pole in longitude, becomes a rotation about the Earth’s axis. The third angle is also a rotation about the projection’s pole, which is now the same axis. Two rotations about one axis add, so only their sum matters (or their difference, at the south pole): the map at longitude λ and turn γ is the map at λ + δ and γ − δ, for any δ at all. This is gimbal lock, the failure every scheme of three rotation angles has at the place where its middle angle reaches a pole, and it is the reason aircraft and spacecraft attitude is kept as a quaternion rather than as three numbers.

On the grid it means that each pole row of 288 cells holds only 24 distinct maps, each twelve times. The twelve copies of any one map lie on a diagonal of the row, spaced two columns and one row apart, and none of them is a neighbour of another: a cell’s neighbours on the row differ from it by one 15° step of longitude or one 30° step of turn, and both of those are different maps.

The copies could meet only through their neighbour #

A basin, in the measurement the earlier essays used, is a connected piece of the set of aspects scoring below a threshold, and a pass is the score at which two pieces join. The pieces are found by adding cells in order of score and joining each to whichever of its neighbours are already in. The height of the pass between two basins describes the sweep, and it is exact for what it is given.

What it was given treated the twelve copies of a map as twelve places. When the lowest map on a pole row enters the sweep, all twelve copies enter together — they have the same score — and each starts a piece of its own, because none of them touches another. Each grows as its neighbours come in. And the twelve pieces, all one basin of the space of maps, can join only when the sweep reaches a cell that touches two of them, which on the row is the map one step round the axis.

So the prediction is exact and it can be checked exactly: wherever the published first pass was copies meeting copies, its score must equal the score of that neighbouring map, not approximately but to the arithmetic’s own precision. For Robinson it does, in 28 of the 36 regions its curve was built from, to nine decimal places. In those 28 the quantity four earlier measurements called “the depth of the pass that fails first” was the score of one particular map: the normal aspect, turned 15° round the Earth’s axis from the best normal aspect on the row.

The other eight regions are genuine merges between different maps, and the correction leaves them alone.

The four regions centred at 38° N show the whole mechanism in one column of numbers. Their best normal aspects on the pole row score 2.12, 1.71, 1.57 and 1.52 times the best aspect of all, for regions 6°, 10°, 14° and 18° across. Turned 15° round the axis, the same maps score 3.13, 2.33, 2.00 and 1.82 times the best — and those four numbers are, digit for digit, the first passes that were published for those regions. Taken as absolute depths above the best score they are 0.0264, 0.0236, 0.0230 and 0.0233: flat, to within fifteen per cent, over a threefold growth of the region. That flatness is what the first break is mostly its denominator found and built on. With the copies joined, the first passes for the same four regions are 0.0260, 0.0211, 0.0167 and 0.0148, falling steadily as the region grows. The constant was a map turned by one grid step, and it was constant because the cost of turning a normal aspect 15° off its best meridian hardly changes as a small region grows.

A curve made of the grid’s own step #

That explains the curve, and the explanation is short. The score of a normal-aspect map turned 15° away from the best normal aspect is the region’s distortion with the projection’s central meridian moved 15° off the one that suits it. How much that costs is precisely how fast the projection’s scale changes across the region as the pattern slides under it — which is the quantity where the valley breaks in two defined as how much of the projection’s own variation the region sees, and against which every curve was drawn.

A quantity plotted against itself, with a grid step in between, gives a clean power law. That is why the fits were so good: $R^2$ of 0.98 for Robinson, 0.999 for the Hammer, 0.996 along most of the family. It is also why the sinusoidal essay found that doubling the grid moved the exponent of a well-behaved member by nearly a quarter: halve the step and the neighbouring map moves closer, and its score falls by an amount that depends on the step. A slope that changes with the grid spacing is the signature of a measurement of the grid spacing.

The repair is to say on the grid what is true of the maps: the copies are one point, so they are neighbours of one another. Joined that way, the twelve pieces are one piece from the moment they enter, no pass is recorded between them, and the sweep reports only merges between maps that differ. Robinson’s curve then disappears. Its pass depth no longer follows what the region sees at all — exponent 0.02, $R^2$ 0.008 — and the depths themselves fall by more than a factor of four, because the fake passes were among the highest in the landscape.

Two curves were made of nothing and three were real #

The same correction on every projection the earlier comparisons measured does not give one answer, and the differences are the useful part.

Robinson and the Winkel tripel had no curve. Their published fits, at $R^2$ 0.98 and 0.91, were made almost entirely of copies, and corrected they fall to 0.008 and 0.050. The exponents six projections have one band and one refusal compared for them, 1.16 and 1.16, were the exponents of a grid step.

The Hammer, Lambert’s cylindrical and Mollweide have one. Copies decided 23 to 30 of their 36 regions too, but the real merges underneath follow the region almost as tightly: 0.952, 0.947 and 0.814. For those three the copies were sitting on top of a genuine curve and flattering it, not inventing it. The Hammer’s corrected exponent is 1.35, not the 0.92 published, because the passes the copies displaced were deeper than the step.

Eckert IV is between, at 0.515 from 0.921.

And the sinusoidal is untouched. Its curve was published as the one refusal among six, $R^2$ 0.150, and corrected it reads 0.152. Copies decided only three of its 36 regions, because on the sinusoidal a real merge higher in the landscape — a small pocket of poor aspects, a few cells across — almost always outranks the step. The projection that looked like the exception was the one measurement the defect did not reach.

The family’s edge was where the copies stopped deciding #

The family essay found the curve holding through nine tenths of the family and coming apart between m = 0.95 and 0.99. The share of passes that were copies falls over exactly that stretch: every one of the 36 regions from m = 0.4 to 0.9, 32 at 0.95, 18 at 0.97, 6 at 0.99 and 3 at the sinusoidal. The published $R^2$ follows the share down, point for point. The edge the family appeared to have is the place where the sinusoid’s approaching shape starts to put a real pocket above the step, and the step stops being the highest merge.

Corrected, the family has no edge because it has no trend. Its curve is present at m = 0 and at m = 0.8, with $R^2$ 0.947 and 0.804, and absent at 0.2, 0.4, 0.6 and 0.9, where the fits read 0.02 to 0.04 and 0.006. Members a tenth apart in a continuous family of maps disagree completely. That is not a property of the maps, which change smoothly along the family; it says that on this grid, with the copies joined, which merge is highest is decided by small things — a pocket a few cells across, one merge edging another — and the power law is present where those happen to line up with the region and absent where they do not.

What seven published numbers become #

Every measurement on this ground that counted pieces or read passes used the same grid, so the copies reached back further than the curves. The table is each published headline computed again with the copies joined, and the measurements divide into those that survive and those that do not.

The fracture itself survives. The threshold at which the near-optimal set first comes apart, and its fall as a region grows, move by a per cent or two: where the valley breaks in two found 2.13 at a 6° region, and corrected it is 2.10. The piece counts do not survive in their published size. The fourteen basins the shape of the valley found at twice the best score are two; twelve were copies. The set does break into many — thirteen pieces by 1.7 times the best — but later than was said.

The pass between the two deepest basins moves closer to its prediction, not further. The argument predicted an exponent of −1; the pass was published at −1.60 and read as moving the answer away from the argument. Corrected, it is −1.26, nearer −1 than the level-set measurement’s −1.33.

Everything built on the pass depth being constant goes. The first break is mostly its denominator found the absolute depth constant to twelve per cent across small regions and drew its conclusion from that; corrected, the spread is forty-three per cent. The share of the threshold’s scaling carried by the denominator survives, 69 per cent against 71. The claim that the pass which fails first stands far above the deep pair — 1.5 to 9.8 times — was mostly copies: corrected, 1.0 to 2.1.

It was seen, and its size was guessed #

The pole-row copies were not invisible. The shape of the valley, the measurement that first counted the near-optimal set’s pieces, listed its five best basins and noticed that three of them sat at a pole latitude of −90° with identical scores. It named the cause correctly — at the pole, longitude and turn combine into a single parameter, and the grid holds many points that are the same map — and then declined to divide it out, because doing so meant treating the poles as a special case and the effect on the counts was, it said, a few pieces out of fourteen.

It was twelve pieces out of fourteen. And the counts were the least of it: every later measurement that read a pass off the same grid read the copies’ meeting as a pass, and four of them fitted power laws to it.

The mistake is worth stating exactly, because it is not carelessness. The degeneracy was diagnosed, correctly, by somebody looking at the right numbers. What was not done was the one computation that would have sized it — run the count once with the copies joined and once without — and a size that is guessed is a size nobody checks again. The antipodal pairing, the other way the grid holds one map twice, was divided out, because its copies are two and far apart and the fix was one line. The pole copies are twelve, close together, on the edge of the grid, and the fix looked like a special case. The easy degeneracy was repaired and the awkward one was estimated.

The two are not quite the same kind of fact, and the difference is worth having. The antipodal pair are two different rotations of the sphere that draw one picture, the second turned upside down — every projected coordinate negated — and they score alike because every projection here is symmetric under that half-turn. The pole-row copies are one rotation written in many ways, identical to the last digit of every coordinate, which is gimbal lock. A search over three angles has to identify both before anything it counts is a count of maps, and report the map, not the parameters is the same rule applied to an answer rather than to a landscape.

How the correction was checked #

The identity must be exact. Every pair of pole-row cells joined as one map must score alike to the arithmetic’s noise; across Robinson over Japan the worst relative difference is $1.2 \times 10^{-11}$.

A copy merge must sit exactly on the step. Where a published first pass was copies meeting, its score must equal the neighbouring map’s to nine decimals. On Robinson all 28 do; a merge that only happened to be close would not.

The correction must matter, and must be refused where it should not. Robinson’s curve must lose most of its $R^2$ once the copies are joined, and it goes from 0.981 to 0.008. The sinusoidal’s copies must decide at most a sixth of its regions, and they decide three of 36; a correction that moved the one curve the copies never reached would be correcting something else.

Nothing else may move. Every figure on this ground that reads the aspect grid was drawn again with and without the copies, 292 of them across forty essays, and the 55 that changed all belong to measurements that count pieces or read passes; none that ranks, scores or fits a map moved.

What the corrected grid still does not settle #

Pockets remain. A merge is counted when both pieces have at least three cells, so a hollow of three poor aspects high in the landscape counts as a basin. On the sinusoidal those pockets decide most of the first passes, and a rule charging a basin for its persistence rather than its size would change what “first” means. That is a choice about the instrument, and it is not made here.

One grid. Everything above is on 24 × 13 × 12. The worst point is not on the grid is the warning that a quantity read off a sampled landscape is a statement about the sample, and the family’s scattered corrected fits say how strongly that applies to a first pass.

The published essays are corrected where they stand. Each of the measurements in the table says in its own text what changed and why, rather than carrying a note pointing here.

Still open: whether a pass depth is a property of anything #

Joined, the copies leave three projections with a curve and three without, and a continuous family whose curve comes and goes between neighbouring members. The first reading is that the first pass is not a stable property of a projection on a grid this coarse. A second is that some of the six have a landscape whose highest merge is a genuine feature — a ridge between two real placements — and others have one whose highest merge is a pocket, and that the difference is real and the family simply passes through both many times.

The measurement that would tell them apart is the one this essay stopped short of: a rule that counts a basin by how deep it is below the pass that ends it rather than by how many cells it holds, run on two grids. Under it, a curve belonging to the projection would survive the change of grid; a curve made of pockets would move with it.