The area weighting was a readership all along

The average was a choice of norm completed a sequence. The rule of thumb scored tested the received advice, the tolerance that decides the verdict showed the threshold was free, which projection a weighting can make best showed the weights were free, the ranking is not an order showed the aggregation could produce cycles, and the norm essay showed the exponent that averages a field is a free parameter with no measurement able to fix it.

Twelve measurements, and each of them freed one ingredient and watched the answer move.

Every one of them held the same thing fixed, and none of them mentioned it. The score at each point was multiplied by $\cos\varphi$ — the area of the ground under it — and the numbers were summed. That is the area weighting, it is what the geometry supplies, and it looks like the absence of a decision.

It is not. It is a statement that every square kilometre of the Earth has an equal claim on the map, which is to say that the people who will read the map are spread uniformly over the globe. Nobody is.

A readership is not the same kind of freedom #

The distinction matters more than the numbers, so it is worth making before them.

A criterion has no fact of the matter. Angular deformation and areal inflation are both real properties of a map and there is no measurement, anywhere, that decides which a comparison should use; the aggregation exponent is the same; so is the tolerance. Those twelve measurements are each of the form this looks like a question about maps and is a question about the person asking.

Who reads a map is not like that. It is a question about the world. It is badly known and it changes and it is not the sort of thing a geometer is used to measuring, but there is an answer, and once it is answered the weighting is determined rather than chosen. So this is the one place in the whole sequence where naming the comparer’s freedom could in principle remove it.

The measurement is what that costs, and it comes out in two parts: how far the verdict moves when the weighting stops being flat, and whether the earlier freedoms matter less once it has.

The five are a flat weighting, a broad land weighting, a concentration on the world’s largest urban regions, a band across the North Atlantic, and one country’s schoolrooms. Each carries a floor set so one reader in twenty is somewhere outside every lobe, which keeps some readers are elsewhere true at every width — and is the difference between a readership and a mask.

How much of the world a map is about #

The number the whole comparison is read against is not the shape of a lobe but its participation share: the smallest share of the sphere’s area carrying half the total weight. A flat field gives exactly a half, by construction. A point gives nearly nothing.

It reads directly as how much of the world this map is really about, and putting the flat weighting on the same axis is most of this essay’s argument. A comparison that weights by area is claiming that half of what matters lies on half the globe. A sailing chart of the North Atlantic is claiming that half of what matters lies on four per cent of it. Both are claims, and only one of them has ever been written down.

Four winners #

Under a flat weighting the library’s first three are Winkel tripel, Robinson and the equirectangular, which is the ordering every general-purpose comparison arrives at and the reason those maps are in atlases — the plate carrée is the third of them, and it earns its place by being wrong evenly rather than by being right anywhere. Under the other four readerships:

• wherever there is land — Lambert’s conformal conic wins, with Albers second and Winkel tripel third;
• the world’s largest urban regions — Robinson wins, Winkel tripel second;
• the North Atlantic trades — the American polyconic wins, the azimuthal equidistant second, and Winkel tripel is fifth;
• one country’s schoolrooms — Winkel tripel wins again.

Four distinct winners from five readerships. The two conics arriving at the top of the land weighting is the recognisable part — a conic is a regional projection and a broad land weighting is a regional question wearing a global library — and designing a grid for one region derives that from the geometry rather than from a ranking.

Nine of ten move, and some of them move a long way. The equirectangular goes from third to fourteenth, Gall–Peters from ninth to sixteenth, Behrmann from tenth to seventeenth. The cylindrical maps lose together and for one reason: a band across one ocean at mid latitudes is answered by a projection that puts its good ground there, and a cylinder’s good ground is the equator.

What the default costs #

Against the cities readership, Winkel tripel is second and two per cent worse than Robinson — a difference nobody would act on. Against the land readership it is third and 10.6 per cent worse. Against the North Atlantic readership it is fifth of twenty and 32.8 per cent worse than the American polyconic.

That spread is the honest shape of the result. The flat weighting is not a systematically bad choice; it is a good answer to one question, and how wrong it is depends entirely on how far the real readership is from being spread over the whole globe. A general-purpose world atlas is close to the flat case and loses little. A chart for one ocean is not, and loses a third. That ordering is worth holding against the projections that are beaten on both counts, which found maps nothing can recommend: a map can be dominated outright, or it can be merely wrong for this reader, and the second is much commoner and much harder to see.

Tightening one readership moves it twice #

The five fields differ in where their lobes are as well as in how tight they are, so the clean test is to take one and tighten it.

From a participation share of 10.8 per cent down to 0.75 per cent the American polyconic holds the top place, at margins between 11 and 39 per cent. Below that Winkel tripel takes it back and keeps it to the tightest field measured.

The reversal is not an artefact. A polyconic map has true scale along every parallel and is built for a region that is wide and shallow, which is what a moderately tight readership is; a readership a tenth of a degree across is answered by whichever map happens to be locally well-behaved at that spot, and the whole shape of the projection stops mattering. So the sweep has three regimes — global, regional, and local — and the middle one is the only place where the answer is about the projection’s construction rather than about the globe or about a point.

The margin’s non-monotonicity says the same thing from the other side. It is widest in the middle of the sweep and narrows at both ends, because at the global end every map is being asked to do the same impossible thing and at the local end every map is nearly right.

Naming the reader does not remove a freedom #

This is the result the essay was written to find and it is the opposite of what it was written expecting.

Varying the criterion over its three natural settings produces three different winners — Winkel tripel, Mercator, Lambert’s cylindrical — which is the freedom five earlier measurements priced. Varying the readership over five stated fields produces four. The readership is at least as large a source of disagreement as the criterion, so naming it does not shrink the space of defensible answers; it adds another dimension of the same size.

What it does change is the kind of disagreement. Two comparisons that differ because one used angular deformation and the other used areal inflation have nothing to settle between them and no experiment that would. Two that differ because one assumed a global readership and the other assumed an Atlantic one have a disagreement about the world, and somebody could go and find out. That is not a smaller problem. It is a different one, and it is the only disagreement in this sequence that is in principle closable.

The adjudicator had taken a side #

There is a version of this argument that is a century old and was not about comparisons at all.

Mercator against Peters sets out the dispute, and the projection that shows true size is where the claim at the centre of it gets measured. The Peters position, stripped of its rhetoric, is a claim about weighting: that a square kilometre in the tropics has exactly the same claim on the page as a square kilometre in Europe, and that a map which enlarges one over the other is making a statement about whose ground counts. Whatever one thinks of how it was argued, it is a coherent position and it is precisely a claim about the weight.

The area weighting says the same thing one level up. It says that a square kilometre of distortion in the Southern Ocean counts exactly as much as a square kilometre of distortion over Paris, when the question is which map is best. That is the Peters position applied to the comparison rather than to the map — and every one of the twelve measurements before this one, including those that scored Peters’ own projection and found it wanting, was computed under it.

This is not an accusation of bias and it is not a defence of Mercator, whose own problems web Mercator is not conformal and conformal does not mean the angles are right set out at length. It is a narrower and more awkward observation: the instrument built to adjudicate a dispute about weighting has a weighting in it, and it is one of the two positions in the dispute. A comparison run under a cities weighting would be running under something much closer to the other. Both are defensible; neither is neutral; and calling one of them “the geometry” is what hid the choice for twelve measurements.

When the flat weighting is right, and it is not never #

The honest counter-argument deserves its own statement rather than a concession in a closing paragraph, because it is good.

A general-purpose map is one whose readership is not known. That is not a gap in the specification — it is the specification: an atlas plate, a classroom wall map, a map in a newspaper about a place the reader has never thought about. For such a map a flat weighting is not a guess about readers spread evenly over the globe; it is the answer to what should be assumed when nothing is known, which is the position of maximum ignorance rather than a claim about the world.

That defence works, and it has two conditions that are worth naming because neither is automatic.

It requires that the readership really be unknown rather than merely unstated. A world atlas published in one country, in one language, to one syllabus has a readership its publisher could describe in a sentence, and unstated is not unknown.

And it requires that the loss be linear in the weight — that being twice as wrong over half as many readers is a wash. The average was a choice of norm is the measurement that makes that assumption visible, because the whole point of a $p$-norm above one is that it is not linear: a map which is terrible in one place and perfect elsewhere scores worse at $p = 4$ than the same total error spread evenly. Under maximum ignorance about where the readers are, a comparer who cared about the worst reader rather than the average one would not use a flat weighting either.

So the flat weighting is defensible for exactly one kind of map, under one aggregation, and the rule scored out of sample already showed how quickly a rule justified for one case gets applied to every other. The result here is not that it is wrong. It is that it is a choice with conditions, and that a comparison which does not state them has spent an assumption it did not declare — which is the same defect, in a different ingredient, that thirteen measurements have now catalogued.

What each number was checked against #

A readership spread over the whole sphere must reproduce the flat weighting exactly. The city field with its lobes widened a thousandfold is required to return the same winner as a constant field. It does. A difference there would mean the sampler and not the readership was deciding.

The participation share must separate the two ends, required below five per cent for the tightest field and above 45 for the flat one. It gives 0.03 and 50.0.

Naming the readership must move the verdict, required as at least three distinct winners across the five fields, because two could be a single swap. It gives four.

And tightening one readership must move it too, so that the effect is concentration rather than the choice of lobe. One country’s field at three widths gives two different winners.

Two defects were found by these controls rather than by inspection, and both had produced a published-looking answer before they were caught.

The first was the sampler. A 60 × 31 latitude–longitude grid has cells about six degrees across, so a readership two degrees wide falls between the samples and the comparison scores a field it never evaluated — the first run reported a cylindrical map winning a readership the size of Wales. The field splits exactly as a constant plus lobes, so the integral splits the same way, and each part is now sampled by a point set suited to it. That is the same lesson which of these numbers are the samplers draws about every estimator in this collection, arriving here from a new direction.

The second was the score. Kavrayskiy’s criterion is $\tfrac12[(\ln a)^2 + (\ln b)^2]$ in the two principal scale factors, which charges a map for being drawn at the wrong size as well as for being the wrong shape — and the library’s projections carry different normalisations. An azimuthal equidistant whose radial scale is $\pi/2$ at its own centre reads 0.32 there against a true distortion of zero. A map-maker chooses a principal scale, so the quantity that means anything is the departure from the best scale for this readership rather than from unity, and the weighted geometric mean of $\sqrt{ab}$ is now divided out before anything is compared.

Where the model stops #

The readerships are stated, not counted. They are analytic lobes at plausible places, and no number here depends on their being right about the world; what is being varied is concentration. A real census would change which projection wins each case and would not change that the winner moves.

Every map is in its normal aspect, and that is a deliberate restriction with a reason. A concentrated readership could be answered by aiming a map at it rather than by choosing a better one, which would be the stronger test — and this collection’s oblique aspects cannot be asked. distortion on an oblique projection returns a spurious Jacobian for every sample on the great circle through the rotation pole: the rotated longitude jumps by $\pi$ across that line, a finite difference straddles it, and the principal scale factor comes back near $10^5$. Aiming a map at its readers puts the readers exactly there. The defect is recorded; the aiming question stays open; and the finding stands on the weaker hypothesis, which is that the winner changes even when no map is allowed to point itself.

One criterion throughout. Kavrayskiy’s, because scoring on angles alone hands every comparison to Mercator — which is conformal, reads exactly zero everywhere, and is a true statement about angles and a useless one about world maps. The criterion is varied only in the one figure that is about varying it.

And no map is being recommended. The ranking is not an order established that the pairwise relation can cycle, and nothing here repairs that. What is added is that a ranking which does not say who the map is for has left out an ingredient the same size as the ones it does state.

Still open: whether anybody’s stated readership would survive being written down #

Thirteen measurements now stand between the received advice and a defensible verdict, and twelve of them freed something. This one is different in kind, and the practical question it raises is not geometric at all.

A publisher choosing a projection for a world atlas has a readership and could state it — a country, a language, a school syllabus. None of them does. What would settle whether that matters is not another sweep over stated fields but a comparison of two real ones: the readership an atlas’s sales figures describe against the readership its choice of projection implies. If those agree, the flat weighting has been a harmless shorthand for a genuinely global audience. If they do not, then the maps in question have been optimised for a reader who does not exist, by an amount this essay can already price — up to a third of the available quality, and a drop of four places in a field of twenty.