The maps with no family are simply better

The family is a symmetry, not a shape measures, from the finished maps and with no formula, which projections in this library are unchanged when the sphere is rotated about some axis. Fourteen are and seven are not, and the seven are Robinson, Winkel tripel, Eckert IV, Mollweide, sinusoidal, Hammer and the polyconic — which is very nearly the set anybody would pick for a world map.

That rung had no account of why, and offered one as a conjecture with a test attached:

A continuous symmetry is a strong constraint on where the distortion can go. A world map’s requirements are not distributed symmetrically. So the freedom the symmetric families give up is exactly the freedom a world map wants. What would test it is measuring how much of the seven’s advantage survives when the criterion is made rotationally symmetric. If the advantage largely disappears, the conjecture holds: their value is in placing distortion, not in reducing it. If the advantage survives, they are simply better maps and the symmetry has nothing to do with it.

This rung runs it.

The conjecture is refuted, and by its own author’s stated test. The second branch is what holds: the seven are simply better maps, and the symmetry has nothing to do with the land.

How the test was set up so it could fail #

A refutation is worth only as much as the fairness of the test that produced it, and four choices here were made to give the conjecture its best chance.

The symmetric maps may re-aim. The aspect is a free choice establishes that any projection can be centred anywhere by rotating the sphere first, and rung eight establishes that a symmetric projection’s axis is its symmetry. So a symmetric map facing an asymmetric criterion is not stuck with the poles: it can point its one axis at whatever the criterion cares about. Every projection here is scored at twelve candidate axes and keeps its best.

That is the whole of the freedom the conjecture said the symmetric family lacks, granted in full.

The weightings run from fully symmetric to fully asymmetric. Uniform over the sphere; a band in the mid-latitudes, which is symmetric about the polar axis; a twofold longitudinal variation; and a single concentration at 20°E 55°N, which has no symmetry whatever. They come from this collection’s own thematic library as stated analytic fields — there is no coastline here and there will not be one — and what matters for the test is their symmetry rather than whether they are the real world.

The criterion charges for shape and area together. Kavrayskiy’s, which is ½[(ln a)² + (ln b)²] in the two principal scale factors. Scoring on angular deformation alone hands the whole comparison to Mercator, which is conformal and reads exactly zero at every point under every weighting — a true statement about angles and a useless one about world maps, and it is what the first run of this test reported for all four bars.

And the comparison is best against best, not average against average. The question is whether the symmetric family contains a map as good as the asymmetric family’s best, which is the only form of the question a map maker cares about.

The instrument, and the one place it had to be corrected #

Every score here comes from sampling the sphere on a 72 × 37 grid to 84 degrees, computing the two principal scale factors at each point from the projection’s four partial derivatives, and averaging Kavrayskiy’s expression with the cosine of the latitude and the weighting field.

That is the same machinery distortion over a region uses, with two differences worth naming.

The weighting is a field rather than a region. A region is a set with a boundary and every point inside it counts alike; a field is a weight at every point of the sphere. The distinction matters here because the conjecture is about how symmetric the weighting is, and a boundary is a blunt instrument for varying that.

And the aspect is swept rather than fixed. Fitting the aspect to the region prices what re-aiming a projection buys for a region; here it is priced for a field, at twelve candidate poles spread over the sphere, and the best is kept. Twelve is coarse and it is enough: a finer sweep can only improve the symmetric maps’ scores, and improving them further would strengthen the refutation rather than weaken it, since the refutation is that they lose anyway.

What the rankings look like #

Three of the seven ahead of every symmetric map, on a criterion with no direction in it at all. That is the refutation in its plainest form.

Why the advantage narrows rather than widening #

The conjecture’s prediction was backwards and the reason is worth following, because it is the account that replaces it.

An asymmetric criterion is a criterion that cares about a small part of the sphere. Weight everything by a concentration twenty-five degrees across and the score is dominated by one region — and over one region every projection is nearly as good as every other, which two projections that cannot be told apart measures directly: below about twelve degrees of extent, two conformal projections cannot be separated from their graticules at all.

So a criterion that looks at a patch is a criterion that discriminates weakly, and the whole library bunches up. A criterion that looks at the whole sphere discriminates strongly, and the differences between projections are at their largest.

That is why the uniform weighting produces the biggest gap. It is not that the symmetric maps do badly where the land is; it is that they do badly everywhere, and the criterion that notices most is the one that looks everywhere.

Where the polyconic sits, and why that matters #

One member of the seven behaves nothing like the other six and it is worth separating before the account is written.

The American polyconic comes thirteenth of twenty under the uniform weighting, at 0.5449 — behind the plate carrée, Miller, Gall–Peters, Behrmann, Mercator and Lambert’s cylindrical, every one of which has a continuous symmetry. Under the concentration it is thirteenth again.

The projections that gave up being one thing is the rung about it: built from a different cone for every parallel, preserving nothing the usual tests look for, with an exact property neither of them measures and sheets that do not fit together. It is asymmetric for a reason that has nothing to do with compromise — it was built by a construction, and the construction happens not to commute with a rotation.

So the seven are two different things wearing one label. Six are compromise maps, fitted against a criterion, and they cluster at the top. One is a construction that happens to lack a symmetry, and it sits in the middle of the table. The classification rung eight measures is real and it does not by itself say which side of that line a map falls.

The account that survives #

If it is not about placing distortion where the land is, what is it?

A continuous symmetry is a constraint on the map, not on the map’s relationship to the land. A projection unchanged by rotation about an axis has every distortion quantity constant along the orbits of that rotation — so on a cylindrical map, the angular deformation and the areal factor are functions of latitude alone. That is a restriction on the shape of the distortion field, and it holds whether or not anybody is weighting by anything.

The restriction is expensive near the axis. Every point of a parallel must carry the same distortion, and a parallel near the pole is short while the latitude band it stands for is thin — so a symmetric map cannot taper. The pole is either a point, in which case the parallels above some latitude are crushed, or a line, in which case a point is drawn as a line; and either way the distortion near the axis is forced up by the constraint rather than by any choice.

An asymmetric map can taper in two directions at once. It has a two-dimensional shape function rather than a one-dimensional one, so it can bring the parallels in and shorten them, which is exactly what Winkel tripel, Robinson and Eckert IV do near their poles.

That account makes no reference to where anything is, which is why it survives the uniform weighting. It is also a special case of something entirely ordinary: constraining an optimisation makes the optimum worse, whatever the objective. The seven are better because they are less constrained, and they were designed by fitting a compromise rather than by imposing a construction — which is the same observation a family is a function, not a list makes one rung down, where searching a family’s free function beats every named member of it by twelve per cent.

What a conjecture with a test attached is worth #

Rung eight could have left the seven unexplained, or offered the conjecture without a test, or quietly asserted it. It did none of those: it stated the mechanism, said plainly that nothing in that rung tested it, and wrote down the measurement that would settle it either way.

That is what made this rung a day’s work rather than a research programme. The test was specified — score the seven against the symmetric family under a rotationally symmetric criterion, and see whether the advantage survives — and running it needed a weighting field, an aspect sweep and a scorer, all of which this collection already had.

It is also what makes the refutation clean. A conjecture without a stated test can always be rescued: the criterion was the wrong one, the weighting was not really asymmetric, the aspect sweep was too coarse. A conjecture whose author wrote down what would refute it cannot be rescued that way, and the honest thing on getting the wrong answer is to say so.

That habit is the one this whole collection runs on — every claim given a test it could fail — applied to a claim about the collection’s own reasoning rather than about a projection.

What the refutation costs and what it buys #

The conjecture was attractive and it explained something. Losing it is a real loss and it is worth being clear about what is left standing.

Rung eight’s measurement is untouched. Fourteen projections have a continuous symmetry, seven do not, and the classification is recovered from the finished maps with no formula. Nothing here bears on it.

The observation is untouched. The seven are still very nearly the set anybody would choose for a world map, and that still wants explaining.

Only the mechanism is gone, and it is replaced by a duller one: they are better because they have more freedom, and the freedom helps for reasons about the sphere rather than about the continents.

The duller account has a testable consequence the attractive one did not. If the advantage is about freedom rather than about placement, then it should scale with how much freedom — and the previous rung supplies that: within a single family, going from one free number to two buys 11.3 degrees of mean angular deformation and going from four to six buys 0.4. The returns to freedom fall away fast, which is why seven maps rather than seven hundred are worth naming.

One thing the conjecture got right #

Refuting the mechanism does not refute every sentence in it, and one of them turns out to be exactly true.

“If a projection is unchanged by rotation about an axis, then every quantity derived from it is constant along each orbit of that rotation.” That is not a conjecture; it is a consequence of the definition, and it is what the account below is built on. Where rung eight went wrong was the next step — assuming that the cost of that constraint is a cost in placement, so that a criterion with no place preference would not charge for it.

It charges for it anyway, because a constraint on the shape of a distortion field is a constraint even when nobody is asking where the field is large. That is the correction, and it is a correction to one inference rather than to the observation the inference started from.

The distinction is worth keeping because it decides what to do next. If the conjecture had been right, the way to a better world map would be to know more about where the readers are. Since it is wrong, the way to a better world map is to have more parameters — which is what a family is a function, not a list does within one family, and what the seven do across all of them.

What is still not explained #

Two things about the seven remain unaccounted for, and neither is what the conjecture was about.

Why exactly seven. Nothing here says the library should contain seven asymmetric maps rather than three or thirty, and the number is a fact about which projections this collection chose to implement rather than about projections. A library built from a different textbook would have a different seven.

Why those seven. Winkel tripel and Robinson lead every ranking here by a wide margin, and Hammer and the polyconic sit well down the list — the polyconic thirteenth of twenty under the uniform weighting, behind six symmetric maps. So “asymmetric” is not a quality mark: it is a necessary condition for being at the top and nothing like a sufficient one.

That second observation is the one worth carrying, because it corrects a reading the hero figure invites. The best asymmetric map beats the best symmetric one; the average asymmetric map does not beat the average symmetric one, and two of the seven are beaten by most of the fourteen.

What the four weightings say about criteria generally #

The four bars carry a second result that is not about symmetry at all, and it is the more portable one.

The gap between the best and the worst map in the library is largest under the uniform weighting and smallest under the concentration: from 0.2635 to 1.3933 under uniform — a factor of 5.3 — and from 0.2658 to 1.3072 under the concentration, a factor of 4.9, with the whole middle of the table bunched much closer. A criterion that looks at a patch of the sphere separates projections weakly; one that looks at the whole sphere separates them strongly.

That is a fact about criteria rather than about maps, and it has a practical edge. Which projection is best establishes that the ordering of projections is a function of the criterion and the region; this adds that the resolution of the ordering is too. A ranking computed over a small region is a ranking in which almost every projection is nearly tied, so the winner is decided by the third digit and moves when anything about the calculation does.

Which means a map maker choosing for a region has a genuinely easier problem than the ranking suggests: on a patch, most of the library is nearly as good as the best of it, and the choice can be made on things a distortion criterion does not measure — what the graticule looks like, what the sheets do at their edges, what everybody else uses.

Where the ladder has reached, and what comes next #

Twelve rungs have asked what a family is. A construction, which turns out to be a story about how projections were discovered. A one-parameter perspective formula, which holds six of eleven. A symmetry, which holds fourteen of twenty-one and explains the taxonomy. A condition, which pins three families and leaves the fourth a function to choose. A table of numbers, which is not a definition at all. And now a function space with a dimension, and a count of what each dimension is worth.

What every one of those has in common is that it is a statement about a single projection or about a set of them, drawn from the sphere to a plane, all at once.

Nothing in the anchor has asked what happens when the sphere is not what is being mapped — when the surface has a different shape, or the map is of one region joined to a map of another, or the thing being drawn is not a surface at all. The families were defined on a sphere and every measurement here assumes one.