Everything else on the page pays for the areas

Three rungs have established that a cartogram meets its areal request exactly, that the map meeting it is one of infinitely many, and that a discretised route to one of them can turn inside out. All three are statements about the construction. This one is about the reader.

A map supports several readings at once, and naming which one it is for is the whole of an honest answer. Somebody looks at it and takes off an area, a distance, a direction, a shape and an order — which place lies between which — and does so without deciding in advance which of the five the map was built for. A cartogram is built for exactly one of them, and this rung measures what happens to the other four.

The calibration is fitted, and it still fails #

The measurement above is deliberately generous, and the generosity is the point.

A reader with a ruler needs a scale, and the honest thing to give them is the map’s nominal one. Instead every page distance here is divided by a factor chosen after seeing the answers, so that the middle pair of the twelve comes out exactly right. That is the best any single number could do on this set: no scale bar printed on this map could beat it, because this one was fitted to the map’s own answers.

Under that calibration the worst of the twelve is still out by thirty-five per cent. There is no scale a cartogram could carry that would make a ruler on it mean anything, and the failure is not a matter of degree — a thirty-five per cent error in a distance is not a distance measured badly, it is a different quantity.

A scale bar is right in one place makes the weaker version of this argument for ordinary projections, where the scale varies because the projection could not help it. Here the variation is the specification.

Bearing goes with distance #

A cartogram’s principal directions rotate — the orientation of an indicatrix is a measurement too — , because the construction has no reason to keep them fixed and every reason to swing them wherever the density’s gradient points.

That the ellipses point differently is the alignment measurement applied to a map whose alignment was never a design goal, and the answer is what that essay would predict: alignment belongs to the pair of map and region, and here the region is the density’s own shape.

Shape is the thing readers think they are keeping #

Ask somebody what a cartogram distorts and they will say shape. That is right and it understates the case, because shape distortion on a cartogram has a property ordinary distortion does not: it is largest where the reader is looking.

An ordinary projection’s worst distortion is at the edge of its useful range — the poles of a cylindrical map, the rim of an azimuthal one — and the subject of the map is usually somewhere else. A cartogram’s worst distortion is on the boundary between crowded and empty ground, because that is where the density’s gradient is, and the boundary between crowded and empty ground is exactly what the map was drawn to show.

So the deformation is not a cost paid at the margin. It is concentrated on the feature the map exists to communicate, and a reader comparing the shapes of two enlarged regions is comparing two of the most deformed patches on the page.

What a distance on a cartogram actually is #

It is worth saying what the ruler is measuring, since it is not nothing.

The page distance between two places on a cartogram is the length of a straight segment in a plane whose area element is the density. Integrated along that segment, what accumulates is neither ground distance nor mass — it is the length of the image of a curve under a map whose local scaling is anisotropic and varies, and the two principal scales along the way are not related to each other by anything.

There is one case where the number means something. If the segment happens to run along a principal direction of the map, and the density is roughly constant along it, then the page length is the ground length times a single scale factor and a reader who knew that factor could recover the ground distance. Neither condition is checkable from the page, and neither holds for eleven of the twelve pairs measured above.

So the honest description of the reading is that a page distance on a cartogram is a real number, computed from real geometry, that stands for no quantity anybody wants. That is worse than an inaccurate distance, because an inaccurate distance can be corrected and this cannot.

Adjacency survives #

Not everything fails, and what survives is worth naming precisely.

A cartogram of a positive density is a homeomorphism, so nothing that is one-to-one on the sphere stops being so locally: continuous, invertible, with a continuous inverse. So nothing is torn, no region is split in two, no two regions swap places, and neighbours stay neighbours. A reader tracing which regions touch which is reading the map correctly, and that is the strongest positive statement available about it.

That is why the family is worth having at all. A construction that broke adjacency would be a picture of nothing — a scatter of blobs, correct in area and unreadable — and the constructions here are chosen from among those that do not.

Betweenness does not #

Adjacency is a statement about touching; betweenness is a statement about a metric, and the metric is exactly what was thrown away. Ten of sixty triples change, and each change is a specific false sentence a reader might take off the map: this place is on the way from that one to the other.

The distinction matters because the two are often conflated in the defence of cartograms. “The topology is preserved” is true and it is a statement about adjacency, and readers do not read topology. They read positions.

The four readings, ranked #

Stated plainly, and in the order a map’s own machinery ranks them.

Area is exact by construction, to five parts in a million on the exact constructions and to between one and twenty-one per cent on the one everybody uses.

Adjacency is exact, structurally, for every construction in the family.

Shape is wrong by up to 139° of angular deformation, concentrated where the density changes.

Distance and bearing are wrong by tens of per cent under the best calibration available, and there is no calibration that fixes them.

Betweenness is wrong on about one triple in six.

That ranking is the map’s honest legend, and no cartogram prints it.

The five readings split three ways, and the split is structural #

The four failures are listed as four findings, and they are one finding, which is worth stating because it says the list is complete.

The areal factor is a pointwise quantity. It is a function of position, so specifying it constrains the map at each point independently of every other point — which is exactly what makes the specification satisfiable for any density at all.

Adjacency is topological. It survives because a homeomorphism preserves it by definition, and nothing about the areal request can threaten a property that does not mention distance.

And distance, bearing and betweenness are all integrals. A page distance is the length of a curve through many points; a bearing is the direction of one at its start; a betweenness is a comparison of two such lengths. Every one of them accumulates the map’s behaviour along a path, and the specification says nothing about paths.

So the three that fail are not three separate casualties. They are the whole of the class of readings that integrate, and they fail together for one reason: a pointwise specification leaves every path-dependent quantity free, and there is no fourth kind of reading for the list to have missed.

That also predicts which repairs are available and which are not. A reading that fails because a pointwise quantity is wrong can be repaired by a correction factor at each point — which is what an areal-factor correction does for a symbol map. A reading that fails because a path quantity is free cannot, because there is no factor: the correction would have to be a different number for every pair of endpoints, which is a table the size of the map squared and is the thing the map was drawn to replace.

Hence the ranking is a hierarchy rather than a list. The one reading a cartogram supports is the one its specification is about; the one it supports for free is the one no map can lose; and the three it destroys are the three the specification was silent on, all of which are the same kind of quantity.

Why this is a purpose argument and not a complaint #

None of the above is an argument against cartograms, and it would be a poor reading of this collection to take it as one.

Which projection is best settled the site’s position at the outset: no map is best, a map is best for something, and naming the purpose is the whole of the answer. A cartogram states its purpose more explicitly than any other map in the library — it is a map for reading areas, and it says so by construction — and the failure is not that it fails at the others. Every map fails at something.

The failure is that a cartogram looks like a map. It has a coastline-shaped outline, place names in roughly the right relations, and a page that invites a ruler. Every other projection in this library carries the same invitation and answers it with a scale bar and a graticule; a cartogram has neither, and nothing on the page tells a reader which of the five readings it will support.

The area is the only reading, and it is a good one #

Against all of that, the reading the map was built for is exact in a way almost nothing else on this site is.

An equal-area projection holds the ratio of any two ground areas, which is a strong property and is what makes the projection that shows true size worth its name. A cartogram holds something stronger: the ratio of any two page areas equals the ratio of the two masses, so a reader comparing two regions by eye is comparing the quantity directly, with no factor to look up and no correction to apply.

That is a comparison ordinary maps cannot support at all. On any projection whatever, comparing two regions’ page areas is comparing their ground areas times two different areal scale factors, and recovering the ratio of the underlying quantity needs the table the map was supposed to replace.

So the trade is real and it is not obviously bad. One reading is made exact and direct; four are made unavailable. Whether that is a good bargain is a question about the audience and the question, which is the only kind of question this collection is willing to answer about best.

What a cartogram could print #

The measurements here are all cheap, and three of them would fit in a legend.

The areal residual — measured rather than named — is what the construction achieved against what it was asked — five parts in a million or seven per cent, and the difference between those two is the difference between two families of algorithm.

The worst distance error under best calibration is one number and it tells a reader not to bring a ruler.

The construction — a name, in the caption, beside the variable — is the one that removes the factor of 1.81 the previous rung measured between two correct maps of the same table.

None of the three appears on any published cartogram known to this collection. The variable appears, the source appears, the year appears; the three numbers that say what the picture will and will not support do not.

Where the density is flat, nothing happens #

The refusal, and it is the one that makes every number above a measurement rather than an artefact of the instrument.

A uniform density is met by the identity. The cartogram of a flat request is the equal-area map underneath it, pair for pair, to machine precision — the same twelve distances, the same six positions, the same sixty triples. Every departure reported here is therefore a departure the density caused, and the machinery is capable of returning nothing and does.

That check has already caught one real error. An early version compared the cartogram’s distance errors against the equal-area base’s and expected the cartogram to be worse; it is not always, because the base map’s own distance errors on this set of pairs are fifty-six per cent, and the redistribution happens to pull two of the twelve pairs back towards the truth. The claim had to be restated as the density moves every pair rather than the density makes every pair worse, which is the true statement and the weaker-sounding one.

The reader’s other instrument #

There is a second thing a reader does with a map that no measurement above covers, and it is the one cartograms are actually judged on: recognition.

A map is searched by shape. A reader finds a place by knowing what its outline looks like and where it sits relative to two or three neighbours, and both of those are among the four readings that fail. That is why a cartogram is slow to read even when it is correct, and why the standard remedy — leaving the outlines recognisable at the cost of some areal accuracy — is a remedy for a real problem.

It is also why the previous rung’s factor of 1.81 between constructions matters more than it looks. Two exact cartograms of one table differ by nearly a factor of two in deformation, and deformation is what recognition costs. So the choice of construction, which the specification does not touch and no publication records, is the choice that decides whether the map can be read at all.

The site’s position on that is the one it takes everywhere: the quantity is measurable, so measure it and print it. cartogramAudit returns the areal residual and both deformation figures in one call over four hundred samples, in under a second. There is no cost argument for leaving them out.

Where the model stops #

Everything here is measured on one density and one construction, and the numbers would move for others. The rankings would not: area is exact for all of them, adjacency is structural for all of them, and the other three fail for all of them because they are the three degrees of freedom the areal request does not touch.

What is not measured is the reading a real audience actually takes. Whether a reader of a cartogram does try to measure a distance, and how often, is a question about people rather than about geometry, and this collection has no instrument for it.

Who found it, and when #

The complaint that cartograms are unreadable is as old as cartograms, and it is usually made as a complaint about shape. The betweenness failure is the sharper version and is rarely stated: it is what makes a cartogram hard to search, because a reader looking for a known place uses its neighbours and its position between landmarks, and one of those two survives.

Waldo Tobler, who spent forty years on the construction, put the defence in the right place: a cartogram is a map projection with an unusual specification, and the honest comparison is not against a good map but against the alternative way of showing the same data, which is a table.

Where the ladder goes next #

The anchor has four rungs and each of them is about a different consequence of one inversion — distortion specified rather than measured. What it has not done is close: the constructions are two of many, the bound is a bound within a symmetry class, and the reading measurements are made on one set of places. A fifth rung would take the variational problem seriously, which is the same deferral Chebyshev’s criterion has carried from the beginning and for the same reason.