A density that asks for no room at all

Every density on this ladder has been positive everywhere. That is not a modelling choice; it is a requirement of the construction, which sends a point to the position its conditional cumulative puts it at — and a conditional cumulative is a quotient with the column’s own mass underneath.

A column with no mass in it is zero over zero.

The failure, measured rather than argued #

Handed a density that is exactly zero away from its islands, the triangular construction returns a map that is not a map. Of seventy-seven probes spread over the drawable part of the rectangle, twenty-eight come back as no number at all.

The refusal that makes that a finding rather than a bug report is the same density with a floor of one fiftieth of the peak added everywhere: zero probes undefined, out of the same seventy-seven, from the same code. What fails is the zero, not the machinery.

It is worth being precise about which zero. A density built from smooth bumps never reaches zero in floating point — a bump of sharpness twenty-four is at 10⁻²¹ of its peak on the far side of the world, which is small and is not nothing, and a quotient of two numbers that size comes back finite and meaningless. The zero here is made real by a cut: below one per cent of the peak the ground is empty. That threshold is stated because the whole rung turns on it.

What a practitioner does, and what it costs #

Nobody ships a map with holes in it, so nobody uses a density with a zero. What is used is a floor: a small constant added everywhere, so that the ocean has a little population and the arithmetic works.

The floor is presented, when it is mentioned at all, as a regularisation — a numerical convenience with no bearing on the answer. It is not.

The arithmetic is not subtle and that is what makes it damaging. The empty ground is most of the sphere, so a floor multiplied by most of the sphere is most of the total, however small the floor is per unit area. A floor of one two-hundredth of the peak — genuinely tiny — still gives the ocean 19.8 per cent of the map.

What the floor says about the sea, stated as a density #

The floor column of that table is quoted as a share of the peak, which is the number a practitioner types and the number that makes it sound harmless. Converting it into what it claims about the ground is one division and the results are not harmless at all.

The empty ground is 83.7 per cent of the sphere and the islands are 16.3. So a floor whose share of the total is s is asserting a mean density over the sea of s/0.837 and over the land of (1 − s)/0.163, both in the same units of total-per-unit-area, and the ratio of the two is what the map is saying about how populated the sea is relative to the land:

Read that way the table stops being a numerical convenience and becomes a set of claims about the world, every one of which somebody would refuse if it were written in a sentence.

At the floor that keeps the map legible, the sea is being drawn as two and a half times as populated as the land. Not slightly populated, not a small correction — denser than the islands the map exists to show, on average over its whole area, by a factor a reader would call absurd. A fifth of the peak sounds like a small number because the peak is a peak; spread over five sixths of the world it is more than everything the data contains.

The middle row is the one to hold. A floor of two per cent of the peak is the smallest number in the table that anybody would describe as negligible, and it says the sea is a fifth as populated as the land — which is a specific, checkable and completely false claim about a real quantity, arrived at by a choice made for the look of a picture.

And the last row is the honest end of the trade, at which the sea is a fiftieth as dense as the land. That is still not zero, it is still a claim, and it costs a hundred and fourteen degrees of angular deformation to make.

The trade, and neither end is available #

Read the two columns together and the choice is stark. A floor large enough that the map keeps its shapes — a fifth of the peak, at fifteen degrees of mean angular deformation — is a floor at which nine tenths of the quantity being mapped is fictional. A floor small enough to be honest about the data — a five-hundredth — costs a hundred and fourteen degrees, which is more than any density on this ladder has cost by any construction.

The trade is not a compromise between two goods. It is a choice about which of two lies to tell, and a published cartogram tells neither the reader nor, usually, its own author which one was chosen.

One caution about reading those ratios. They are means over each region, so a floor that makes the sea denser than the land on average does not mean any point of the sea is drawn denser than any point of an island — the islands’ peaks are far above anything the floor produces. What the ratio measures is the total, which is the quantity a cartogram’s areas are shares of, and the total is the only thing the areas can be read against.

The islands, which is where the request is real #

It is worth saying what the map does succeed at, because it is not nothing.

Nothing in the construction degrades gracefully towards the zero. It is exact right up to it — the areal residual on a floored density is the same five parts in a million as on any other — and then it stops existing. That is the shape of the failure worth carrying: the floor is not buying accuracy, it is buying existence, and every number the map reports is exactly right about a density that is not the one anybody meant.

What is happening to the empty ground #

At a floor of zero the smallest areal factor is zero, and that is the whole failure in one number. A map with a vanishing Jacobian on a set of positive area is not injective there — it has sent a region of ground to a curve — so it has no inverse, and a reader cannot ask where on the map a place is.

That is the same failure a map that cannot be read backwards prices for a projection, arriving from the opposite side. There the loss of an inverse is a property of a formula that somebody wrote down for other reasons; here it is a direct consequence of a request, and the request is the honest one.

It is also worth separating from folding. A map that meets its target can fold is about a determinant that goes negative, which a discretisation does and the continuum construction cannot. This is a determinant that goes to zero, which the exact construction does, on purpose, because it was told to.

The three maps, drawn #

The middle panel is the one that looks like a published cartogram, and its ocean holds 72 per cent of the mapped quantity.

The three panels are also a warning about reading a cartogram by eye. Nothing in the pictures marks which is which: all three meet their own stated density exactly, all three are drawn by the same code from the same islands, and the difference between them is a constant that was never on the page. A reader shown the left panel would conclude the islands are a small part of the world, and shown the right one that they are most of it, and both conclusions are about the floor.

What a cartogram could do instead #

Three options, and the arithmetic above prices all three.

State the floor. It is one number, it belongs in the caption beside the density, and with it a reader can compute the ocean’s share for themselves. Every figure in this anchor names its density in the same breath as it names its projection, which is the habit this collection was built on; a floored density is a different density and should be named as one.

Cut the empty ground out. A cartogram of the land alone has no zero in it, because land is inhabited everywhere at some level — though the boundary between two features is then doing work it was not drawn to do. The price is that the map is then interrupted — the ocean is a tear rather than a region — which is the trade giving up continuity prices for an ordinary projection, arriving here for a different reason.

Or draw the empty ground at a stated size rather than at a stated density: fix the ocean’s total page area by construction and redistribute only the rest. That makes the areal factor piecewise-defined and the map non-smooth at the coastline, which is a real cost and is at least a cost somebody chose.

None of these is free, and the point of pricing them is that the fourth option — a floor nobody states — is not free either and is usually the one taken.

Why the ocean cannot simply be ignored #

The tempting answer is that nobody cares what the ocean looks like, so let it be squeezed and move on. That answer fails on the areas, which is the one thing a cartogram exists to get right.

A cartogram’s areal factor is relative: a region is drawn at an area proportional to its share of the total. So the ocean’s share is in the denominator of every island’s area. At a floor of a fifth of the peak the islands are sharing 7.5 per cent of the page between them; at a floor of a five-hundredth they are sharing 91.1. The same two islands are drawn at areas differing by a factor of twelve, from the same data, by a choice nobody wrote down.

That is a stronger statement than the shape argument and it is easier to check. Two cartograms of one dataset, made by two people who each picked a sea density they liked the look of, do not merely differ in their outlines — they differ in the ratio a reader is being asked to read off them, and neither map says so.

The comparison Mercator against Peters draws between two projections is a comparison of two stated formulae. Here the two maps have the same formula and different unstated inputs, which is worse in exactly the way an unstated input always is.

What a reader can check from the outside #

None of this needs access to the code that made a cartogram, which is worth saying because most published ones do not come with any.

The ratio of the largest drawn region to the smallest is measurable off the page, and comparing it with the ratio in the data is a one-line test. If the drawn ratio is much smaller, a floor has been applied and its size can be recovered from the difference.

The empty ground’s drawn area is measurable off the page too, and the honest question — what share of the mapped quantity does this map assign to places that have none of it — is answerable by anybody with a planimeter and the total.

And the smallest drawn region says whether the construction was near collapse: a region drawn at a fiftieth of its ground area is a region whose Jacobian is within a factor of fifty of zero, and the map beside it is correspondingly deformed.

Three readings, none of them requiring the algorithm, all of them absent from every published cartogram this collection has found. What makes them worth listing is that the defect they detect is invisible: a cartogram with a large floor is a well-behaved picture, and the thing wrong with it is what it means rather than how it looks.

Where the model stops #

The threshold that makes the zero real is stated at one per cent of the peak and the qualitative failure does not depend on it: any cut that leaves a column with no mass in it breaks the construction, and 83.7 per cent is the share at this cut.

A cartogram is judged on more than its areas, and everything the fourth rung measured about distance, bearing and betweenness gets worse as the floor falls, in step with the angular deformation rather than independently of it.

The floors here are applied uniformly. A practitioner who knows the data may floor only the empty ground, which changes the shares in the table and changes nothing about the trade — the empty ground is still most of the sphere, so a floor on it is still most of the total.

And the construction is the triangular one throughout. A diffusion cartogram handles a zero differently and no better: its flow has the density in the denominator too, so a zero is an infinite velocity rather than a division by zero, and what it produces is a map that has not converged rather than one that is undefined.

The generalisation #

This is the same accounting the attribute is a claim about the geometry asks for on the other side of the collection: a number stored beside a shape means nothing without the area it was divided by, and here the area has been quietly altered before the division.

A regularisation added to make an algorithm run is a change to the question, and its size is not the right measure of how large a change. What matters is the regularisation’s total weight over the domain it is applied to, and when it is applied to the part of the domain that is largest, a small constant becomes most of the answer.

Stated that way it is not about cartograms at all. It is the same failure as a prior that is flat over an unbounded parameter, or a smoothing term applied over a region where the data is absent — and in every case the diagnostic is the same one this rung uses: compute what fraction of the result the regularisation is responsible for, and report it beside the result.

For a cartogram that fraction is a single number, it costs one integral, and no published implementation prints it.

Who found it, and when #

The problem is as old as the technique. Every cartogram of population meets an ocean, and the published algorithms handle it: the diffusion method’s original description adds a uniform “sea” density outside the region of interest and says so, with the observation that its value affects the result. That is the floor, named, in the paper that introduced the most-used construction.

What has not been done, as far as this collection can tell, is the accounting. The sea density is chosen for the look of the output and its share of the mapped total is not reported, so a reader of a published cartogram cannot tell whether they are looking at a map of a population or a map of a regularisation.

Where the ladder goes next #

Six rungs in, this anchor has priced what a request costs, who can meet it, what the cheapest map is, and what happens when the request has a hole in it. What it has never asked is what happens when the density is not known exactly — when it is an estimate with an error bar — and whether a cartogram’s areas inherit that error or amplify it.