Every density can be met and none is free

The previous rung built a map to a stated density and measured what it cost. It ended with a warning, and this rung is the warning made into a measurement: the map it built is one of infinitely many, and it is close to the worst of them.

Composing a cartogram with any area-preserving map of the page gives another cartogram of the same data. The areal factor of a composition is the product of the two areal factors, so multiplying by one changes nothing about the areas and everything about the shapes. That is not a loophole; it is the same freedom every equal-area map is every other one is about, arriving one condition further along.

What the freedom actually is #

Fixing the Jacobian determinant of a plane map fixes one function. A plane map is two functions of two variables, so what is left is a whole function of two variables’ worth of choice — an infinite-dimensional family, not a handful of parameters.

The simplest members to write down are compositions with area-preserving shears, which preserve the areal factor and nothing else. Take the finished cartogram and follow it with $(u, v) \mapsto (u + \varepsilon\sin\pi v,\; v)$. Its Jacobian is $\begin{pmatrix} 1 & \varepsilon\pi\cos\pi v \\ 0 & 1\end{pmatrix}$, whose determinant is one at every point, so the areas are untouched. Every shape is not.

The second family is more interesting because it needs no composition at all. The triangular construction conditions on $x$ first and then repairs $y$; run it the other way and it is an equally exact cartogram of the same density, built by the same two quadratures on the transposed rectangle. Nothing in the request distinguishes the two orders, and the two maps are not the same map.

Why area-preserving is a weaker condition than it sounds #

It is worth being precise about how large the family is, because “area-preserving” sounds restrictive and is not.

A map of the plane is a pair of functions. Requiring the Jacobian determinant to be one is a single first-order partial differential equation in those two functions, and a single equation on two unknowns leaves one unknown free — a whole function of two variables, not a finite list of parameters. That is the same arithmetic the site did for the conformal case and got the opposite answer to, and the difference is worth stating plainly.

Conformality is two equations on two unknowns, and it happens to be a system whose solutions are rigid: fix the boundary values of a conformal map and the interior is determined. That is why a conformal request can be refused. The equal-area condition is one equation, and one equation never determines two functions, so it can never refuse and it can never be pinned down.

The practical form of that statement is the one this rung is built on. Given any cartogram of a density and any area-preserving map at all, the composition is another cartogram of the same density — and there are as many area-preserving maps of the page as there are functions on it.

Four constructions, one density #

The spread is the result. A reader handed a cartogram and told that its areas are proportional to something has been told the truth and has been told nothing about the picture, because the picture was decided by a choice nobody recorded.

Three of the four are exact in area to between $5 \times 10^{-6}$ and $9 \times 10^{-6}$, which is the noise floor of the difference operator rather than a property of the constructions. The fourth is out by seven per cent and is the one in every atlas.

The one place the least cost can be written down #

“None is free” is a slogan until somebody produces a number that nothing can get under. Over the whole family that number is not available — the family is infinite-dimensional and a minimum over it is a calculus-of-variations problem this site does not solve. Within a symmetry class it is available exactly.

Take a radially symmetric density on a disc, where the azimuthal family is one function and the same reduction applies. The map that meets it and keeps the rotational symmetry sends radius $r$ to radius $R(r)$ with

$$R(r)^2 = \frac{2}{\bar f}\int_0^r f(s)\,s\,\mathrm{d}s,$$

and within the symmetric class that map is unique up to a rotation. Its principal scales are $R'(r)$ radially and $R(r)/r$ tangentially, so its angular deformation is a closed form at every radius — and because the map is unique, that is not one construction’s price. It is the price.

The numbers are 13.4° at a contrast of two, 29.9° at four, 49.2° at eight, 70.0° at sixteen and 90.6° at thirty-two. That is a bound, and it is the kind this site prefers: not a bound on what somebody’s algorithm achieves, but a bound on what any map of that class could achieve, derived rather than measured.

The bound is a real bound and not a fit #

The radial curve is drawn rather than fitted, and the difference matters for what can be claimed from it.

At radius $r$ the two principal scales are $a = R'(r) = f(r)\,r / (\bar f R(r))$ and $b = R(r)/r$, both in closed form from the density’s own cumulative moment. Their ratio is what the angular deformation is a function of, and every step from the density to the number is algebra. No sampling, no search, no optimisation over a family of trial maps.

What makes it a bound rather than one more measurement is the uniqueness. A radially symmetric density on a disc, met by a map that keeps the rotational symmetry, admits exactly one answer up to a rotation — because the symmetry reduces the two-dimensional problem to a one-dimensional one, and a one-dimensional area-preserving map with a fixed centre has no freedom left. The infinite family collapses to a point as soon as the symmetry is imposed.

The bound therefore says something a measurement cannot: for a symmetric request, no cleverness gets under this curve. Every construction in the fleet, every construction not yet written, and every construction that will never be written, all sit above it.

The bound is linear in the logarithm of the contrast #

The five numbers on the radial curve — 13.4°, 29.9°, 49.2°, 70.0° and 90.6° at contrasts of 2, 4, 8, 16 and 32 — are increments of 16.5, 19.3, 20.8 and 20.6 degrees per doubling. Above a contrast of about four the increment has settled, and the bound is

$$\omega \approx 20.2\,\log_2 C - 10.5 \ \text{degrees},$$

which reproduces the last four points to within a degree. Every doubling of the contrast costs twenty degrees of angular deformation and no construction can buy it back, for a symmetric request.

That form has a ceiling, and the ceiling is the useful part. Angular deformation is bounded above by 180°, which is the degenerate case where one principal scale has gone to zero and the map has collapsed a direction. Setting the expression to 180 gives

$$\log_2 C = 9.43, \qquad C \approx 690.$$

A radially symmetric density with a contrast above about seven hundred cannot be drawn as a cartogram at all — not badly, not with effort, at all: the bound says every map meeting it has points where one direction has been squeezed to nothing, and a map with a vanishing principal scale is a map with no inverse.

Seven hundred is not a large contrast for the quantities cartograms are drawn of. A population density runs from a city centre to an empty desert over five or six orders of magnitude, so a cartogram of raw population density is past the ceiling by a wide margin before anybody chooses a construction.

Which explains something the practice does without justifying. Every published cartogram floors, smooths or aggregates its density first, and the reason is usually given as numerical stability or legibility. The bound gives a stronger reason: past a contrast of a few hundred there is nothing to be stable about, because the requested map does not exist as a diffeomorphism. The flooring is not a convenience — it is what brings the request back inside the range where an answer exists, and what that costs is the rung this ladder pays it in.

Contrast is the variable, and it is not the only one #

The bound rises with the contrast, and so do the measurements: 86.0° for a contrast of seven, 103.8° at eleven, 126.1° at twenty-five, 139.1° at twenty-six, 143.8° at eighty-one.

But the ordering is not exact, and the pair at twenty-five and twenty-six is where it shows. Those two densities have nearly the same contrast and differ by 13° of worst-case deformation, because one has two bumps and the other four, so the second has twice as much boundary between crowded and empty ground.

That is the same lesson the cost of a density is set by its gradient rather than its value taught one rung down, arriving as a population statement rather than a pointwise one. Contrast is the first-order variable and the number of transitions is the second, and a report that quotes only the first is quoting half.

A choice nobody records #

The practical consequence is a documentation failure rather than a mathematical one.

A published cartogram states its variable, its source and sometimes its year — never the purpose the map is best for, and never its construction. It does not state its construction, and the construction is responsible for a factor of 1.81 in the only quantity a reader judges the picture by. Two teams handed the same table produce two maps that a reader would not identify as the same data, and both are correct.

This is report the map, not the parameters applied to a family the site had not met. The projection ladder learned it about aspect searches: the parameters that come out of an optimisation are not reproducible and the map they produce is, so the map is what should be published. A cartogram has the same shape of problem one level up — what should be published is not the algorithm’s name but the deformation it chose, and the deformation is measurable.

Why the exact ones are the ugly ones #

The triangular constructions are asymmetric by design. The whole correction in the first coordinate is applied globally, from a marginal that has integrated the other coordinate away, and only then is the second coordinate repaired conditionally. A density with structure in both directions is therefore handled well in one and badly in the other, and which is which depends on which quadrature ran first.

The two orders are the cleanest demonstration of the non-uniqueness available, because they are not a composition with anything. They are two applications of one recipe with one arbitrary decision made two ways, and they differ by half as much again in cost.

What a smoother construction buys #

The diffusion map is the gentlest of the four on the mean and it is not the gentlest on the maximum — on a two-bump density its worst point is 126.9° against the triangular map’s 123.9°, while being the better of the two over ninety-four per cent of the map. A maximum over a sampled grid is a statement about one sample; the area-weighted mean is the number a reader’s eye integrates.

The refusal, which is the flat density #

Every measurement in this ladder has to be able to return zero, and the case that returns it is the uniform density.

A constant areal request over a fixed region, once the total is matched, is the identity — the map is asked to draw every cell at the size it already is, and it does. The measured angular deformation of the redistribution is then zero to machine precision, on every construction, and the total deformation of the map is exactly the base projection’s own.

That is the refusal the cost measure needs. Without it, a figure reporting seventy degrees of deformation for a cartogram would be reporting an instrument that always reports seventy degrees, and the reader would have no way to tell. With it, the instrument is known to be capable of the answer nothing, and every non-zero reading is a statement about the request rather than about the arithmetic.

The same test catches a subtler failure. An early version of the audit reported a uniform density as costing 133°, which is a real number and is entirely the equal-area rectangle’s own deformation at the edge of the sampled band. The redistribution was costing nothing and the total was large, and reading the total as the cartogram’s price would have made every figure in this anchor a figure about the Lambert cylindrical projection.

Where the model stops #

The bound is a bound within a symmetry class, and the constructions measured against it are not symmetric. So what the figure shows is a curve nothing can get under for a symmetric request and a set of points that are not answering that request — which is honest and is weaker than it looks.

The true minimum over all maps meeting a general density is the solution of a variational problem: minimise an integral of $\log(a/b)$ subject to $ab$ prescribed. That is a real optimisation with a real answer and this site does not compute it, for the same reason the minimum-distortion solver for an arbitrary region has been deferred since the collection first met it. What is available is an upper bound from the best construction tried, a lower bound from the symmetric case, and the honest statement that the two are not close.

What this does to a comparison #

The consequence for anybody comparing two cartograms is sharper than it first appears.

Suppose two maps of the same variable disagree — one shows a region large and the other shows it larger still. The natural reading is that the underlying numbers differ, and it is very often wrong: both maps can be exact cartograms of the same table, differing only in which member of the family their software chose. The areas cannot disagree, because both are exact; what disagrees is every judgement a reader makes that is not an area.

So the comparison that a cartogram invites — this region has grown since the last edition — is only available if the two editions were built the same way, and almost nothing published says how it was built. A reader has no way to separate a change in the data from a change in the construction, and the second is capable of moving a region’s apparent shape more than any plausible change in the first.

The site’s own habit is the fix, applied here: every figure in this anchor names its construction beside its density, in the caption, in the same place a projection is named. Two of the four constructions above differ only in the order two quadratures were run, and a caption that omitted that would be omitting the whole reason the pictures differ.

Who found it, and when #

The non-uniqueness is Moser’s, from 1965, and it is the reason his theorem is stated as an existence result and never as a construction: the proof produces a diffeomorphism by integrating a vector field, and any divergence-free field added to it produces another.

The observation that this matters for cartograms is much later and much more grudging. The literature’s usual framing is that different algorithms produce different maps and that some are better than others, which treats the spread as a defect of the immature ones. It is not a defect. It is a whole function of freedom that the specification does not touch, and no amount of algorithmic maturity will remove it.

Where the ladder goes next #

Two of the four maps here were built by integrating something. The exact flow being integrated cannot fold — it is a diffeomorphism at every time — and every discretisation of it can, and does. The next rung is about the gap between those two sentences, and about the fact that the fold is exactly locatable rather than a numerical accident.