A mean that does not exist can still be printed

The previous rung dealt with an estimator that returns a lower bound and is presented as a value. This one deals with an estimator that returns a number and is presented as a measurement, when the quantity it is estimating does not exist.

The projection is Mercator and the quantity is the average of its areal distortion — the single most quoted number in any argument about map projections, and the thing every complaint about Greenland is a statement of. It has a closed form. The closed form has no limit.

The closed form #

Mercator’s areal factor is $\sec^2\varphi$. Averaging it over the sphere means weighting each place by the ground it covers, which puts a $\cos\varphi$ in the integral, and the cosine cancels one power of the secant:

$$\bar{s}(\Phi) = \frac{\int_{-\Phi}^{\Phi}\sec^2\varphi\,\cos\varphi\,\mathrm{d}\varphi}{\int_{-\Phi}^{\Phi}\cos\varphi\,\mathrm{d}\varphi} = \frac{\operatorname{artanh}(\sin\Phi)}{\sin\Phi}.$$

The numerator is $\int\sec\varphi\,\mathrm{d}\varphi$, which is the integral that produces Mercator in the first place — the projection’s own $y$ coordinate, evaluated at the edge of the band. So the average areal distortion of a Mercator map over a band is, to within a factor of $\sin\Phi$, the height of the map. And the height of a Mercator map of the whole sphere is infinite, which is the one fact about the projection that everybody already knows.

Every halving of the remaining gap to the pole adds the same amount, and the amount is $\ln 2 = 0.6931$. Measured over six halvings the increments are 0.6694, 0.6853, 0.6907, 0.6924, 0.6929 — converging on $\ln 2$ from below, which is what a logarithm’s increments do and what a convergent sequence’s increments never do.

What a sampler returns instead #

A sampler is not given a band. It is given a sphere and a resolution, and it returns a number.

A cell-centre rule with $n$ samples in latitude puts its outermost sample half a step from the pole, where the areal factor is finite, so nothing overflows and nothing warns. The number comes out, it has three or four plausible digits, and it is the answer to a question nobody asked.

The relation is $\bar{s} \approx \ln n + 1.512$, fitted slope 1.0010 against $\ln n$. The number is the logarithm of the sample count. Any value on that line can be published as the mean areal distortion of Mercator; a coarse study and a careful one will differ by a factor of three; and the more careful study will report the larger distortion, which reads as the more alarming result and is nothing of the kind.

Where the 1.512 comes from #

The fitted line is quoted as ln n + 1.512 and the intercept is left as a fitted constant. It is not one: it is two terms, and separating them says something the slope does not.

A cell-centre rule with n cells in latitude has a step Δ = π/n and puts its outermost sample half a step from the pole. The numerator of the mean is Σ sec φᵢ · Δ, and it splits into two pieces. Everything up to the last cell approximates ∫ sec φ dφ out to π/2 − Δ, which is ln(2/Δ). The last cell contributes Δ · sec(π/2 − Δ/2) = Δ / sin(Δ/2), which tends to exactly 2 — the same 2, whatever the resolution, because the cell’s width shrinks at precisely the rate its integrand grows.

The denominator is 2, so the mean is

$$\ln\frac{2}{\Delta} + 2 \;=\; \ln n + \ln\frac{2}{\pi} + 2 \;=\; \ln n + 1.548,$$

against a fitted 1.512. The small shortfall is the asymptotic being asymptotic; the structure is exact.

Two readings follow and both are sharper than the fitted line.

Half the constant is the poles and it never goes away. The two polar cells contribute exactly 2 to a mean whose whole value at eight samples is 3.569 — so at that resolution the two outermost cells out of sixteen supply more than half the answer. Refining the sampler adds ln n to everything else and leaves the polar contribution at 2 forever, which is a stronger statement than the number grows: the number grows because the rest of the sphere is slowly catching up with two cells that never change. The two cells the reader would most confidently call negligible are the ones carrying the statistic.

And the crossover has a place. The polar term exceeds everything else while ln(2n/π) < 2, which is n < πe²/2 ≈ 11.6. So for any sampler with fewer than about a dozen divisions in latitude — which is a coarse but entirely ordinary choice for a quick figure — the reported mean areal distortion of Mercator is mostly a measurement of the two cells nearest the poles, and changing nothing but the resolution changes which cells those are.

That is the sharpest form of the rung’s complaint. A convergent statistic sampled coarsely is a rough answer to the right question; this one sampled coarsely is a precise answer to a question about the grid.

The comparison that makes it a finding #

The awkward part is that Mercator’s other summary numbers are perfectly well defined, so a reader has no way to tell from the shape of the output which kind of number is being handed over.

The Kavrayskiy number is the root-mean-square of the logarithms of the two principal scale factors. For a conformal projection the two are equal, so for Mercator it is the root-mean-square of $\ln\sec\varphi$, and the integrand is $\ln^2(\sec\varphi)\cos\varphi$. At the pole $\cos\varphi$ goes to zero linearly while $\ln^2\sec\varphi$ grows like the square of a logarithm, and a logarithm squared loses to a linear factor every time. The integral converges.

It settles at 0.52124 and stays there. So one summary of Mercator’s distortion over the sphere is a number and another is not, and they are computed by the same code, from the same derivatives, at the same points, and printed in the same table.

What was computed, and how #

The closed form for the areal mean is elementary and is checked against the quadrature at every band: at ±85° the formula gives 3.14329 and a two-hundred-thousand-point rule gives the same to six figures. That agreement is what licenses using the formula in the assertion, where a numerical mean of a divergent quantity would prove nothing.

The Kavrayskiy mean has no closed form used here and is computed by a cell-centre rule at two hundred thousand samples, checked by doubling. It is convergent, so doubling settles it.

The assertion that carries the rung is a statement about increments rather than about values, because a value cannot distinguish the two cases. It requires that the areal mean’s increment per halving stay above 30 per cent of its first one — a divergent sequence’s increments do not decay — and that the Kavrayskiy mean’s fall below 35 per cent of its first. A file that had mixed up the two integrands would fail it rather than print a plausible average.

The cutoff that is doing the work #

Every finite answer for the areal mean is the answer for some band, and the band’s edge is a choice.

The most consequential such choice in the subject is Web Mercator’s, which cuts the world off at 85.0511287798066° north and south. That latitude was not chosen for anything to do with distortion. It is the latitude at which the Mercator $y$ coordinate equals $\pi$, so that the map of the world is exactly square and a tile pyramid can be built on it — which is what the square costs the poles is about.

The mean areal factor over that band is 3.153. Over ±80° it is 2.474 and over ±89° it is 4.742. So the most widely quoted average distortion of the most widely used map in the world is a consequence of a decision about tile arithmetic, and it would be a different number if the tiles had been chosen a different shape. Zoom is a ladder prices the rest of what that decision fixed.

That is not an argument against the cutoff, which is a good decision for its own reasons. It is an argument for saying what the number is a number about. A figure quoted as the average areal distortion of Mercator is quoting a band it does not name, and the band is doing more work than the projection.

Which projections this happens to #

The divergence is not a property of being cylindrical, and it is not a property of having no pole. It belongs to one specific construction.

The equirectangular’s areal factor is $\sec\varphi$ rather than $\sec^2\varphi$, so after the area weight the integrand is bounded and the mean exists. The Miller cylindrical compresses the Mercator latitude coordinate by a factor of four fifths before stretching it back, which is a device with no property to recommend it except that the poles arrive at a finite height — and this is a consequence nobody lists among its advantages.

So the only member of the family without a mean areal factor is the conformal one, and it has to be. Conformality on a cylinder forces the parallel spacing to be $\int\sec$, which is the integral that diverges. The projection is the integral. Why Mercator exists derives that requirement from the rhumb line, and this is the same requirement read as a statement about statistics.

Why the divergence is not an artefact of the sphere #

An obvious objection: the areal factor is infinite at the pole, the pole is one point, and a single point should not be able to spoil an average. Sets of measure zero do not usually matter.

The objection is right about the point and wrong about the neighbourhood. The integrand near the pole is $\sec\varphi$ after the area weight has cancelled one power, and $\sec\varphi$ behaves like $1/(90° - \varphi)$, whose integral is a logarithm. A logarithm diverges. It diverges slowly — which is exactly why the sampled numbers look so reasonable and why nobody notices — but slowly is not the same as not.

The comparison with the Kavrayskiy integrand settles it. That one also has a singularity at the pole, and it is integrable, because taking a logarithm before squaring is enough to beat the area weight’s zero. The difference between the two cases is one logarithm, and that is the whole of it.

What survives, and what does not #

This does not touch the properties that make the invariants worth having. What survives a change of coordinates establishes that the areal factor is a property of the map rather than of the parameterisation, and it still is. The areal factor at a point is a perfectly good number at every point.

What fails is a specific operation applied to it — averaging over an unbounded quantity’s whole domain — and the failure is a property of the statistic, not of the quantity. So the repair is not to stop using the areal factor. The repairs available, in order of how much they cost:

Quote a band. “Mean areal factor 2.47 over ±80°” is a complete statement and costs four words. Every finite number in this essay becomes correct the moment its band is printed beside it.

Use a statistic that exists. The Kavrayskiy and Airy numbers were designed as summary measures of distortion and they converge for Mercator. The reason is not luck: both take logarithms of the scale factors before averaging, and taking a logarithm first is the standard repair for a multiplicative quantity with a heavy tail.

Report a quantile. Half of the sphere’s area lies below 30° in latitude, where Mercator’s areal factor is at most 1.33, and that number is stable under any refinement whatsoever. It is also a much more honest description of what most of a Mercator map is doing, and it is the statistic distortion over a region would report if it were asked for one number rather than four.

Never report a mean areal factor without saying which. The ellipses are a sample, drawn at a size somebody chose found the same shape of problem in the drawn indicatrix field: a published instrument with two unstated choices in it, both of which move the reader’s number. This is a third such choice and it is larger than either.

The number the argument actually needs #

None of this touches the comparison the Mercator argument is really about, and the distinction is worth being precise about because it is the difference between a broken statistic and a working one.

“Greenland looks as big as Africa” is not a claim about an average. It is a ratio of two areal factors integrated over two bounded regions, and both integrals converge, because neither region contains a pole. Mercator against Peters states it as a ratio of closed forms over latitude–longitude cells, and the projection that shows true size does the same arithmetic from the other side. Those numbers are exact, they do not move when the sampler is refined, and nothing in this essay weakens them.

What fails is only the attempt to compress the whole map into one figure by averaging. That operation is the one with the unbounded integrand, and it is also the one that gets quoted in a headline, because a single number is what a headline has room for. The well-posed version of the complaint is a comparison and the ill-posed version is a summary, and the ill-posed one is the one in circulation.

The same decomposition explains why the closed form and the quadrature agree so well over a band. Cut the sample off at 85° and the polar cell is gone, the integrand is bounded, and the midpoint rule is an ordinary well-behaved quadrature converging at second order. Everything unusual in this rung lives in one cell, and stating a band is exactly the act of removing it.

Where the model stops #

The sphere is the easy case. On the ellipsoid the areal factor of Mercator is $\sec^2\varphi$ times a bounded correction, so nothing changes about the divergence, and every number here moves in the fourth decimal. Nothing rests on that.

Nothing here is about the maximum. Mercator’s maximum areal factor over the whole sphere is infinite, which nobody has ever tried to quote as a number, and the previous rung’s estimator would report the value at the outermost sample — again the logarithm of the sample count, in a different disguise. The two failures are the same failure looked at through two statistics.

Who found it, and when #

The divergence of $\int\sec$ is the oldest computation in the subject. Edward Wright published the tables that make Mercator’s construction usable in 1599, by summing secants at one-minute intervals — a numerical quadrature of exactly this integrand, three quarters of a century before the calculus that would name it — and everybody who has used the projection since has known that the map has no top.

What is new here is nothing about the integral and everything about the statistic. The step from “the map has no top” to “the map has no average areal distortion” is one line, and it is a line the literature does not take, because the average is quoted in an argument about fairness and the infinite height is quoted in a discussion of construction, and those are different chapters. The ellipses are a sample found that Mercator’s placements-averaged areal factor is 5.45 against a ground-weighted 3.04. Both of those numbers are on the line $\ln n + 1.512$ at different $n$, which is the sharper way to say what that comparison was measuring.

Where the ladder goes next #

Two rungs have taken estimators apart while holding the sample fixed — a grid, a cell-centre rule, one particular arrangement of points. The next asks what the arrangement itself costs. There is no equal-area lattice on a sphere, so every sampler trades uniformity of area against uniformity of spacing, and the trade has a rate. That rung also reproduces, from the sampler’s own geometry, the one occasion on which this collection printed a measurement that beat a proved bound.