The square costs the poles

85.0511287798066 degrees. The number appears in every description of the web tiling scheme, is usually given to two decimal places, and is usually explained as approximately where Mercator becomes unusable.

It is not approximate and it is not about usability. It is the solution of one equation, and the equation is about the shape of an image.

The equation #

Mercator’s northing, in the spherical form the scheme uses, is

$$y(\varphi) = R\ln\tan\left(\frac{\pi}{4} + \frac{\varphi}{2}\right)$$

and the world is $2\pi R$ wide, because longitude runs from $-\pi$ to $\pi$ and the projection multiplies it by $R$. For the projected world to be square, the northing at the cut must be half that width:

$$R\ln\tan\left(\frac{\pi}{4} + \frac{\varphi_{\max}}{2}\right) = \pi R$$

The radius cancels, which is the first thing worth noticing — the cut does not depend on the size of the Earth — and the rest is a rearrangement:

$$\varphi_{\max} = 2\arctan\left(e^{\pi}\right) - \frac{\pi}{2} = 85.0511287798066°$$

The site’s gate asserts the round trip rather than the formula: it computes $y(\varphi_{\max})$ and requires it to equal $\pi R$ to within $10^{-14}$ of a relative part, which it does. That is a check on the projection’s own code as much as on the constant, and it fails if either drifts.

What the cut costs, exactly #

Everything above the cut is a spherical cap, and a cap has a closed-form area: $2\pi R^2(1 - \sin\varphi)$. Two of them, over the whole sphere’s $4\pi R^2$, gives a fraction of

$$1 - \sin\varphi_{\max} = 0.0037279$$

which is 0.373 per cent of the Earth’s surface, 1,901,487 square kilometres. No dataset is involved, which matters here for the reason it matters throughout this site: the number has no error bar and no simplification level, and anyone can check it with a calculator.

What is not computed here is how much land that is, and the omission is deliberate. Land requires a coastline dataset, a coastline has a generalisation level, and an area read off one is partly a measurement of the vendor’s choices — the reasoning set out at length when this site refused a shapefile at foundation. The fraction of the sphere is exact; the fraction of the land is a different kind of claim and is left to someone with a different kind of source.

What is inside the caps #

The geography can be named without measuring it, and it is worth naming because it makes the trade concrete.

North of 85.05° is the central Arctic Ocean, including the pole itself and a great deal of the sea ice that is the most closely monitored surface on the planet. South of it is the interior of Antarctica: the South Pole, the ice divide, and most of the East Antarctic plateau.

So the scheme’s cut removes, from the default map of the world, the two regions whose science is most active and whose mapping is most in demand per square kilometre. Every polar data centre therefore serves its own tiles in a polar stereographic projection with its own scheme — which is the correct answer, and which means the two are separate mapping worlds that share no tile, no zoom level and no coordinate.

The alternatives, and the mistake in the usual argument #

The standard defence of the cut is that a square world is what a quadtree needs. That is half right, and the missing half changes the arithmetic by a factor of five hundred.

What a quadtree of square tiles needs is that the projected world be covered by an integer grid of equal squares at its root. Quartering any of them then gives squares, forever. A root of one tile forces a square world. A root of $m \times n$ tiles allows any world whose aspect ratio is exactly $m/n$ — and the geographic tiling, on the plate carrée, has used a two-tile root and a $2{:}1$ world since before the Mercator scheme existed.

So the question is not what cut makes a square but what cut makes a rational aspect, and the answers are computed from the same equation with $\pi$ replaced by $(m/n)\pi$:

A factor of five hundred for one extra tile #

The 2:1 row is the one worth staring at. A Mercator pyramid with two root tiles side by side — the arrangement the geographic scheme already uses — would have cut at 89.786° and dropped 3,558 square kilometres instead of 1,901,487.

That is a factor of 534, bought with one additional image at level zero.

The steepness is Mercator’s own doing. The northing climbs without bound near the pole, so extending the map by half of its own height in projected units buys 4.7 degrees of latitude, and the area of a cap falls as $1 - \sin\varphi$, which near the pole falls quadratically in the angular radius. Two multiplicative effects in the same direction give a five-hundredfold difference for a modest change in the shape of the picture.

What that would have cost instead #

It would be dishonest to present the alternative as free, and the costs are real if unglamorous.

Level zero stops being one image. A great deal of tooling assumes the root of the pyramid is a single tile at $0/0/0$, and every quadkey gains a prefix naming which root it descends from. The tree is still a tree; it is a forest of two.

The map is twice as tall as it is wide. A viewport showing the whole world either shows half of it, or shows it at half the zoom with wide empty margins east and west. That is an interaction cost on the one view a reader spends the least time in.

And the extra latitude is nearly all stretched beyond use. At 89.786° the scale factor is $\sec\varphi = 280$: a kilometre of ground occupies as much of the picture as 280 kilometres does on the equator, and the areal factor is 78,000. The ground would be present and would be unreadable, which is a different failure from being absent and arguably a worse one.

That last point deserves the weight, because it is the honest defence of 85.05°. A cut at 89.786° does not give a usable polar map; it gives a map on which the last four degrees of latitude occupy as much paper as the first sixty. The polar regions need a different projection, not more Mercator — which is what the polar data centres concluded independently.

The same decision, taken for a different reason, in the survey world #

There is a second truncated Mercator in wide use and its cut is at 84° north and 80° south. The near-coincidence of the numbers is a coincidence: nothing about the two choices is the same.

The zone system stops at 84° because the transverse Mercator’s series is a local construction and its accuracy degrades away from its own central meridian; the limit is a tolerance, chosen so that the published scale factor stays inside the accuracy the system claims, and beyond it a different projection takes over — the polar stereographic, in its own grid with its own conventions. UTM and the zone system works the mechanism through, and what a standard parallel buys is the argument for the scale factor the zone carries.

So: one truncation is a statement about error and the other is a statement about the shape of a picture. The first has a tolerance in it and can be argued about with numbers; the second has an aspect ratio in it and can only be argued about with a data structure. That two systems arrive within a degree of each other from opposite directions is the sort of coincidence worth naming, so that nobody concludes there is a natural limit to Mercator at 85°. There is not.

The asymmetry of the survey limits — 84° north against 80° south — is the other tell. It is there because the ground is not symmetric: the Arctic Ocean has traffic and the Southern Ocean at 80° has an ice shelf. A limit chosen for use is allowed to be lopsided. A limit chosen to make a rectangle square is not.

What the poles get instead #

The projection that actually serves the polar regions is the stereographic, and the reasons are worth setting out because they are the mirror image of the reasons Mercator serves the rest.

A polar stereographic is conformal, like Mercator, so the same pleasant behaviour under rotation holds. Its scale factor grows away from the pole rather than towards it, which puts the stretched part of the map at the latitude a polar user cares least about. And its natural extent is a disc rather than a strip, which suits a region that is a cap.

What it cannot do is join up with the rest of the world. There is no continuous, distortion-free way to hand a reader from one projection to another as they pan south, so every polar map is an island — the same structural problem as a lobed projection, taken apart in giving up continuity, where the tear is placed deliberately and its size is measured.

That is the honest summary of the cut: not that 0.373 per cent of the Earth is missing, but that the map is two maps, and the seam between them is a discontinuity nobody has a good answer to. The aspect is a free choice shows how far a rotation can move a projection’s good region around the sphere; what it cannot do is give one map two good regions at opposite ends.

The cut is not where the projection fails #

It is worth separating three latitudes that get conflated, because only one of them is about the projection.

85.0511° is where the tiling stops, and it is a fact about the image’s shape. Nothing about Mercator changes there.

84° is where this site’s own library stops sampling most projections, a working limit chosen so that a figure’s extent is finite and its distortion measures do not diverge.

Nowhere is where Mercator fails. It is conformal at every point of the sphere except the two poles themselves, and its scale factor is finite everywhere in between. There is no latitude at which the mathematics degrades; there is only a latitude past which the numbers on the page stop being useful, and that is a judgement rather than a discovery.

The conflation matters because “Mercator can’t show the poles” is usually said as though it were a defect of the projection. The projection maps the pole to infinity, which is a property of the map, and the cut is a decision about a rectangle. Why Mercator exists shows the property that made the projection worth having, and that property holds all the way up.

The other end of the same choice #

There is a symmetry worth stating: the cut is at the same latitude north and south, and nothing requires that.

A scheme could cut at 85° north and 60° south, or anywhere, and the world would still be some rectangle. What it would not be is symmetric about the equator, which would put the equator off the centre of the root tile and make the coordinate arithmetic carry an offset. Symmetry is chosen for the same reason the square is: it keeps the index simple.

This is the same class of decision as the false origin of a national grid — a shift chosen so that the numbers a user handles come out convenient, described in a grid has an origin that is not there. There the offset makes every coordinate positive; here the symmetry keeps the equator at $y = 0$. Both are conventions with no geometry in them, and both are routinely reported as though they were properties of the Earth.

What the cut is evidence of #

Two things, and the second is the one this field keeps finding.

The first is that a data structure can reach a long way into what a reader sees. The requirement was that images quarter cleanly. The consequence is that the default map of the world has no poles on it — a cartographic fact of some consequence, arrived at from a constraint with no cartography in it at all.

The second is that the constraint was over-stated and nobody checked. The square is not required; a rational aspect is. That distinction costs one root tile and would have saved 99.8 per cent of the discarded area, and the reason it was never taken is presumably that the square looked like the constraint. This site’s habit is to compute a bound rather than accept one, and the arithmetic here took ten lines.

Whether the choice was right is a separate question, and the answer is probably yes for the reason given above — the extra latitude is unreadable. But right for the reason usually given it is not, and the difference between those is the whole of what a derivation buys over a quotation.

What a derivation buys over a quotation #

The rung’s second finding — that the constraint was over-stated and nobody checked — is the one worth generalising, because the shape of it is common and the cost of it here was unusually large.

A quoted constraint arrives without its derivation. Tiles must be square is repeated in every account of how a tile pyramid works, and it is nearly true: what is actually required is that a tile quarter cleanly into four tiles of the same aspect, which any rational aspect ratio satisfies. The square is one solution and it was mistaken for the requirement.

The mistake is invisible because the quoted version works. A square pyramid does quarter cleanly, does tile the plane, does everything the system needs. Nothing fails, nothing is slow, and no user complains — so there is no signal that would prompt anybody to go back and ask whether the constraint was the constraint.

And the cost of the over-statement is not paid by the system. It is paid by the map: the poles are gone, a fifth of a per cent of the world is discarded, and the default world map has an arbitrary top and bottom edge. That cost falls on readers, who have no way to attribute it, rather than on the engineers who could have avoided it.

Which is why a derivation is worth ten lines even when the answer will be the same. Here it changes the conclusion’s status rather than the decision: the square is probably still the right choice, for a reason about readability rather than about tiling, and knowing that is the difference between a design and an inheritance.

It is worth naming what would have to be true for that to be taken up. A pyramid’s aspect ratio is as much an interface as its tile size, so the alternative is available to a new system and not to an existing one — which puts this finding in the same category as the projection choice one rung along: a measurement for the next designer rather than a proposal for the current one.

And it leaves the next system free. Somebody building a pyramid for another body, or for a domain where the polar regions matter, now has the actual constraint rather than the folklore one — and the actual constraint permits an arrangement that costs one extra root tile and keeps the poles.

The distinction matters because the two are argued about as though they were one thing, and only one of them is still open.

Where this ladder goes next #

The pyramid’s geometry has now been taken as far as it goes. The next rung goes down to the pixel — a place with a size, in which a coordinate is rounded and two coordinates may or may not survive as two — and the one after that to what a tile does not know about its neighbours.