One pair of numbers, a hundred and twenty places

Ten rungs of this ladder have taken a grid coordinate apart: its origin, its units, its scale factor, its convergence, its zone boundaries, its ordering. Every one of them has treated the coordinate as a pair of numbers in a stated zone.

This one drops the zone, which is what a spreadsheet with two columns does.

The count #

A UTM coordinate has three components: an easting, a northing and a zone number from 1 to 60. The hemisphere is a fourth, carried either as a letter or as a convention about the northing — north of the equator the northing is measured from it, and south of the equator it is measured from a false origin ten million metres south.

Two of the four are numbers that go in a column. Two are not.

So a pair of numbers with nothing attached is a valid coordinate a hundred and twenty times over, and the ambiguity is not a degradation of anything. Each of the hundred and twenty is a complete, correct, unambiguous position; there is simply no way to tell from the numbers which one was meant.

The shape of the ambiguity #

The structure of those hundred and twenty is the part worth having, and it follows from what the zone system is.

Every UTM zone is the same transverse Mercator projection with its central meridian moved six degrees. Not a similar projection — the same one, with the same scale factor at the axis, the same false easting, the same ellipsoid. So the same numbers put the point at the same latitude and the same offset from the axis in every zone, and the sixty candidates differ only in where the axis is.

The candidates are therefore a ring at one latitude, spaced exactly six degrees of longitude apart. That is what the hero figure shows, and it is checked rather than observed: the sixty latitudes agree to $10^{-12}$ degrees and the fifty-nine longitude gaps are six degrees to $10^{-9}$.

The southern ring is at a different latitude, and that is the false northing at work. Read as northern, 5,678,000 metres puts the point 5,678 kilometres north of the equator, at 51.25°N. Read as southern, it puts it 10,000,000 − 5,678,000 = 4,322 kilometres south of it, at 39.04°S. The hemisphere error is not a reflection; it is a ninety-degree error in latitude, and it is the only one of the four kinds of confusion here that a glance at a map would catch.

What the numbers themselves would have to look like #

It is worth asking whether the ambiguity is visible in the numbers at all, because if it were, a validation rule could catch it. It is not, and the reason is arithmetic rather than luck.

An easting in UTM runs from about 166,000 to 834,000 metres — the false easting of 500,000 plus or minus the half-width of a zone at the equator — and every value in that range is legal in every zone. A northing runs from 0 to 9,329,000 in the north and from 1,116,000 to 10,000,000 in the south, and the two ranges overlap almost entirely. So every legal pair is legal in all hundred and twenty readings, and no range check, no checksum and no plausibility rule can narrow the field.

There is one exception and it is instructive. A northing below 1,116,000 cannot be a southern reading, because the southern convention’s false origin puts the equator at ten million and 84° south at 1,116,000 — so a northing of 400,000 is unambiguously northern, and it is the only piece of information the numbers carry about themselves. Everything else a validator might want — the zone, the hemisphere, the units, the datum — is outside the pair, and a format that carries two columns carries none of it. That covers latitudes below about 3.6° north, which is a strip of the world nobody works in disproportionately.

A grid reference names a square is about a different kind of under-determination in the same numbers — a truncated reference names a hundred-kilometre square rather than a point — and the two compound rather than cancel: a four-figure reference with no zone names a hundred and twenty squares.

How far apart the nearest two are #

At the equator a wrong zone is obvious: the point lands six hundred kilometres away, usually in a different country and often in a different ocean.

At eighty degrees north it lands a hundred and sixteen kilometres away, which is inside the same territory, on the same kind of ground, at a plausible-looking place. The zone system’s ambiguity is at its most dangerous exactly where the zones are narrowest, and sixty zones was a decision about one latitude is the reason the zones are the width they are — a decision taken to bound the scale error at the equator, which is where this ambiguity is least harmful.

Why a second measurement does not help #

The obvious response is to bring more information: measure a bearing against grid north, or a distance against a grid distance, and see which candidate is consistent.

It does not help, and the reason is the design.

Grid convergence is a function of the offset from the central meridian and the latitude. The point scale factor is a function of the same two. Every candidate has the same values of both — because every candidate has the same offset and the same latitude, which is exactly what “sixty identical zones” means. Measured across the ring, the convergence agrees to $1.3\times10^{-9}$ degrees, which is five nanoseconds of arc, and the scale factor to $9\times10^{-16}$.

So a surveyor who measures the convergence perfectly learns nothing about which zone the coordinate came from. The uniformity that makes the zone system usable is the uniformity that makes the ambiguity irreducible, and the two are the same property.

There is one geometric quantity that does differ between candidates, and it is not one anybody measures: the deflection of the vertical and the local gravity field are properties of the ground rather than of the grid, so an astronomical observation would separate the candidates immediately. That is a measurement of where the observer is, not of the coordinate, which is the point — the ambiguity is in the numbers and no amount of arithmetic removes it.

What each stated word is worth #

The asymmetry is the practical content. The zone is the expensive word, and it is a single small integer that fits in any format anybody has ever used. A grid has an origin that is not there makes the same observation about the false origin: the most consequential part of a grid is the part that is a convention rather than a measurement. It is also the one most often lost, because it is not a coordinate — it is metadata about a coordinate, and metadata is what a two-column export drops.

The military grid reference system exists to solve exactly this, by putting the zone and a pair of hundred-kilometre square letters into the reference string itself. That works and it introduces a repetition of its own: the hundred-kilometre letter pairs cycle every three zones, so an MGRS reference with its zone number stripped repeats every eighteen degrees of longitude — twelve hundred kilometres at mid-latitude, and a hundred and twenty places reduced to twenty.

The failure this has actually caused #

The shape of the accident is worth stating, because it is not the one the arithmetic suggests.

The dangerous case is not a dataset with no zone at all — that is obvious the moment anybody plots it, since the points land in a ring round the world and the ring is unmistakable. The dangerous case is a dataset in which most rows are in one zone and a few are in another, and the zone column is dropped or ignored. Then the stray rows land six degrees away, inside the same plot, at the same latitude, and they look like data.

That is the situation a country spanning two or three zones produces every day. Spain spans zones 29, 30 and 31; Norway spans 31 to 36; the conterminous United States spans 10 to 19. Where two zones meet prices what happens to a feature crossing a boundary; this is what happens to a feature that does not cross one and whose zone is nevertheless lost.

The second shape is a coordinate transcribed between systems. A northing of 5,678,000 is a plausible UTM northing and a plausible national-grid northing and a plausible state-plane northing in feet, and two grids over the same ground shows what the second kind of confusion costs: not six degrees but a few hundred metres, which is far harder to see and far more likely to be believed.

What was computed, and how #

The candidates come from one inverse transverse Mercator solve. Because every zone is the same projection, solving in zone 31 gives the latitude and the offset from the axis, and the other fifty-nine are that answer with the axis moved — so the sixty candidates are not sixty computations and their exact six-degree spacing is a property of the construction rather than of the arithmetic.

Four assertions carry the rung and each could fail alone. There must be a hundred and twenty candidates. Every candidate in a hemisphere must be at the same latitude to $10^{-12}$ degrees — if the latitudes differed, the zone system would not be sixty copies of one projection and everything else here would be wrong. The longitudes must be exactly six degrees apart. And the convergence and the scale factor must be identical across the ring, which is the statement that no geometric measurement chooses between them, expressed as a check.

The separation figure’s assertion is a shape rather than a value: it must fall monotonically with latitude, exceed six hundred kilometres at the equator, and fall below a hundred and thirty at the zone system’s limit — because the finding is that the ambiguity gets worse where the geometry gets better.

The same numbers on a map of one zone #

Drawn twice on two projections, the ring is the same set of places and it looks different, which is a reminder worth having at the end of a rung about what a coordinate does not say: the picture of the ambiguity is itself a projection, and the six-degree spacing that is exactly uniform on the ground is not uniform on either page.

Where the model stops #

This is UTM and not every grid. A national grid has one zone, so its coordinates are unambiguous within the country and ambiguous only against other countries’ grids — which is a different failure, and a worse one, because two national grids can put the same numbers a few kilometres apart rather than six degrees. A coordinate without its system is not a location is the general form.

The polar zones are not counted. UTM stops at 84° north and 80° south and the universal polar stereographic takes over, with its own conventions and its own two zones. Including them would raise the count and complicate the ring structure without changing anything.

The units are assumed to be metres. The units are part of the coordinate prices the other convention that is not in the numbers, and a state plane coordinate in survey feet read as metres is a third kind of ambiguity that multiplies with this one rather than overlapping it.

And this is about a bare pair, not about a bad one. Nothing here concerns a coordinate that is wrong; every candidate is right. The failure is that the numbers under-determine the answer by a factor of a hundred and twenty, and no check on the numbers can detect it, because there is nothing to detect.

The generalisation #

A coordinate system with identical repeated cells is a coordinate system whose ambiguity cannot be resolved from inside a cell. That is the whole of it, and the zone system is the cleanest example the subject has: sixty cells, each an exact copy, so every measurable property inside a cell is the same in all sixty.

The trade is real and the zone system took the right side of it. Sixty identical zones mean one set of formulas, one scale factor, one table of series coefficients, one convergence rule — which is why UTM is usable at all, and why designing a grid for one region produces something better for that region and useless everywhere else. The price is that the zone number is doing all the work of distinguishing sixty places, and it is carried in a field that is not a coordinate.

The general rule for anybody designing such a system: if a periodic construction is chosen for uniformity, the period is the ambiguity, and the label that resolves it must be inside the value rather than beside it. MGRS’s designers understood that; the plain easting-and-northing convention it replaced did not; and the two live side by side in every dataset, which is why this failure is still current.

One line that would have prevented it #

A coordinate reference system identifier — an EPSG code, or the zone written into the column name — is a few bytes and it removes the whole of this. It is not part of the coordinate and it is not carried by any format that carries two numbers, which is why it is lost.

The same defect, quieter, in every coordinate #

The zone case is loud because the ambiguity is thousands of kilometres and the resolving label is a small integer somebody obviously forgot. It is worth seeing that the rule it produces convicts systems nobody thinks of as broken.

The identifiers that obey the rule do it by construction. A geohash carries its precision in the length of the string, so no separate field says how many cells it means. A quadkey carries its zoom level the same way, in the number of digits. A tile reference written z/x/y puts the level first. MGRS puts the zone and band ahead of the easting. In each case the resolving label is inside the value — the value cannot be copied with the label left behind, because there is no value without it.

The one everybody uses does not. A latitude and longitude pair carries no datum, and the same pair of numbers names places up to a few hundred metres apart depending on which datum is meant. That is not sixty possibilities but several hundred, most of them within a few hundred metres of each other, which is precisely what makes it the quieter failure: a zone error puts a point in another country and is caught in a minute, while a datum error puts it down the road and is caught never.

So the same rule, applied to the coordinate that every dataset in the world is written in, says that a bare pair of degrees is under-specified in exactly the way a bare easting and northing is. The difference is only in the size of the ambiguity, and the size of the ambiguity is what decides whether the mistake is discovered.

Which suggests the uncomfortable reading of the zone failure. It survives not because it is subtle but because it is nearly always caught — so the practice that produces it is never punished hard enough to change, and the same practice, applied where the error is small, produces mistakes that are still in the data.

Where the ladder goes next #

Eleven rungs have priced what a grid is: its origin, its units, its scale, its ordering and now what it leaves out. All of them have assumed the grid is being read by somebody who wants a position. A grid is also read by machines that want to sort — a spatial index, a tile key, a database ordering — and what those want from a coordinate is not a position at all but a total order with locality, which is a property this ladder has never measured and which the applied field’s cell anchor has been circling from the other side.