Sixty zones was a decision about one latitude

Every rung of this ladder so far has stood inside one zone. A grid has an origin that is not there; its scale factor was chosen; an area on the grid is not an area on the ground; where two zones meet is a category error rather than a small one. Twelve essays, and not one asks how many zones there are or why.

Six degrees is a width, and a width meets a tolerance at one latitude. The ground a degree of longitude covers falls as cos φ, the scale error across a zone goes as the square of that ground width, so a fixed-width zone is tight at the equator and increasingly generous towards the pole. What that costs, and what a system that spent the tolerance evenly would look like, is a measurement this ladder owes.

The tolerance UTM meets, which is not the one it is quoted at #

Before asking what tolerance would justify a zone width, it is worth measuring the one UTM delivers, because the number in circulation is not it.

One part in 2,500 is quoted for UTM everywhere. That is 400 parts per million, and it is the deficit the 0.9996 constant introduces at the central meridian — a scale factor of 0.9996 is 400 ppm below one, by arithmetic.

The worst error across a zone is a different number:

At the equator the zone edge is 981 ppm above true scale while the centre is 400 below — a spread of 1,381 ppm across one zone, which is exactly what the tangent construction would have cost at the edge before the constant was applied, redistributed. Above about 45° the edge error has fallen below 400 and the worst point in the zone is the middle: the constant is now over-correcting, and the zone is uniformly too small.

So the honest statement of UTM’s tolerance is 981 ppm, achieved at the equator, and the number usually quoted describes the best part of the worst zone. The difference matters to anyone reducing a measured distance: a 981 ppm scale factor is a metre in a kilometre, and a surveyor who applied 400 because that is what the specification says would be out by six hundred parts per million on a baseline at the edge of an equatorial zone — which is a fifth of a metre on a three-hundred-metre line, and is exactly the kind of correction a stated tolerance is supposed to decide whether to apply.

The zone profile, at two latitudes #

Two profiles say the whole thing before any zone counting starts.

At the equator the tangent construction reaches 1,382 ppm at the zone edge, and multiplying by 0.9996 splits that into 981 above and 400 below — a worst case of 981 instead of 1,382, which is the trade a secant grid makes and is worth making.

At 60° north the zone is physically narrow — six degrees of longitude is 334 kilometres rather than 668 — so the tangent construction reaches only 343 ppm at the edge. Subtracting 400 turns a 343 ppm error into a 400 ppm one. The constant is a net loss, and this ladder found that in its second phase and asserted where the crossover is: at cos²φ = 400/1382, which is 57.5°.

That crossover is the same fact this rung is about, seen from inside one zone rather than across the system. Above 57.5° the zone is too narrow for its own scale factor; equivalently, at the tolerance the constant implies, the zone could be wider.

What that tolerance buys at other latitudes #

Take 981 ppm — no, take the fair comparison. Rebalance the scale factor for each candidate width, so that what is compared is the best grid of one width against the best grid of another rather than a tangent map against a secant one. At the equator that rebalancing meets 690 ppm with a three-degree half-width, which is the tolerance this rung uses throughout.

The half-width that meets 690 ppm:

The equator row is the check that the instrument is measuring the right thing: at the tolerance UTM meets there, the width UTM uses there is the width that comes out. Everything else is the finding. At 60° north a zone could be twice as wide; at 85°, twelve times.

The system that would have spent it evenly #

Divide the northern hemisphere into eighteen bands and give each the widest zone its own latitude allows:

Fifty-one at the equator against UTM’s sixty — because the rebalanced scale factor buys a little width over the fixed 0.9996 — and eight at 82°, where UTM still uses sixty.

That is the shape of the finding: UTM’s zone count is the answer to its tolerance at one latitude and is between two and seven times too many everywhere else. Not because the designers were careless, but because the thing designed was a width and the thing that matters is a ground width, and the two differ by cos φ.

What the arithmetic is, in one line #

The whole latitude dependence is one approximation and it is worth writing down, because it says why the effect is as large as it is.

At a half-width of Δλ, the ground half-width at latitude φ is R Δλ cos φ. A transverse Mercator’s scale factor grows as the square of the ground distance from its central meridian: k ≈ 1 + (Δλ cos φ)² / 2. Rebalancing the constant halves the worst departure, so the tolerance ε is met when

$$(\Delta\lambda \cos\varphi)^2 / 4 \approx \varepsilon$$

and therefore Δλ ∝ 1/cos φ at fixed tolerance. At 60° north cos φ is a half, so the allowed width doubles; at 85° it is 0.087, so the width goes up by a factor of eleven and a half.

The measured curve rises slightly faster than 1/cos φ — 37.05° at 85° against the 34.4° that formula gives — because at those widths the quadratic approximation is no longer good and the higher terms of the Krüger series matter, which is where the series stops being the map arriving from a different direction. The measurement uses the projection rather than the approximation, which is why the two can be compared at all.

Why a width and not a count #

It is easy to read the table above as a criticism and it should not be. Three reasons a fixed width was the right decision, and all three are about what a zone system is for.

A zone has to be findable from a coordinate. Which of the sixty zones a longitude falls in is a division and a floor: zone = ⌊(λ + 180)/6⌋ + 1, computable in the head and unambiguous. A variable-width system needs a table, and the table has to be shipped with every coordinate that uses it — which is a version of the problem a projected coordinate declares five things and hides four of them already names.

Zone boundaries have to be the same at every latitude, or they are not boundaries. A system whose zones widen with latitude has boundaries that move, so a zone is no longer a region of the Earth — it is a region at a latitude, and a north–south line crosses several. Working across a boundary is already the worst thing that happens on a grid, and multiplying boundaries by making them latitude-dependent would multiply it.

And the pole is where the argument stops mattering. The bands where a variable system would save most are 70° to 84°, which is where UTM stops anyway and the polar stereographic takes over. The saving is largest exactly where there is least to survey.

So the fixed width is a decision about usability, paid for in tolerance, and the price is now measured: a factor of two to seven in zone count, or equivalently a scale error two to seven times tighter than needed above 45°.

Where the count came from historically, and why it is 60 #

The number is not arbitrary and it is not derived from a tolerance either — it is derived from a chart series, which is a third thing.

Six degrees of longitude by four of latitude is the sheet layout of the International Map of the World, adopted in 1913, and UTM’s zones inherited the six. The tolerance came afterwards: the scale factor 0.9996 was chosen to suit a zone whose width had already been fixed by a paper size, not the other way round.

That order of decisions explains the pattern this rung measures. If the width had been derived from a tolerance it would vary with latitude, because the tolerance is a ground quantity and the width is an angular one. It was derived from a sheet, sheets are laid out in degrees, and degrees are angular — so the system inherited a constant angular width and a scale factor fitted to it at the worst case, which is the equator.

The same inheritance is visible in the sheet layout itself: the International Map’s sheets are four degrees tall and six wide at the equator, and above 60° they are merged in pairs precisely because a six-degree sheet becomes absurdly narrow on the ground. The map series solved the cos φ problem for its own purposes and the coordinate system built on it did not, because merging a sheet is easy and merging a coordinate system is not — a zone boundary is where a coordinate stops meaning what it meant, and a boundary that appears at one latitude and not another is worse than one that is everywhere.

Which way the slack actually runs #

The slack is not free precision — it is a constraint that binds elsewhere, and this is the practical half of the rung.

Above 45° every UTM zone’s worst error is the central-meridian 400 ppm, which is a uniform deficit rather than a spread. A uniform deficit is the easiest error there is to remove: multiply. That is exactly what a grid scaled to the ground does, and it explains why the practice is common in high-latitude countries and rarer near the equator — at 60° north the projection’s own variation across a zone is small enough that a single constant fixes nearly all of it, and at the equator it is not.

It also explains why national grids beat UTM by so little. A country with a grid of its own beats the international zone by a factor of 1.51, and that factor is small precisely because UTM is already loose at the latitudes most countries sit at: the national grid’s advantage is not that it is cleverer but that it is allowed to be wider, and above 45° UTM had width in hand it did not spend.

What a modern design would do differently #

Two things, and neither is a variable width.

Publish the worst error rather than the constant. A specification that states k₀ states the best point in the zone, and every user needs the worst. UTM’s tolerance is stated as its scale factor, which is the error at the best point in the zone. Every essay on this ladder has had to measure the worst point, and every user of the system needs the worst point. Nothing about the design would change; the number in the specification would.

Choose the constant per zone rather than per system. The rebalanced k₀ that meets 690 ppm at the equator is 0.99931, not 0.9996, and at 60° north the best constant for a six-degree zone is much nearer one. A per-zone constant costs one extra number in the zone table — which already carries a central meridian for each zone — and would remove the over-correction above 45° entirely.

There is a third thing a modern design would not do, which is to keep the count. Sixty is now carried in file formats, EPSG codes, military grid references and a great deal of software, so the count has become an interface rather than a design parameter — and an interface with a wrong number in it is cheaper to keep than to change. This rung is not a proposal; it is a measurement of what the number costs, which is the thing nobody had written down.

The second recommendation is what low-distortion projections do, one region at a time, informally and without a system. Doing it inside the zone system is a smaller change than replacing the zone system, and this rung’s numbers are the case for it.

What a variable width would break, beyond usability #

Usability is the right answer for why the width is fixed and it is vaguer than it needs to be. There are two specific properties the fixed width buys, and both are lost outright by any latitude-dependent design — including the one this rung has just priced.

A zone is computable rather than looked up. With a fixed width the zone number is $\lfloor(\lambda+180)/6\rfloor + 1$: one division, no table, no dependence on latitude, and the same answer from every implementation that has ever been written. A width that varies with latitude makes zone membership a function of both coordinates, which means a table, which means a version of the table, which means two datasets that disagree about which zone a point is in because they were built against different revisions. The zone system’s greatest practical virtue is that nobody has ever had to ask which edition of it they hold.

And a zone is a rectangle in latitude and longitude. That is why a zone’s extent can be stated in two numbers, why a dataset’s coverage can be tested against it by comparing bounds, and why the boundary between two zones is a meridian — a curve that every projection in use draws as a straight line, that every gazetteer can state, and that a surveyor can identify on the ground.

Under a variable width, a zone boundary would run along a meridian for part of its length and then jump sideways as the allowed width changed with latitude. A zone would be a staircase, its bounding box would be much larger than the zone itself, and a feature spanning latitudes could cross a zone boundary twice at the same longitude. Every convenience above disappears, and they disappear together.

So the eleven-fold slack at 85° is not a straightforward waste, and the rung’s number should be read with that attached. It is the price of a rule simple enough that sixty years of software has implemented it identically — which is a real thing to have bought, and which is the reason the count survives as an interface after it has stopped being a design.

What the measurement does establish is that the price is known rather than assumed, and that a design taking the other side of the trade for one region — a national grid, a low-distortion projection for one city — is not being fussy. It is spending slack the international system leaves on the table by construction, and now there is a figure for how much.

What this rung establishes #

UTM’s worst scale error is 981 parts per million, at the edge of an equatorial zone, against the 400 always quoted — which is the central meridian’s deficit, is the best point of the zone, and is the worst point only above 45° where the constant over-corrects.

Six degrees is the answer at one latitude. At the tolerance the system meets at the equator, the allowed half-width rises from 3.00° there to 37.05° at 85°, and the zones needed to encircle the world fall from 51 to 8.

The width scales as 1/cos φ at fixed tolerance, which is one line of arithmetic and is why the effect is a factor of eleven rather than a few per cent — and the measured curve rises faster still at the top, where the quadratic approximation the line rests on stops holding.

And the fixed width is a decision about usability rather than an oversight, paid for in a tolerance two to seven times tighter than needed above 45° — slack that shows up as the ease with which a high-latitude national grid can be scaled to the ground, and as the smallness of the advantage a national grid holds over the international one.