A grid reference names a square

Seven essays of this ladder are about the geometry a grid imposes: where its origin is, what its scale factor buys, what an area on it is worth, where two zones meet, and how good a grid a country could have had.

None of them is about how a coordinate on it is written, and that is a second design, made by different people for a different reason, with arithmetic of its own.

A grid reference is not a rounded coordinate. It is a truncated one: the digits are the leading digits of the easting and northing, so dropping digits from the right always names a larger square that contains the same ground. That property is why the notation survives a radio, and it is the reason grid references are read aloud and geographic coordinates are not.

It is also a systematic displacement of every reference towards the grid’s own south-west, of exactly half a cell in each axis, which no amount of care in the field removes.

The bias, measured against its closed form #

For a population of points uniform across a cell of side s, the offset in each axis is uniform on [0, s). So the mean is −s/2 and the standard deviation is s/√12, both of them closed forms, and both are what the measurement has to reproduce rather than discover.

Over twenty thousand points in a hundred-metre cell:

• truncated: mean 50.24 m west and 50.20 m south, against a predicted 50.0;
• rounded: mean 0.17 m and 0.18 m, against a predicted zero;
• scatter: 28.73 m and 29.00 m truncated, 29.08 and 28.82 rounded, against a predicted 28.87 for both.

The mean displacement of a truncated reference is 71.0 metres, against (s/2)√2 = 70.7. The mean displacement of a rounded one is 0.25 metres, which is the sampling noise of twenty thousand points.

That is the whole of the first result, and its importance is in the word bias. Scatter shrinks when observations are averaged; a bias does not. A hundred grid references averaged together still sit fifty metres south-west of the mean of the hundred points they came from, and a thousand still do.

What truncation costs, exactly #

The bias and the scatter are usually quoted separately, and putting them together gives the price of the design in one number.

For a uniform population over a cell of side s, the per-axis error of a truncated reference has mean s/2 and standard deviation s/√12, so its root mean square is √((s/2)² + s²/12) = s/√3. A rounded reference has no bias, so its root mean square is s/√12. The ratio is

$$\frac{s/\sqrt3}{s/\sqrt{12}} = 2$$

exactly two, at every cell size, in each axis, and — since both errors are isotropic in the plane — for the two-dimensional displacement as well. On a hundred-metre cell that is 81.6 metres root-mean-square for a truncated reference against 40.8 for a rounded one.

So the containment guarantee costs a factor of two in error and not a penny more. That is a clean price for a design decision, and it is a good one: doubling the error of a hundred-metre reference to buy the property that every shortening of it stays true is a trade almost any user would take, and the exactness of the two means the trade does not change with the precision quoted.

The same two numbers say how quickly the bias overtakes the noise, which is the sharper form of a bias does not average out. Over n references the scatter’s contribution to the mean falls as s/√(12n) while the bias stays at s/2, so the bias is the larger of the two as soon as

$$\sqrt{n} > \frac{\sqrt{12}}{2} = \sqrt3, \qquad \text{that is, } n > 3.$$

Four references are enough. With four the systematic south-westward displacement already exceeds the standard error of their mean; with a hundred it exceeds it by a factor of five; with the ten thousand records of a gazetteer, by fifty. A dataset large enough to be worth analysing is a dataset in which the bias is the dominant error and the scatter is invisible beside it — which is exactly the regime in which nobody looks for a systematic offset, because the numbers have stopped scattering and started looking precise.

Why the notation chose it anyway #

The bias is not an oversight. It buys the property the notation exists for.

Under truncation, dropping a digit from each axis takes a reference naming a hundred-metre square to one naming a kilometre square that contains it. The coarser reference is a strictly weaker statement about the same point, and any user can drop digits at any time without knowing anything about the point.

Under rounding, that fails. A point at easting 449,960 rounds to 4500 at ten-metre precision and to 450 at hundred-metre precision, and the hundred-metre square whose corner is 450 does not contain the point — the point is forty metres outside it. Rounding at one precision and rounding at another are different operations that can disagree about which cell a point is in, and a reference read over a radio and shortened by the listener would sometimes name a square the speaker did not mean.

So the design is a deliberate trade: a known, constant, correctable half-cell bias in exchange for a guarantee that every truncation of a reference is a true statement. Stated that way it is obviously the right choice, and stated that way it is also obvious that the bias should be corrected by anybody converting references to positions — which is exactly the step that gets skipped, because a reference looks like a coordinate.

The square is a square on the grid #

A grid reference names a square on the grid. On the ground that square is s/k metres on a side, where k is the point scale factor, so a hundred-metre square is:

• 100.0399 m at 52°N on the central meridian, where k = 0.9996013;
• 99.9390 m at 50°N and 6°W, where k = 1.0006104;
• 99.9784 m at 55°N and 1.5°E.

Forty millimetres is nothing for the purpose a three-figure reference serves. It is quoted because it comes with a companion fact that is not nothing: the square is still a square. Its four sides agree with each other to within 0.07 of a millimetre over a hundred metres at every place tried.

That is not a coincidence and it is not a property of grids. It is conformality, doing the one thing conformality does. A conformal projection has no angular deformation, so a small square on the grid is a small square on the ground rotated slightly and scaled slightly, and the notation’s implicit promise — that a reference names a square of ground — holds because of a choice made about the projection a century before anybody wrote the reference.

On a grid built from an equal-area projection the same notation would name a parallelogram whose shape varied across the sheet, and every statement of the form “within a hundred-metre square” would mean something different at different places.

Which way is south-west #

The bias points south-west on the grid, and grid south-west is not ground south-west. The angle between them is the convergence, and on the same national grid it is zero on the central meridian, −3.07° at 6°W and +2.87° at 1.5°E.

So a population of references at the western edge of the grid is displaced 71 metres in a ground direction three degrees away from where a naive reading would put it. Three degrees of 71 metres is 3.7 metres, which is smaller than the bias itself by a factor of nineteen and larger than the ground-size effect by a factor of ninety.

That ordering is the useful part. Anybody correcting the half-cell bias correctly gets the whole of it; anybody correcting it in the wrong azimuth leaves a residual of a few metres; and anybody worrying about the scale factor first has picked the smallest of the three effects. Grid north is not north is usually a statement about bearings, and this is the same statement about the direction of a systematic error.

How many places answer to the same digits #

The digits of a reference are the low-order digits of the coordinate, so a reference of three figures per axis repeats every hundred kilometres. Within a six-degree zone at 52° north that is four hundred-kilometre squares across and nine up: 36 places with the same digits, and 2,160 across the world’s sixty zones.

The count falls towards the pole because a zone is narrower there — 412 kilometres wide at the equator, 229 at 70° — which is a small and pleasing consequence of the zone system’s own geometry.

The letter pair in a military reference exists to name which hundred-kilometre square, and it is a separate token from the digits: it is what makes the reference absolute rather than local. A reference quoted without it is a statement about a place within a known region, which is exactly how it is used in the field and exactly what makes it dangerous when it leaves the field.

What was computed, and how #

The population is drawn from a stated generator with a stated seed and spread uniformly over a hundred-kilometre block of the grid, and the offsets are computed as the difference between the point’s own easting and northing and the corner the truncation names.

The three quantities are compared with closed forms rather than with each other, which is what makes the measurement a check: the mean must be half a cell, the scatter must be s/√12, and the rounded population’s mean must be zero. All three are properties of a uniform distribution and none of them is a property of the grid, so a disagreement would be a defect in the arithmetic rather than a finding.

The ground size of a square is computed by inverting the four corners back to latitude and longitude and measuring the four sides with a geodesic on the grid’s own ellipsoid — not on WGS84, because the British grid is defined on Airy 1830 and measuring its squares against another ellipsoid’s metric would be wrong by 1.7 parts per million everywhere.

The claim that the sides are equal is asserted as a ratio, at 2 × 10⁻⁵ of the side, rather than as an absolute number of millimetres, so it says the same thing at any cell size.

The comparison the notation was designed to win #

Set the two notations side by side and the design becomes an argument rather than a habit.

A geographic coordinate truncated to a stated number of decimal places names a cell that is 1.112 metres tall at every latitude and between 1.112 and 0.097 metres wide depending on where it is. Its shape changes by a factor of eleven between the equator and 85°, so the same written precision means two different things at two places and there is no single sentence describing what a five-decimal coordinate is worth.

A grid reference truncated to a stated number of digits names a cell that is square, the same size everywhere in its zone to within a part in a thousand, and whose size is a round number in metres. “Six figures is a hundred metres” is true everywhere the grid is used, which is a sentence a person can hold.

Both notations are quantisations of a coordinate and both carry a half-cell bias. The difference is entirely in the shape and constancy of the cell, and that difference is inherited from the projection: a conformal projection with a scale factor near one over the whole zone gives square cells of near-constant size, and a graticule gives neither.

So the argument for a grid reference is not that it is shorter or easier to say. It is that the projection underneath it makes the quantisation uniform, and uniform quantisation is the property that lets a precision be quoted as a single number.

Where the model stops #

One notation. The measurement is of truncation to a stated number of digits per axis, which is what a national grid reference and a military grid reference both do. Other schemes quantise differently — a geohash interleaves bits and halves alternately, and its cells are not square — and the bias arithmetic there is a different calculation with the same shape.

A uniform population. Real references are not made at points spread evenly over a cell; they are made at road junctions, summits and buildings, which are not uniformly distributed at any scale. The closed form is the uniform case, and a real population’s bias is between zero and a full cell depending on where its points sit inside their cells.

Nothing here is about the accuracy of the position. A reference is exact about the square it names. Whether the point was in the right square is a survey question that this ladder’s other rungs deal with.

The convergence figures are one grid’s. A zone-based system has convergence up to three degrees at its edges by design; a single-zone national grid has whatever its extent gives it. The ordering of the three effects is what transfers.

The generalisation #

Three quantities, three sizes, one written number:

• the half-cell bias: 71 metres for a three-figure reference, and the same fraction of the cell at every precision;
• the direction of that bias: grid south-west, which is a few degrees from ground south-west;
• the ground size of the cell: within a few tens of millimetres of its nominal size, and square.

The first is a systematic error the size of the precision being quoted, which means it is exactly the error a user is entitled to ignore if and only if they treat the reference as a square. The moment a reference is converted to a point — put in a database, plotted, differenced with another position — the bias becomes a real displacement, and it is in the same direction for every record in the file.

That is the shape of failure this ladder keeps finding. The units are part of the coordinate, the origin is part of the coordinate, and now the quantisation rule is part of the coordinate — and none of the three travels with the numbers.

Who found it, and when #

Alphanumeric grid references are a product of artillery survey. The British Ordnance Survey adopted a letter-pair system in the 1920s and 1930s; the military grid reference system in its current form was adopted by the US Army in 1947 and became a NATO standard soon after.

The truncation rule is stated explicitly in the field manuals, usually as an instruction — “do not round” — with the containment property given as the reason. It is the sort of rule that survives because it is drilled rather than because it is derived, and the half-cell displacement it implies is not usually mentioned, because in the field it is smaller than the plotting error and the whole point of the reference is the square.

The bias becomes visible only when references are treated as data. A gazetteer of grid references converted to a coordinate list carries a uniform fifty-metre south-westward shift, and it would take a comparison against an independent source to see it, because nothing in the file looks wrong.

Where the ladder goes next #

Eight rungs have taken a grid from its origin to the body it is defined on, and this one leaves it in a written form. What remains unexamined is the grid’s own lifetime: a national grid is realised by marks in the ground, the marks move, and a readjustment moves every coordinate on the grid without changing the projection, the ellipsoid or the notation by a single parameter.

That is a question about realisation rather than geometry, and the collection already has the essay it would build on.