Four cities that cannot be drawn to scale

The argument this field is built on is about a surface. A sphere has Gaussian curvature 1/R², a plane has none, curvature survives any map that preserves distances, and therefore no map from one to the other preserves them. No map is faithful makes it, and fifteen essays since have followed it out into scale factors, angles, areas and second derivatives.

Every quantity in that account is a derivative. A scale factor is a ratio of two lengths in the limit as both go to zero; an angle is a ratio of derivatives; an areal factor is a determinant of them. The theorem is about what happens to pairs of points infinitely close together, and it is proved by comparing two ways of differentiating.

A reader of an atlas has never met a pair of points infinitely close together. What a reader has is four cities.

Six numbers. The question is whether four dots can be put on a sheet of paper so that the six separations between them, measured with a ruler, are those six numbers in some fixed proportion. It is a question about a finite table, it has a yes-or-no answer, and the answer is no.

Three places never pose the question #

Take any three of the four. They have three distances between them, and three distances are the sides of a triangle — so long as no one of them exceeds the sum of the other two, which for distances measured on any surface at all is guaranteed, because a path from A to B via C is a path from A to B.

So a triangle can always be drawn. Every triple of places on the Earth, however far apart, has an exact flat picture: three dots, three ruled separations, no error anywhere.

That figure is the shape of the whole ladder. The triangle inequality is a real constraint and the sphere satisfies it, so three places are free. The moment a fourth arrives, the picture acquires a constraint the sphere does not satisfy, and the size of the violation is what the rest of this essay is about.

The counting is worth doing before the geometry. Four dots on a plane carry eight coordinates. Sliding the whole picture about costs two of them, turning it costs one, so five numbers in the arrangement are genuinely free. There are six distances. Five free numbers cannot in general satisfy six equations, and the sixth equation is the one that fails.

The best flat picture, and how wrong it has to be #

Nothing above says how wrong. A constraint that cannot be satisfied may be missed by a millimetre or by half the sheet, and the difference is what decides whether anybody should care.

The honest measurement is the smallest error any flat arrangement can be forced down to. “Drawn to scale” is a claim about ratios, so the quantity is the worst of the six ratios between drawn separation and true distance, after the picture has been allowed the one free choice a scale bar represents. Minimising that over every arrangement of four dots gives a single number, and it is not reached by the standard method: classical multidimensional scaling minimises a least-squares criterion in the squared distances and lands about twice as high, so the number below comes from a pattern search started at the scaling solution and driven downhill until no coordinate move helps.

Six-tenths of one per cent. On the London to Sydney edge that is a hundred and six kilometres; on London to New York it is thirty-five.

Whether that is large depends entirely on what the picture is for. Nobody navigates from a drawing of four dots. But the number is a floor rather than a typical case: it is what the best possible arrangement achieves, and it is what every map of those four places is measured against by the essay after this one. The best projection in this collection’s library manages 8.28 per cent on the same four cities, thirteen times worse, and that gap is a rung of its own.

The figure is also the first place in this collection where a picture of places carries no map at all, and it is worth pausing on how strange that is. Every other figure on this site is drawn in a projection and says so in its own footer, because the site’s rule is that a picture of the sphere has a page and the page has properties. This one has no page. It is a set of dots whose separations are as nearly right as separations can be made, and asking which projection it is in has no answer.

The test that decides it exactly #

The optimisation above says how wrong. It does not say why, and it cannot be used to prove that no arrangement does better — a search that stops finding improvements has found a local minimum and nothing more.

The exact statement is older than any of the machinery. In 1841 Arthur Cayley wrote down a determinant in the squared distances of a set of points, and Karl Menger showed a century later what it decides. For four points the determinant is five by five: the matrix of squared distances, bordered by a row and a column of ones with a zero in the corner. Its value is 288 times the square of the volume of the tetrahedron those distances make.

That single number sorts the possibilities into three, and the three are worth separating carefully because two of them are usually run together.

Zero. The four points are coplanar. A flat picture exists and is exact.

Positive. The four points make a tetrahedron of real volume. No flat picture exists, but a picture in space does, and the failure is that the sheet has one dimension too few.

Negative. There is no tetrahedron either. The six numbers are not the distances of four points in space, or in four dimensions, or in any number of them. The failure is not about the sheet at all.

For London, New York, Tokyo and Sydney the determinant is −3.344471 × 10²³, in square kilometres to the sixth power. It is the third case.

That is a stronger statement than the one this essay set out to make, and it deserves its own rung; the fifth of this ladder prices the escapes and finds that buying a dimension buys nothing. What matters here is that the sign is decided, not estimated, and that it is decided by six numbers with no geometry anywhere in the calculation.

Cayley, Menger and Schoenberg #

The three pieces of this arrived a century apart and for three different reasons, which is worth recording because none of them was thinking about maps.

Cayley published the determinant in 1841, as a relation among the mutual distances of points — an identity in algebra, generalising Heron’s formula for the area of a triangle from its three sides to the volume of a simplex from its edges. Menger took it up in the 1920s as the foundation of a metric geometry that starts from distances and never mentions coordinates at all, which is why the criterion carries both names.

The eigenvalue form is Isaac Schoenberg’s, in 1935, and it is the one this ladder runs on. Double-centring the matrix of squared distances turns it into a matrix of inner products about the centroid, and a matrix of inner products is positive semidefinite exactly when the vectors it came from exist. So the number of positive eigenvalues is the dimension the set occupies, and a negative eigenvalue says the vectors do not exist at all. Everything in the figures above is one of those two statements.

The reason a cartographer never met any of it is that the subject grew up around the continuum. A projection is a pair of functions and the natural instruments for a pair of functions are derivatives, so the field’s whole vocabulary — scale factor, indicatrix, angular deformation — is differential, and what can be unrolled is stated as a condition on curvature rather than on any finite set of points. There is nothing wrong with the vocabulary. It simply has no word for six numbers.

The same instrument, made to pass #

A determinant of six numbers each around ten thousand is a number around 10²³ almost whatever the six are. “It is not zero” is therefore no evidence at all until the same code has been shown to return zero when zero is the truth.

There is a set of six numbers for which it is the truth, and it is the same four cities measured differently. The chordal distance between two places is the straight line through the rock: shorter than the great circle, and equal to it in the limit of small separations. Four places on a sphere are four points in space, so their chordal distances are the distances of four points in three dimensions by construction, and the determinant must come out positive with a real volume attached.

The chordal determinant is +8.299415 × 10²³ and the spectrum has no negative eigenvalue, so the machinery passes the case it must pass. The geodesic one fails it. The difference between the two runs is the surface: a great circle is longer than its chord by a factor that grows with the separation, and stretching six distances by six different factors is exactly what takes a set of four points out of Euclidean space.

Two instruments are used here rather than one, and it is worth saying why both are kept. The determinant is a single number and is the classical statement, but a single number cannot say how the failure happens. The spectrum is four numbers and can: it distinguishes “these four need three dimensions” from “these four need none that exist”, and the determinant’s sign is only a compressed reading of it.

How the failure scales with the span #

Nothing so far has explained why anybody ever draws a map of a town and does not worry.

The measurement that explains it varies one thing. Take the same four cities, keep every bearing from their centroid fixed, and shrink the whole configuration towards that centroid — the same arrangement of places, spanning less and less of the sphere. Then ask the determinant what it says, normalised by the sixth power of the set’s own diameter so that only shape is left and size divides out.

A slope of two is Gauss’s theorem arriving as a number about a finite set. Curvature is a second derivative of the metric, so its first effect on a distance between two places is quadratic in how far apart they are; a configuration a quarter the size has departures a sixteenth as large. That exponent is measured directly and much more sharply in the rung after this one, where the quantity is the flat-picture error rather than a determinant and the fit comes out at 2.0088.

It is also the reason the whole subject is usually invisible. Nobody notices that a town cannot be drawn to scale, because at a town’s size the failure is a part in ten million.

That pair of figures is the whole practical content of this rung. The impossibility is exact and it is a function of span, so the question a map maker faces is never whether — it is always how much, against a tolerance somebody has to state.

What radius the six numbers were computed on #

Every distance in this essay is a great-circle distance on a sphere of radius 6,371.0088 kilometres, which is the arithmetic mean of the Earth’s three semi-axes and the number this collection uses throughout. Four radii of the Earth measures what that choice costs: the mean radius is right for an area to half a part per million and wrong for a distance along a meridian by 559 parts per million, which on the London-to-Sydney edge is nine and a half kilometres.

That does not touch a single conclusion here, and the reason is worth stating rather than assumed. Every quantity above is a ratio — the determinant is normalised by the diameter, the eigenvalues are reported against the largest, the flat-picture error is a percentage. A radius that is uniformly 559 parts per million too small rescales all six distances by the same factor and cancels out of every one of them. What it would touch is a claim in kilometres, and the two such claims in this essay — a hundred and six kilometres on one edge and thirty-nine metres on another — carry that error and are quoted to the precision it permits.

The distances are also spherical rather than ellipsoidal, which is a larger effect than the radius: geodesics on the ellipsoid finds the two differ by up to a fifth of a per cent on routes of this length, with a sign that changes depending on which way the route runs. Recomputing this ladder on WGS84 would change the six numbers in the fourth digit and change nothing else, because a set of six numbers from an ellipsoid is not the distance set of four coplanar points either — an ellipsoid is not flat, and the whole argument needs only that.

What is different about arguing this way #

Fifteen essays in this field argue from Theorema Egregium and every one of them is about a map — a rule that assigns a page position to every point of the sphere, including the ones nobody asked about. How small is flat enough prices the same trade-off, and prices it as a field: at what size does a region stop being flat to a stated tolerance.

The argument here gives that up and buys something for it.

It gives up everything about the places that are not in the set. A flat picture of four cities says nothing whatever about a fifth. It has no graticule, so there is nowhere to hang a coastline; it has no inverse, so a point on the sheet is not anywhere; and adding a fifth city moves all four of the original dots, because the optimisation is over the whole configuration. A projection is exactly what this is not — a rule fixed in advance, applied to whatever arrives.

In exchange it answers exactly. No tolerance appears anywhere above. The determinant is negative, the eigenvalue is negative, the best arrangement is off by 0.6286 per cent and no arrangement is better. There is no sampling, no quadrature and no region to choose a weighting over, which is the step the ranking depends on the region shows turns every regional measurement into a judgement. Sixteen numbers go in and one comes out.

It is a question about the whole set at once. How big a triangle it takes and measuring curvature from inside both recover curvature from measurements made on the surface, and both do it with three points and an angle sum. Three points are exactly the case that always works here, which is not a contradiction: an angle sum is a comparison against what a flat triangle with those sides would have, so it uses the triangle’s angles as well as its sides. This ladder is handed sides and nothing else, and with sides alone the fourth point is the first thing that says anything.

And it is about the quantity a reader actually reads off a map. The places where a map is exactly right finds the curves on which a projection’s scale factor is one, and a scale factor of one at a point does not mean that any two places either side of it are the right distance apart. What a reader does with a map is put two fingers on it, and that is a distance between two named places thousands of kilometres apart — a quantity no derivative describes and no essay in this field had measured. A circle of a distance is not a circle is the nearest the collection had come, and it asks the question of one place and a radius rather than of a set.

Where the four cities came from, and why any four would do #

The four in this essay are the ones the collection already uses, and nothing about them was chosen. Any four places on a sphere fail, unless they happen to lie on one great circle — in which case they are four points on a line, the determinant vanishes for the same reason a degenerate triangle’s area does, and the flat picture is the line itself.

That exception is exactly as large as it sounds: a set of measure zero. Four places picked at random on the Earth lie on a common great circle with probability zero, and four cities picked because they are cities lie on one with probability zero as well. The set of four-place configurations that can be drawn to scale is the boundary between two open regions, and nothing lives on a boundary by accident.

There is a second exception hiding in the first and it is worth naming. Four places arranged so that the determinant is positive rather than negative would need three dimensions and not four, which is the ordinary situation for four points in space and the reason the tetrahedral reading is the one most people expect. On a sphere it does not happen: stretching every chord into its arc does not merely lift the four points out of a plane, it takes them out of Euclidean space entirely, and two charts are enough, and one is not is the same fact told about coverings rather than about distances.

It is worth noticing what this does to a familiar reassurance. It is often said that a map of a small area is essentially exact and only the large ones are compromised. The first half of that is a statement about tolerance and is true at any tolerance somebody names; the second half suggests there is a threshold, and there is not. The determinant is non-zero at every span in the figure above and would be at every span below it, all the way down to two towns and a village. What changes is only the number of digits before the error appears.

The question the six numbers cannot answer #

This rung has established that no flat picture of four places is exact, and has measured how far from exact the best one is. Two things it has not touched, and one of them is uncomfortable.

The comfortable gap is the size law. The exponent of two above is read off a determinant, which is a proxy: it is the volume of a tetrahedron that does not exist, in units nobody can interpret. The quantity a reader cares about is the flat picture’s own error, and measuring that against the span — with the minimax optimisation run at every size rather than a determinant evaluated — is the next rung, and it produces a much sharper number.

The uncomfortable one is this. A flat picture of four places was found here to be wrong by 0.6286 per cent, and a projection of the same four is wrong by 8.28. The free picture is thirteen times better and no cartographer has ever used one. That looks like a straightforward indictment of projections until the same comparison is run on sixteen places instead of four, at which point the free picture’s advantage falls to eight per cent — and the reason it collapses is a counting argument about degrees of freedom that this essay has already made in passing and not noticed it was making. Five free numbers against six equations is a special case of 2n − 3 against n(n − 1)/2, and the second of those grows twice as fast as the first.