The escape is not a dimension

Four rungs of this ladder take the sheet of paper as given. They measure what a flat picture of a set of places cannot do, how badly, against what alternative and with how much freedom — and every one of them treats flatness as the constraint and looks for the best behaviour inside it.

There are three ways to relax the constraint. Give the picture more dimensions; give the page curvature; or give up that the reader measures it with a ruler. This rung prices all three, and the first has an answer everybody expects and it is wrong.

The dimension that is not the problem #

Four points that refuse to lie in a plane lie in a space. That is the ordinary situation for four points, it is what the Cayley–Menger determinant’s positive branch means, and it is what almost anybody would guess is happening to four cities.

It is not what is happening. Four cities that cannot be drawn to scale computes the determinant as −3.34 × 10²³, and the sign is the whole of the matter: positive is a tetrahedron of real volume, zero is coplanar, and negative is neither. Negative means the six numbers are not the distances of four points in space, or in four dimensions, or in any number of them.

Two numbers identical to fifteen decimal places is a stronger statement than a small improvement would have been. The third dimension is not merely unhelpful; it is not there. Four points always span at most three dimensions, the double-centring uses one of them up, and of the three that remain two are positive and one is negative — signature (2, 1). The four cities want a plane and an imaginary axis.

Why a metric can fail to be Euclidean #

The mechanism deserves a paragraph because it is not the same as curvature and it is what makes this rung different from the four below it.

A set of points in a Euclidean space of any dimension has a matrix of inner products about its centroid, and a matrix of inner products is positive semidefinite — always, because its diagonal entries are squared lengths and its quadratic form is a squared length of a combination. Schoenberg’s construction runs that backwards: double-centre the squared distances, and if the points existed, the result is that matrix. So a negative eigenvalue is a proof that the points do not exist, and no amount of room helps, because room is what positive eigenvalues buy.

The great-circle metric fails the test and the chordal one passes it, and the difference between them is a monotone stretch: the arc is the chord times ρ/(2 sin(ρ/2)), a factor that runs from one at zero separation to π/2 at the antipode. Stretching every distance by a different factor is exactly the operation that can take a metric out of Euclidean type, and this one does.

That is a statement about the metric rather than about the surface. It is why the failure survives any number of dimensions while Gaussian curvature — the obstruction the rest of this field is built on — is a statement about a surface embedded in a space and would be relieved by enough dimensions if the obstruction were only about fit. What can be unrolled asks the surface question and answers it with curvature; this rung asks the distance question and answers it with a sign.

Eight negative eigenvalues out of sixteen is not a marginal failure. It is the ordinary state of a large set of places on a sphere, and it says that the sphere’s geodesic metric is not a distorted Euclidean metric in any sense — it is a different kind of object.

That first row is the sharp version of a claim the rung at the foot of this ladder makes and cannot demonstrate at every size. There is no threshold. A set of five towns forty miles apart has a metric that no Euclidean space holds, exactly as a set of sixteen cities does, and the only difference is by how much. Where the two differ is that the county’s failure is smaller than the ink and the continent’s is larger than the picture.

The instrument’s own non-monotonicity #

One feature of the dimension figure is a defect in the instrument rather than a fact about the places, and it is worth naming before the figure is used for anything.

Classical multidimensional scaling in k dimensions keeps the k largest eigenvalues, and when some eigenvalues are negative the k largest are not necessarily the k most useful. On the sixteen-city set the best three-dimensional picture reads 27.3 per cent and the best four-dimensional one reads 63.1 — worse, with more room. That is not a statement about the sphere. It is that the fourth eigenvalue is small and positive, adding it stretches the picture along a direction carrying almost no structure, and a minimax score notices immediately.

So the dimension ladder is drawn on the four-city set, where the spectrum is (+, +, 0, −) and the reading is unambiguous, and the sixteen-city case is reported here in prose rather than in a figure. The honest version of the claim is the eigenvalue one: no number of dimensions reaches zero, because a negative eigenvalue is never spent. How the error behaves on the way there is a property of the estimator, which is exactly the distinction how wrong a flat picture has to be draws between an exponent that belongs to the sphere and a constant that belongs to the scoring rule.

The page that does hold them #

The second escape is a page that is not flat, and it works — for exactly one page.

The instrument is the trilateration from the rung below, moved onto a sphere of stated radius. Put the first place at the pole of the chart, the second down the zero meridian at the colatitude its distance implies, and place every further point by intersecting two circles. That holds 2n − 3 distances exactly by construction on a curved page just as it does on a flat one, and the distances it decides rather than holds are the reading.

Three things in that figure are worth separating.

The exactness at R is a check and not a finding. The distances came off a sphere of that radius, so a construction that reproduces them there is doing arithmetic correctly and nothing more. What makes it worth drawing is that the same code is wrong everywhere else on the axis, so the zero is a zero rather than a floor the instrument cannot see below.

The refusal below 0.9 R is geometry, not failure. On a sphere too small, the two circles a place must sit on do not intersect: the distances are simply too large for the page. That is a hard refusal with no fitted answer behind it, and it is what a construction has that an optimiser does not — an optimiser handed impossible distances returns its least-bad arrangement and says nothing.

The curvature is pinned to a couple of per cent. A reader who wanted a curved page with a convenient radius — a globe small enough for a desk, say, holding true distances — cannot have one. Two per cent off and the decided distances are wrong by nine per cent, which is worse than several projections manage on the same set.

The honest summary of the second escape is therefore that it is not an escape. A page of the right curvature holds the distances because it is the surface they were measured on; a globe is not a better map, it is the thing a map is a map of. What the figure adds is a number for how right the globe has to be, and two per cent is tighter than the phrase “use a globe” suggests.

What the globe’s tolerance means for a physical one #

Two per cent of the Earth’s radius is a hundred and twenty-seven kilometres, which sounds enormous until it is expressed as what it is: a globe whose radius is two per cent wrong is a globe at the wrong scale, and a globe at the wrong scale is exactly as right about every distance as a globe at the right one, because the reader divides by the scale bar.

The figure is measuring something else, and the difference is worth being careful about. The construction holds 2n − 3 distances in kilometres rather than in proportion, so changing the page’s radius while holding those distances fixed does not rescale the picture — it changes the ratio of the distances to the page’s own curvature, which is a real geometric change and not a change of units. What the axis therefore varies is the relationship between how far apart the places are and how curved the surface they are drawn on is.

Read that way, the tolerance says something about the Earth rather than about globes. If the Earth’s radius were two per cent different and the distances between cities were what they are, the cities would not fit. That is a statement of the same kind as the bound on the body the country is on, which prices what a country’s grid could achieve given the size of the planet under it: the planet’s radius is a parameter in every map problem and it is almost never treated as one.

The third escape, and why nobody has taken it #

The remaining freedom is the ruler. Every rung of this ladder has assumed that the reader of a picture measures the separation of two dots with a straight edge, and that assumption is a convention rather than a law.

Drop it and the constraint dissolves immediately. A picture in which distance is read off a nomogram, or from a scale that varies with position, or from a table beside the drawing, can be exactly right about every distance — the last of those trivially, since a table of distances is every distance exactly right. What it cannot be is a picture anybody reads at a glance.

That trade is worth stating precisely because it is the one this whole ladder measures the ends of. A distance table has n(n − 1)/2 exact distances and no geometry. A flat picture has 2n − 3 exact distances and complete geometry. An azimuthal equidistant map has n − 1 exact distances, complete geometry, and works for places nobody has listed. Those are three points on one axis and the axis is how much of the reading a reader has to do arithmetic for.

This collection has met the trade once before from the other side. A projection defined by a table has an interpolation in it is about a map whose definition is nineteen pairs of numbers rather than a formula, and the finding there is that the table does not define the map — the interpolation between its entries does, and three published interpolations disagree. A distance table has the same gap and hides it better: the table is exact at its entries and says nothing at all between them, and a reader wanting a distance to a place not listed has no rule to apply.

Two intermediate positions exist in practice and neither is usually described this way. A grid scaled to the ground is not a map prices the first at the scale a surveyor works at: a grid distance and a ground distance differ by a factor the user is expected to apply, and applying it is arithmetic the reader does rather than geometry the page provides. A map with a distance scale that varies — a graphic scale printed for several latitudes, which many older world maps carry — is trading a little reading effort for a lot of accuracy. And a map with a table of city distances in its endpapers is carrying both objects side by side, with the picture doing the geometry and the table doing the distances, which is the whole trade admitted rather than resolved.

The escape that is not on the list #

One candidate is missing from the three and its absence is deliberate: a page with negative curvature. If a positively curved page holds the distances of places on a sphere, a hyperbolic one might be thought to hold something.

It holds nothing, and answering that by argument rather than by measurement would be exactly the failure this collection exists to avoid. So the construction was rewritten with hyperbolic cosines in place of cosines — one line changes, the law of cosines — and run.

The hyperbolic branch is monotone, has no minimum anywhere, and is worse than the flat sheet at every curvature: 240.7 per cent at k = R against the flat sheet’s 152.3, 184.9 at k = 2R, 153.8 at k = 10R. Curving the page the other way makes the picture worse in proportion to how much it is curved, and the reason is the direction of the failure. The spectrum’s negative eigenvalue says the places are too close together for a Euclidean page — a plane has too much room between them, not too little — and a hyperbolic page has more room still.

That is worth recording because the hyperbolic page is a real and useful object elsewhere. It is the natural page for a tree, for a hierarchy, and for any set whose distances grow exponentially with depth, and there is a body of work on drawing such sets in it. What it is not is a page for the Earth, and the reason has nothing to do with cartography: the Earth’s places have too little room between them rather than too much.

What is left of the impossibility when all three are spent #

Putting the three together gives the statement this anchor exists to make.

Distances between places on a sphere cannot be drawn to scale on a flat sheet, and the failure is not about the sheet being flat rather than curved, nor about it having two dimensions rather than three. It is about the metric: great-circle distances are not the distances of any set of points in any Euclidean space, so the shortage is not of room. The only exact picture is the sphere itself at its own radius, and the only exact table is the table.

That is a sharper and narrower statement than Theorema Egregium, and it is worth having beside it rather than instead of it. The classical theorem is about a surface and a map and is proved with second derivatives; it says a map cannot preserve every distance in a neighbourhood, however small the neighbourhood. This says a finite set of named places cannot have every one of its distances right on a sheet, however few the places, and it is proved with the sign of an eigenvalue.

It also sits at a different place in the collection’s own argument. No map is faithful is where this field starts and its proof is Gauss’s; every rung after it prices a consequence in scale factors, angles or areas. What no essay before this anchor had is a proof that does not use curvature at all — and the eigenvalue argument does not. It never mentions a surface, a derivative or an embedding of the sphere in space. It takes a table of distances, does linear algebra, and reports a sign.

The two are related and neither implies the other. A metric of Euclidean type can still come from a curved surface, and a flat surface always gives one — so curvature is what makes the metric fail here, and the finite statement is what a reader of an atlas actually meets.

What the ladder has not asked #

Five rungs have treated the set of places as given. Somebody hands over a list of cities, the distances follow, and everything after that is arithmetic.

Nothing has asked where the list came from, and there is a question there. The five sets used in this ladder differ in their constant by a factor of seventeen at the same span, and the difference is entirely in how the places are arranged on the sphere — a set strung along a great circle is cheap and a set spread over a cap is expensive. That means a map maker handed a region and asked to show the places in it has a choice nobody has priced: which places to show. Dropping one city from a set of sixteen can only lower the least error, and dropping the right one may lower it a great deal.

Whether the drop is worth what it costs is not a question about geometry, and it is the first question in this anchor that is not.

There is a second gap and it is squarely geometric. Every set in this ladder is a set of points — cities treated as dots, with no extent. A real map shows regions, and the distance between two regions is not a number but a range, from the closest pair of their boundaries to the furthest. The shortest route between two coasts is the paths field asking that question about a route; asked about a picture, it changes the count, because a region contributes two coordinates and a shape, and a picture that is right about a range rather than a number has different freedom. Nothing in these five rungs applies to it.