The error belongs to a few of the places

Five essays measure what a flat picture of a set of places cannot do. Four cities that cannot be drawn to scale proves it cannot be exact; how wrong a flat picture has to be finds the error growing as the square of the set’s span; the best flat picture is not a map finds a free arrangement of dots beating every projection; five distances of six, and never more counts what a picture can hold; and the escape is not a dimension closes the ways out.

Every one of them takes the list of places as given. Somebody hands over sixteen cities, and everything after that is arithmetic about those sixteen.

A map maker is not handed a list. A map maker chooses one, and the choice turns out to matter more than anything the arithmetic can do with it.

Leaving a place out can only help #

Before the numbers, one thing about them is certain in advance, and it is what makes the rest measurable rather than noisy.

Any flat picture of all sixteen cities, with one dot erased, is a flat picture of the other fifteen, and it has the same worst error or a smaller one — the erased dot’s pairs are simply no longer counted. So the least error of fifteen can never be larger than the least error of sixteen. Leaving a place out can help or do nothing. It cannot hurt.

That turns the bars into a clean question: how much does each city’s presence cost? The answers run from exactly nothing to more than half of everything.

The error quoted is the same one the earlier essays use. A picture is given its one free choice of scale, every one of its 120 pairs is compared with the great-circle distance, and the worst relative error over all of them is the score. The least score any picture can have is found by moving the dots one at a time until no move helps, starting from classical scaling and from the layout of every projection in the library with its azimuthal members centred on the set — so the free picture cannot come out worse than a map, since each map’s layout is one of the pictures the search is allowed.

Six pairs hold the picture at its worst #

A worst-case score is a statement about a few pairs, not about all of them, and the few are identifiable.

At the best picture, the worst error is reached by six pairs at once, and no fewer. If it were reached by one, that pair could be improved a little and the score would fall. The search stops only when the worst error is held at its level by several pairs pulling in opposite directions — some drawn too long, some too short — so that relieving any one of them strains another past it.

The six are London–Reykjavík, Sydney–Reykjavík and Santiago–Reykjavík, all drawn too short, and Sydney–Cape Town, Sydney–Santiago and Santiago–Buenos Aires, all drawn too long. The rest of the 120 pairs are slack: moving any of their cities a little changes nothing about the score.

Reading the bars against that list explains most of them. The five cities whose absence changes nothing — New York, Tokyo, Anchorage, Singapore and Los Angeles — belong to none of the six. Sydney belongs to three, and its absence halves the error.

Membership is not the whole account #

Three cities belong to three of the six pairs, and only one of them is expensive.

Leaving out Santiago brings the error only to 51.5 per cent, and leaving out Reykjavík to 50.7. Both sit in as many of the pairs that bind as Sydney does, and neither buys much. The reason is that the binding pairs are a property of one picture, and removing a city changes the picture. Take Reykjavík away and the pairs it was in go slack — but the dots rearrange, and a different six pairs of the remaining cities take over at a level only a little below the old one.

And one city belongs to none of the six and still helps. Leaving out Honolulu brings the error to 48.7 per cent. In the best picture of all sixteen, Honolulu’s pairs are all slack; without it, the search finds a different arrangement altogether, one that was unavailable while Honolulu’s dot had to be placed somewhere.

So the pairs that bind a picture say which cities are certainly free — the ones in none of them — and say nothing reliable about how much the others cost. That has to be measured, one omission at a time.

Where on the sphere the strain sits #

The six pairs are not scattered, and where they fall says what kind of picture the best flat arrangement of sixteen cities is.

The centroid of the sixteen lies in the Norwegian Sea, at 69° north. In the best picture the city drawn nearest the middle is Reykjavík, twelve degrees from that centroid on the sphere; London and Moscow come next. The cities drawn furthest out are Sydney, Santiago, Buenos Aires and Cape Town, in that order — and they are the four furthest from the centroid on the sphere as well, between 103 and 139 degrees round it. The free picture has arranged itself as a disc centred on the far north Atlantic with the southern hemisphere laid round its rim. That is the arrangement an azimuthal equidistant map centred there makes, and that map is the best projection in the library for this set.

A disc drawn that way has two things it must get wrong, and the six binding pairs are exactly those two things.

Round its rim it must stretch. The southern cities lie near a circle of the sphere that curls under the south pole, and a flat disc lays that circle out at its full radius, so everything running round the rim is drawn longer than it is: at 117 degrees from the centre the stretch is a factor of 2.3, and at 139 degrees 3.7. Sydney–Cape Town, Sydney–Santiago and Santiago–Buenos Aires are all rim pairs, and all three are drawn too long.

Having drawn the rim long, a picture with a single scale has to draw something else short to balance it, and what it shortens are pairs running from the middle. All three short pairs end at Reykjavík: to London beside it, and out to Santiago and to Sydney on the rim.

That reading accounts for the city whose absence halves the error. Sydney is the furthest of the sixteen from the centroid, so it sits where the rim’s stretch is largest and at the far end of the longest pair from the middle; it belongs to both kinds of binding pair at once. Leave it out and the rim draws in to Santiago at 117 degrees, the stretch the picture has to absorb falls from 3.7 to 2.3, and everything relaxes.

It also accounts, less neatly, for why Santiago and Reykjavík are cheap to lose. Santiago’s place on the rim is taken by Buenos Aires, a few degrees further in; Reykjavík’s place in the middle is taken by London and Moscow, a few degrees further out. Each has a near neighbour ready to take its role, and Sydney, alone in the far south-west of the Pacific, does not.

Four cities, and why any three are exact #

The same measurement on the smallest set that cannot be drawn to scale is the check that the instrument returns zero when zero is the answer.

Four places have six distances and no flat picture holds all six. Three places have three distances, a flat picture of three dots has three free numbers once the rigid motions are spent, and any three great-circle distances obey the triangle inequality, so a triangle with those sides exists. Leaving out one of four cities therefore buys everything, and it does so whichever city goes.

The sixteen-city bars could not be read without this. A search that returned small non-zero errors for three cities would be a search that stops early, and every bar above would be measuring where it stopped. It returns zeros.

One city at a time #

If one omission halves the error, the natural next question is what a second, third and fourth buy.

The first omission is the big one and the rest are steady: each of the next three takes a further fifth or so off what is left. Four cities out of sixteen buy a picture whose worst distance is wrong by an eighth rather than by a half.

The procedure is greedy, and that is a real limitation rather than a formality. The best four cities to leave out together are found by trying all 1,820 foursomes; these four were found in fifty-eight searches by always taking the best single step. A greedy sequence can miss a pair of cities that help little alone and a great deal together, and nothing here rules that out. What the sequence establishes is a floor on how much omission can buy: at least a factor of four, for a quarter of the places.

The order in which places matter is not a ranking #

The greedy sequence contains a surprise that is worth a figure of its own.

Singapore is one of the five cities whose absence from the full set changes nothing. It is also the best city to leave out once Sydney has gone, by a margin: 20.4 per cent without it, against 20.7 without Cape Town and 21.6 without Honolulu.

Nothing about Singapore changed. What changed is which pairs bind. With Sydney present, the worst of the picture was pinned by pairs running across the southern oceans from Sydney, and Singapore’s pairs were all slack. With Sydney gone, the picture relaxes into a new arrangement whose worst pairs run through Singapore instead.

So there is no list of cities ordered by cost that a map maker could consult. The cost of a city depends on which others are in the picture, and the city that is free to include in one selection is the one to exclude in another. Any rule of the form “leave out the most distant place” or “leave out the most isolated place” would have to be re-evaluated after every omission, and the evidence here is that the re-evaluation changes the answer.

Choosing the places against choosing the picture #

The three earlier essays on flat pictures are all about choosing a picture for a fixed set of places. The size of what they found can now be set beside the size of what choosing the places does.

The best flat picture is not a map finds that a free arrangement beats every projection by a factor of thirteen on four cities and by only eight per cent on sixteen, and explains the collapse by counting: a picture of n places has 2n − 3 free numbers against n(n − 1)/2 distances, so on sixteen places it has almost no freedom to spend. That eight per cent is the most any choice of picture can buy for this set.

Leaving out one city more than halves the error, where the best possible picture takes eight per cent off it. Leaving out four cities divides it by four. The largest decision in a flat picture of many places is not how it is drawn but what is in it.

That is not an argument for dropping inconvenient cities from maps. It is a price. A world map that must show Sydney accepts an error in its worst distance that is twice what it would otherwise be, and the choice of projection cannot buy any of it back.

What an atlas already does about it #

Map makers do not leave Australia off world maps. They do something that has the same effect on the arithmetic: they break the picture.

An inset — a separate panel for a place that would strain the main picture — is a flat picture of a smaller set, joined to the main one by a caption rather than by geometry. What a cut buys prices the equivalent move for a projection, where interrupting a map into lobes buys shape at a cost in cut length. For a finite set of places the move is cleaner: the places are partitioned, each part gets its own flat picture, and the distances between the parts are no longer drawn at all.

Two charts are enough, and one is not counts the fewest sheets a whole sphere needs, and how many sheets an atlas needs counts how many a stated tolerance needs. A set of places is the finite version of both questions, and the arithmetic here supplies its first term. An atlas of these sixteen cities with Sydney on a sheet of its own has a worst distance error on its main sheet of 25.5 per cent, where a single sheet had 53.3 — a factor of two for one inset, before any cleverness about which sheet should hold what, and before any sheet is given a projection.

The measurements here say where such a partition pays. It pays when one place, or a few, sits in the pairs that hold the picture at its worst and nowhere else — the Sydney case — and it buys nothing when the cost is spread, as it is for Santiago and Reykjavík, whose absence merely hands their role to others.

Where the model stops #

The score is a worst case. A picture scored on its mean or root-mean-square error spreads its cost over every pair, and no single city would then carry half of it. The worst case is the right score for a promise — “no distance on this sheet is wrong by more than” — and the wrong one for an impression. The average was a choice of norm shows how completely the choice between a mean and a worst case can reorder a comparison of projections; whether it would reorder these cities, and name a different city as the costly one, is not measured here.

The searches are searches. Every least error here is the best a pattern search found, started from classical scaling and from every projection’s layout. A search can stop above the true optimum and never below it. Two facts guard the numbers: every leave-one-out value comes out at or below the full set’s, as it must, and every three-city value comes out exactly zero.

The four omissions are greedy. They give a floor on what omission buys, not the best four places to omit.

And the sixteen cities are one set. They were chosen, for an earlier essay, to cover every continent; a set chosen differently would have different binding pairs and a different Sydney. What is general is the shape of the finding — a worst case held by a handful of pairs, most places free and a few expensive — and not the name of the city.

Still open: whether the costly place can be named without searching #

Every omission here was priced by searching for the best picture without the place. That is sixteen searches for one step and fifty-eight for four, which is cheap on sixteen places and not on sixteen hundred.

What would make the finding usable is a rule that names the costly place from the geometry alone. The obvious candidate — the place in the most pairs that bind — named Sydney correctly and named Santiago and Reykjavík, which cost almost nothing, just as confidently. A better candidate might weigh each binding pair by how much the picture has to strain to hold it, or look for the place furthest from where the rest of the set’s great circles run. Whether any such rule survives a second set of places is not something one set can show.