A place with a size can be drawn to scale

Every set in the essays on flat pictures is a set of points. Four cities that cannot be drawn to scale treats London as a dot with no extent, measures the great-circle distance from that dot to the dot that is Tokyo, and proves that no four dots on paper can have the six distances four such points have.

London is not a dot. It is tens of kilometres across, and the distance from London to Tokyo is not one number but a range — from the nearest pair of their edges to the furthest. A picture that is right about that range is right about the cities, whatever it does to their centres.

That changes the question from whether a flat picture can be exact to how large the places have to be before one can.

A distance between two places is a range #

Give every place in a set the same radius ρ and treat it as a disc. The distance between two such places, measured from any point of one to any point of the other, runs from d − 2ρ to d + 2ρ, where d is the distance between their centres. A flat picture draws the centres; it is right about the pair when the distance it draws, at its chosen scale, falls inside that range.

So a flat picture is right about every pair of places of radius ρ exactly when its largest absolute error, over all the pairs and at its best scale, is no more than 2ρ. The least radius at which some picture is right about the whole set is therefore half the least worst absolute error any flat picture can have.

That is a different quantity from the one the earlier essays measure, and the difference is the point. They score a picture on its worst relative error, because “drawn to scale” is a statement about ratios. A place’s extent is not a ratio: a city forty kilometres across is forty kilometres across whether the other city is five hundred kilometres away or fifteen thousand. So the score here charges every kilometre the same.

The picture best for points is not the picture best for places #

The hero figure carries two pictures of the same four cities, and they are different arrangements.

The hollow dots are the picture from the best flat picture is not a map, the one whose worst relative error is least. Its worst absolute error falls on the longest pairs, because a small relative error on a distance of seventeen thousand kilometres is a large number of kilometres, and it would need places of 51.6 kilometres to be right about every range.

The solid dots are a picture found for this question instead. It accepts a larger relative error on the short pairs, where a per cent is few kilometres, in exchange for a smaller one on the long pairs, and needs places of only 37.7 kilometres. Neither picture is better. They answer different questions, and a map maker who wants dots of a given size to be honest needs the second.

The radius grows as the cube of the set #

The exponent follows from a result already measured. How wrong a flat picture has to be finds the least relative error falling as the square of the set’s angular span. An absolute error is a relative error times a distance, and the distances in a set scale with its span, so the least absolute error — and with it the radius — scales as the span cubed.

Read off the four cities shrunk step by step, the law is concrete. The arrangement shrunk to 8,807 kilometres across needs places of 3.3 kilometres; to 4,414, of 400 metres; to 2,208, of 49 metres; to 1,104, of 6.2 metres; to 552, of 77 centimetres. The constant in front of the cube is 4.6 × 10⁻¹² per square kilometre for this arrangement: radius equals that constant times the diameter cubed.

The eight-city arrangement runs parallel to the four-city one and sits above it. That is the separation how wrong a flat picture has to be draws between an exponent that belongs to the sphere and a constant that belongs to the arrangement, arriving again for a different score: the eight cities need larger places at every size, and by the same factor at every size.

The same cube, one field over #

How small is flat enough asks a question that sounds the same and is not. It asks how large a patch of ground can be before treating it as flat puts a measured distance out by more than a tolerance, and finds the error growing as the square of the patch — a relative error, fixed by a surveyor’s instrument.

The question here is what size the objects on a picture must be for the picture to be honest about them, and the answer grows as the cube because the objects’ extent is absolute. A surveyor’s tolerance is a fraction of what is measured. A place’s size is not.

The radius is a promise the picture can keep #

A dot on a map is a claim about where something is, and the claim has a width. A tolerance is a promise about the picture makes that point about simplifying a line: a tolerance is not an error allowance hidden in the data but a statement to the reader about what the ink can be trusted to within. The radius measured here is the same kind of statement for distances.

Read that way, the threshold is not a size places must have but a size a picture can honestly promise. A picture of the four world cities can promise that every distance between them is right to within the width of places 38 kilometres in radius, and no finer; a picture of five British towns can promise six metres. Anyone reading distances off either picture with more precision than that is reading the drawing rather than the ground, and the essays on flat pictures before this one are about exactly that mistake made for points.

The promise has a direction worth noticing. It is weaker for a large set, which sounds obvious, and it is weaker in proportion to the cube of the set’s size, which does not. Doubling the extent of a map’s coverage does not double what its distances can promise. It multiplies the width of that promise by eight.

Set by set, at their own sizes #

Five towns in Britain need places of six metres. Any flat picture of Edinburgh, London, Cardiff, Norwich and Plymouth that treats each town as a disc a dozen metres across can be made right about every distance between them — which is to say that for a map of a country, the impossibility of drawing its towns to scale is smaller than the towns.

The world sets are a different matter, and the two of them that share a diameter show why. Four cities and five cities both span 16,994 kilometres; adding Cape Town to London, New York, Tokyo and Sydney raises the radius needed from 38 kilometres to 575. The set did not get larger. It got harder to arrange, because Cape Town’s distances to the other four cannot be fitted without straining pairs the four-city picture had left comfortable. That is the arrangement’s constant, and at world scale it varies by more than the cube law does.

What the arrangement puts into the constant #

The cube law fixes how the radius changes as one arrangement is shrunk or enlarged. It says nothing about which arrangement, and the constant in front of the cube carries all of that.

Two arrangements measured through a whole sequence of sizes show how wide the constant runs. For the four cities the constant is 4.6 × 10⁻¹² per square kilometre; for the eight it is 1.7 × 10⁻¹¹, nearly four times larger. At a diameter of a thousand kilometres, that is the difference between places of five metres and of seventeen.

The refusal below marks one end of the range exactly. Places strung along a single great circle have a constant of zero: they need no extent at any size, because their distances are a line’s. The escape is not a dimension observes the same ordering for relative error, finding a set strung along a great circle cheap and a set spread over a cap expensive. The arrangement’s constant is how far a set is from a line, measured in the one currency a flat picture charges.

And the constant does not stay put as the set grows to the size of the world. The five-city set is the four-city set with Cape Town added, at exactly the same diameter, and its radius is fifteen times larger — so at world scale the constant has stopped being a property of an arrangement that can be shrunk and become a property of which places are present. That is the finding of the error belongs to a few of the places arriving from the other side: for points, one city carried half the error; for places, one city multiplies the size they must have by fifteen.

The values for the five-, eight- and sixteen-city sets are quoted as upper bounds, and the reason is the instrument. The least absolute error is found by the same kind of search the relative error is, and a search can stop above the true optimum. On the small sets the values fall on a line of slope three to three decimal places across five halvings, which a search stopping at arbitrary points would not do; on the world sets two runs of the eight-city search stopped 3 per cent apart. The small sets carry the law.

What each objective costs the other #

The picture that is best about ratios is never best about places, and the penalty for using it is between a quarter and three quarters of the radius.

The penalty does not follow the set’s size. It is largest on the smallest set, the British towns, and smallest on the five world cities. What it follows is how different the set’s distances are from one another: the relative score cares about the shortest distances as much as the longest, and the absolute score cares about them hardly at all, so a set whose distances run across a wide range of lengths gives the two scores the most to disagree about. Britain’s towns run from about 160 kilometres apart to 620, a range of nearly four; the five world cities’ shortest distance is already several thousand kilometres.

Places along one great circle need no size at all #

Every measurement above has to be able to return zero, or the radii are measuring the search.

Places strung along a single great circle, within half of it, have great-circle distances that simply add: the distance from the first to the third is the distance from the first to the second plus the distance from the second to the third. Those are the distances of points on a straight line, and a straight line is a flat picture. The measured radius is zero to thirteen decimal places.

Lift the middle place off the circle and additivity fails, a little at first. The radius needed grows as the square of the lift — sixteen times larger for a lift four times larger — which is the local version of the same curvature every other set here is paying for. A quarter of a degree off a great circle costs places of twenty-three centimetres.

What a printed map already does #

A dot on a printed map has a size, and the size is a radius on the ground.

On a world map at a scale of one to fifty million, a millimetre on the paper is fifty kilometres. The four world cities need places of 38 kilometres, so a dot a millimetre and a half across — an ordinary city symbol — is already large enough for some flat picture to be right about every distance between the four cities to within the ink. The impossibility this whole field is built on is, for these four, smaller than the symbol that marks them.

The eight-city set is not so kind. It needs places of up to 1,198 kilometres, which at the same scale is a dot nearly five centimetres across. No symbol makes eight world cities honest on one sheet, and the picture that tries has to let some distances fall outside anything the dots could excuse.

For a national map the arithmetic is the other way round. At one to a million a millimetre is a kilometre, and a country’s towns need places of metres. The ink is a hundred times larger than the error, and the question of drawing to scale does not arise at all.

The scale at which it starts to matter #

Between those two ends there is a size of map at which the dot and the radius are the same, and it is worth locating, because it is where a map of distances stops being honest by default.

Take the four-city arrangement and a printed dot one millimetre across, and ask at what scale and coverage the dot’s radius on the ground equals the radius the arrangement needs. At a continental coverage of 4,400 kilometres the places must be 400 metres in radius; a half-millimetre dot is 400 metres on the ground at one to 800,000, a scale no continental map is drawn at. At 8,800 kilometres the places must be 3.3 kilometres; the same dot covers that at one to six and a half million, where a map of that coverage would be more than a metre across. At the full 17,000 kilometres the dot must cover 38 kilometres, which it does at one to seventy-five million — a world map about half a metre wide. A page-sized world map is printed at half that scale, where the same dot covers twice the ground.

So for this arrangement the ink of a printed map is always larger than what it has to excuse, at every scale anybody prints a map of that coverage at. The same is not true of the eight-city set, whose places at world size need a radius thirty times larger, and the difference between the two is not their size but which cities are in them. The bound on the body the country is on finds the size of the planet setting a limit on what a national grid can achieve; here the size of the planet sets how quickly the limit on a picture of places tightens, and the arrangement sets where it starts.

Where the model stops #

Every place has the same radius. Real places differ in size by orders of magnitude, and a set with a large city and a small town would be scored more fairly with a radius for each — which is a different and harder problem, with a set of radii to choose rather than one.

A place is a disc. The range of distances between two real places depends on their shapes and their orientation relative to each other, and a long coastal city is nearer some places than its radius suggests. A disc is the shape that makes the range depend on one number.

The scale is free. A picture here is allowed whatever scale suits it best, which is the same freedom a printed scale bar represents. A picture held to a stated scale would need larger places.

And the world sets’ radii are upper bounds, found by searching. The cube law is established on sets small enough that every starting picture agrees.

Still open: places of different sizes #

The honest version of this question gives London a radius of twenty kilometres and Reykjavík one of five, and asks whether some flat picture is right about every range those sizes allow.

Allowing each place its own radius changes what is being asked. Five distances of six, and never more counts what a flat picture of points can hold exactly — 2n − 3 distances — and a picture of places with a radius each has n more numbers to spend, one per place, which is enough in principle to hold every distance if the radii may be as large as they like. It is no longer a threshold but a trade: a picture might be exact if the big cities are allowed their full extent and the small towns are held to a point, or it might need the towns to be larger than they are. The natural score is the total area of the places rather than the largest radius, and whether the pairs that fix the answer are still a handful — as they are for one radius, and as the error belongs to a few of the places finds they are for points — is something one shared radius cannot say.