Cartographic Projection
every map is wrong, and by exactly how much
Figure

What a map centred on a place spends, and what it leaves unspent

Drawn here at the parameters it defaults to, with every essay that calls it.
What a map centred on a place spends, and what it leaves unspent. An azimuthal equidistant map centred on London is exactly right about every distance FROM London — measured here at 6.7e-16 on three of them, which is rounding — and about nothing else: the worst of the other 3 is out by 32.3%. Those n − 1 exact distances are the star, and the plane allows 2n − 3. The gap between the two curves is n − 2 distances a flat picture could have held and no projection centred on a place does, because a projection is a rule about the whole sphere and cannot spend its freedom on the places that happen to be in the set.

An azimuthal equidistant map centred on London is exactly right about every distance FROM London — measured here at 6.7e-16 on three of them, which is rounding — and about nothing else: the worst of the other 3 is out by 32.3%. Those n − 1 exact distances are the star, and the plane allows 2n − 3. The gap between the two curves is n − 2 distances a flat picture could have held and no projection centred on a place does, because a projection is a rule about the whole sphere and cannot spend its freedom on the places that happen to be in the set.

It is one of 78 figures drawn by the same construction, and the drawing above is this one with nothing changed.

1 essay calls it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

The share of its distances a flat picture can hold. A picture of n places on a plane has 2n coordinates and is unchanged by two translations and a rotation, so 2n − 3 numbers in it are genuinely free; each distance held exactly is one equation. So at most 2n − 3 of the n(n−1)/2 distances can be right, whatever the places are and however hard the map maker tries. The share falls as 4/n: five of six for four places, thirteen of twenty-eight for eight, and 197 of 4,950 — 4.0% — for a hundred. The dashed curve is 4/n, which the count approaches from below and never crosses. The impossibility

Five distances of six, and never more

A flat picture of n places has 2n − 3 free numbers and n(n−1)/2 distances, so it can be exactly right about 2n − 3 of them and no more — five of six for four places, twenty-nine of a hundred and twenty for sixteen, and 3.98 per cent for a hundred. The bound is reached by a construction with no fitting in it, and an azimuthal equidistant map spends only n − 1.

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