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

The extra constraint is one, and stays one however many places there are

Drawn here at the parameters it defaults to, with every essay that calls it.
The extra constraint is one, and stays one however many places there are. The largest number of constraints a flat picture can hold exactly, against the number of places, for distances alone, bearings alone and the two mixed. Distances give 5, 7, 13 at four, five and eight places and bearings give the same, which is 2n − 3 in both cases. Mixed sets give 6, 8, 14 — exactly one more, at every size. The gain does not grow with the picture because it is not a share of anything: it is the single freedom that neither pure kind can see, and there is only ever one of it.

The largest number of constraints a flat picture can hold exactly, against the number of places, for distances alone, bearings alone and the two mixed. Distances give 5, 7, 13 at four, five and eight places and bearings give the same, which is 2n − 3 in both cases. Mixed sets give 6, 8, 14 — exactly one more, at every size. The gain does not grow with the picture because it is not a share of anything: it is the single freedom that neither pure kind can see, and there is only ever one of it.

It is one of 65 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.

Six of eighteen, where either kind alone can hold only five. London, New York, Tokyo, Sydney, drawn holding five distances exactly — the thick edges — and one bearing exactly, the arrow from London to New York. Six exact constraints, where five distances is the most any flat picture of four places can hold and five bearings is the most it can hold of those. All six are satisfied to 1.9 × 10⁻¹¹. The sixth is not free room found by luck: a picture is blind to where it sits, how it is turned and how large it is, distances see the last of those three and bearings see the second, and a picture holding both kinds is blind only to where it sits. Tokyo–Sydney, dashed, is the distance left out; the picture is wrong about it by 6.9%. The impossibility

A mixed picture holds one more, and what it is holding is north

A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.

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