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

The sixth constraint chooses a turn, and not the turn the other bearings would choose

Drawn here at the parameters it defaults to, with every essay that calls it.
The sixth constraint chooses a turn, and not the turn the other bearings would choose. Five distances fix the shape of a picture of four cities and leave it free to turn. The worst of its twelve bearing errors, against how far it is turned. The best turn is 342°, where the worst bearing is 132.1° out and the mean is 65.5°. The turn that holds London to New York exactly is 83°, where the worst is 165.9° and the mean 96.3°. So the sixth constraint costs nothing in distances — the shape does not move — and it is not free in bearings: it spends the turn on one of them, and the eleven others pay 33.8° more in the worst case than they need have.

Five distances fix the shape of a picture of four cities and leave it free to turn. The worst of its twelve bearing errors, against how far it is turned. The best turn is 342°, where the worst bearing is 132.1° out and the mean is 65.5°. The turn that holds London to New York exactly is 83°, where the worst is 165.9° and the mean 96.3°. So the sixth constraint costs nothing in distances — the shape does not move — and it is not free in bearings: it spends the turn on one of them, and the eleven others pay 33.8° more in the worst case than they need have.

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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