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

Three of the five arrows may point either way, and two may not

Drawn here at the parameters it defaults to, with every essay that calls it.
Three of the five arrows may point either way, and two may not. A flat picture of London, New York, Tokyo, Sydney holding five bearings exactly. All eight drawable choices hold the same five pairs and differ only in direction, and they agree on two of them: London to Tokyo and New York to Tokyo, drawn heavy, point that way in all eight. The other three — London–Sydney, New York–Sydney, Tokyo–Sydney — are drawn light and appear four times each way, which is what makes eight rather than two. The two forced pairs are the two of largest bearing asymmetry, 124.6° and 127.9°, against 78.5° for the largest of the free ones. London–New York, dashed, is held by none.

A flat picture of London, New York, Tokyo, Sydney holding five bearings exactly. All eight drawable choices hold the same five pairs and differ only in direction, and they agree on two of them: London to Tokyo and New York to Tokyo, drawn heavy, point that way in all eight. The other three — London–Sydney, New York–Sydney, Tokyo–Sydney — are drawn light and appear four times each way, which is what makes eight rather than two. The two forced pairs are the two of largest bearing asymmetry, 124.6° and 127.9°, against 78.5° for the largest of the free ones. London–New York, dashed, is held by none.

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.

Two pairs of ten are in every picture that can be drawn. London, New York, Tokyo, Sydney, Cape Town, drawn holding seven of their ten pairs' bearings exactly — the most any flat picture of five places can hold. Of the 15,360 ways of choosing seven pairs and a direction on each, 152 give a picture whose held arrows all point forwards, and London–Tokyo and Sydney–Cape Town, drawn heavy, appear in every one of them. The three pairs this particular choice leaves out are dashed. The price is paid by the rest: the worst, New York to Sydney, is 142.1° from the truth. The impossibility

The pair that cannot be held is not the one the geometry names

Four world cities have exactly one pair of the six that no drawable picture holds, and the obvious candidate — the pair whose bearing there and bearing back disagree most — is not it. That quantity predicts something else exactly: whether a held pair's arrow has a forced direction. It separates the two groups with a clean gap and nothing in between. And at five places there is no forbidden pair at all, so the four-place case was a statement about how little room five constraints leave rather than about the map.

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