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

Every bearing from two places, and how many point backwards

Drawn here at the parameters it defaults to, with every essay that calls it.
Every bearing from two places, and how many point backwards. For each set, every ordered pair of centres: hold every bearing out of the first and every bearing out of the second, 2n − 3 in all. The bar is the fewest arrows any pair of centres leaves pointing backwards, as a share of those held. Five towns in Britain: none, and all 20 pairs of centres drawable. Four cities: 1 of 5. Eight: 3 of 13. Sixteen: 8 of 29, from Santiago and Buenos Aires, and not one of the 240 pairs of centres drawable.

For each set, every ordered pair of centres: hold every bearing out of the first and every bearing out of the second, 2n − 3 in all. The bar is the fewest arrows any pair of centres leaves pointing backwards, as a share of those held. Five towns in Britain: none, and all 20 pairs of centres drawable. Four cities: 1 of 5. Eight: 3 of 13. Sixteen: 8 of 29, from Santiago and Buenos Aires, and not one of the 240 pairs of centres drawable.

It is one of 51 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.

Five bearings of four cities, held exactly. A flat picture of London, New York, Tokyo, Sydney holding five one-way bearings exactly, each drawn as an arrow from the city it is measured at: London to Tokyo, Sydney to London, New York to Tokyo, New York to Sydney, Tokyo to Sydney. Five is the most any flat picture of four places can hold, and of the 192 ways of choosing five pairs and a direction on each, this is one of eight that can be drawn with every arrow pointing forwards; all eight leave London–New York, dashed, unheld. The price is paid by the other seven bearings: the worst, London to New York, is 165.0° from the truth. The impossibility

Five bearings of twelve, and only eight ways to draw them

A flat picture of four places can hold five of their twelve one-way bearings exactly, the same count as distances, and the five are found by solving equations rather than searching. The equations cannot tell a line from its reverse. Of the 192 ways to choose five, eight can be drawn with every arrow pointing forwards, all eight leave London and New York unheld, and on sixteen cities no choice made at random can be drawn at all.

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