Figure

Four cities cannot be drawn to scale, and any three can

Drawn here at the parameters it defaults to, with every essay that calls it.

The least worst-case error of a flat picture of London, New York, Tokyo and Sydney is 0.629%. Leave any one of them out and the three that remain have an exact flat picture — the error is zero to the arithmetic in all four cases — because three distances that satisfy the triangle inequality are always the sides of a triangle. That is the one set where leaving out a place is guaranteed to buy everything, and it is the check that the instrument returns zero when zero is the answer.

It is one of 44 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.

Sixteen cities, each left out in turn. The least worst-case distance error any flat picture of these sixteen cities can have is 53.3%; the best projection in the library, Azimuthal equidistant, manages 57.6%. Each bar is that least error with one city left out. Leaving out Sydney brings it to 25.5%; leaving out Cape Town, the next most costly, to 46.7%; and leaving out any of five — New York, Tokyo, Anchorage, Singapore, Los Angeles — changes it not at all. The note on a row counts the pairs holding the picture at its worst that the city belongs to.

The impossibility

The error belongs to a few of the places

The least error of any flat picture of sixteen cities is 53.3 per cent, and it is not spread over the sixteen. Six pairs hold it there and Sydney is in three of them: leave Sydney out and it falls to 25.5, while leaving out any of five other cities changes it not at all. Choosing which places to draw is worth more than any choice of how to draw them.

The whole library