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

The radius each set needs at its own size

Drawn here at the parameters it defaults to, with every essay that calls it.
The radius each set needs at its own size. How large every place must be for a flat picture of each set to be right about all its distances, on a logarithmic bar. Five towns, 623 km across: 6.2 m. Four cities, 16,994 km across: 38 km. Five cities, 16,994 km across: 575 km. Eight cities, 17,685 km across: 1,198 km. Five towns and four cities differ by a factor of twenty-seven in diameter and six thousand in the radius they need — about the cube of the one, with a factor of three belonging to their different arrangements. Four cities and five cities span the same diameter and need radii fifteen times apart, which is the arrangement rather than the size. The world sets' values are the best a search found, so they are upper bounds; the smallest set's agrees from every starting picture.

How large every place must be for a flat picture of each set to be right about all its distances, on a logarithmic bar. Five towns, 623 km across: 6.2 m. Four cities, 16,994 km across: 38 km. Five cities, 16,994 km across: 575 km. Eight cities, 17,685 km across: 1,198 km. Five towns and four cities differ by a factor of twenty-seven in diameter and six thousand in the radius they need — about the cube of the one, with a factor of three belonging to their different arrangements. Four cities and five cities span the same diameter and need radii fifteen times apart, which is the arrangement rather than the size. The world sets' values are the best a search found, so they are upper bounds; the smallest set's agrees from every starting picture.

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.

A radius of 37.7 km makes a flat picture of four cities exact. Each row is one pair of London, New York, Tokyo, Sydney. The bar is the range of distances between two places of radius 37.7 km — the true centre-to-centre distance, plus or minus 75.4 km — and the solid dot is where the best picture for places of that size draws the pair. Every dot is inside its bar, and the pairs at the bars' ends are the ones that fix the radius. The hollow dots are the picture that is best about ratios instead: its worst pair is 103.3 km off, so it would need places of 51.6 km. The best picture of points is not the best picture of places. The impossibility

A place with a size can be drawn to scale

Points on a sphere have distances no flat picture holds. Places are not points: give every place a radius and the distance between two of them becomes a range, and some flat picture is right about every range once the radius passes a threshold — 38 kilometres for London, New York, Tokyo and Sydney, six metres for five towns in Britain, growing as the cube of the set.

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