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

The determinant that has to be zero, and is not

Drawn here at the parameters it defaults to, with every essay that calls it.
The determinant that has to be zero, and is not. The Cayley–Menger determinant of the four places, divided by the sixth power of their diameter so that only the shape is left, against how much of the sphere they span. Zero is what a flat picture requires; the chordal distances give it to rounding at every size, and the geodesic ones never do. The fitted slope is 2.149 on the log axes, so the departure falls as roughly the square of the span — the same exponent every other measurement in this ladder finds, arriving here through a determinant rather than through an optimisation.

The Cayley–Menger determinant of the four places, divided by the sixth power of their diameter so that only the shape is left, against how much of the sphere they span. Zero is what a flat picture requires; the chordal distances give it to rounding at every size, and the geodesic ones never do. The fitted slope is 2.149 on the log axes, so the departure falls as roughly the square of the span — the same exponent every other measurement in this ladder finds, arriving here through a determinant rather than through an optimisation.

It is drawn by curvature-figure with show: "distance-determinant" — one member of a family of 29 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

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.

The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes. The impossibility

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

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