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

The exponent belongs to the sphere and the constant to the arrangement

Drawn here at the parameters it defaults to, with every essay that calls it.
The exponent belongs to the sphere and the constant to the arrangement. The square law fitted separately on five configurations of places — from five towns inside one country to sixteen cities on every continent — each shrunk through its own ladder of spans. The exponents agree: 2.025, 2.016, 1.999, 2.013, 1.978. The constants in front of them do not, and span a factor of 17. That separation is the point of running five sets rather than one: how badly a particular arrangement of places resists a flat sheet is a fact about the arrangement, and that the resistance falls as the square of the span is a fact about the surface they sit on.

The square law fitted separately on five configurations of places — from five towns inside one country to sixteen cities on every continent — each shrunk through its own ladder of spans. The exponents agree: 2.025, 2.016, 1.999, 2.013, 1.978. The constants in front of them do not, and span a factor of 17. That separation is the point of running five sets rather than one: how badly a particular arrangement of places resists a flat sheet is a fact about the arrangement, and that the resistance falls as the square of the span is a fact about the surface they sit on.

It is drawn by curvature-figure with show: "distance-sets" — 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 least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not. The impossibility

How wrong a flat picture has to be

The rung below proves no flat picture of four places is exact and leaves the size of the failure to a determinant nobody can read. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.

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