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The thread: A theorem, not a limitation — page 5

No map is faithful, and this is not a shortcoming awaiting a cleverer cartographer. Gaussian curvature is intrinsic, a sphere has some and a plane has none, and that settles it. Essays 97 to 98 of 98.

The points a projection cannot move. The critical points of a stated field, found twice: once on the sphere from its own gradient, and once in Lambert cylindrical's page coordinates from the page's own numbers, with nothing shared between the two searches. seven points, 3 maxima, 2 saddles and 2 minima, and the two sets agree in position to 2.3e-7 degrees and in type at every one. The page's Jacobian is invertible wherever the map is a map, so it sends a zero gradient to a zero gradient and cannot move a critical point anywhere.

Measuring distortion

What the page cannot move

Six rungs measure what a page does to a field's readings and every one of them moves. The critical points do not: found on the sphere and found again in a projection's own page coordinates with nothing shared between the searches, they agree to 10⁻⁷ degrees and in type at every point, on every projection. What the page does move is their shape, by a factor bounded exactly by the indicatrix's axis ratio squared.

The doubling ladder is the one sequence that cannot see it. The largest departure of a small circle from its own indicatrix that a grid of n latitudes over 10° to 70° north finds on the Robinson projection. The filled marks are 4, 8, 16, 32, 64 and 128 — the doubling ladder every convergence study runs — and they rise smoothly to about 4.66e-4 with the increments halving, which is what a convergent first-order sequence looks like. The open marks are grids whose samples land on the projection's five-degree table entries. They report 1.28e-2, twenty-seven times higher, and whether a grid does that is decided by whether n is a multiple of four.

Measuring distortion

A refinement that stops moving

Doubling the sample and watching the answer settle is how every quadrature in every field is checked. On the Robinson projection the doubling ladder — 4, 8, 16, 32, 64, 128 — converges beautifully, with its increments halving at every step, on a limit that is wrong by a factor of twenty-seven. Whether a grid finds the answer is decided by whether n is a multiple of four.

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