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

Five pages of different curvature, and the one that works

Drawn here at the parameters it defaults to, with every essay that calls it.
Five pages of different curvature, and the one that works. The worst decided distance on a page of each stated radius, for London, New York, Tokyo, Sydney. A page too small is refused rather than fitted — the construction reaches a step where two circles that must cross do not, which is a geometric fact rather than a failure of a solver. A page too large is drawn and wrong. Exactly one radius is right, and it is the radius the distances were measured on, which is the whole of what "the escape is a curved page" amounts to: the page has to be the sphere.

The worst decided distance on a page of each stated radius, for London, New York, Tokyo, Sydney. A page too small is refused rather than fitted — the construction reaches a step where two circles that must cross do not, which is a geometric fact rather than a failure of a solver. A page too large is drawn and wrong. Exactly one radius is right, and it is the radius the distances were measured on, which is the whole of what "the escape is a curved page" amounts to: the page has to be the sphere.

It is drawn by curvature-figure with show: "distance-curved" — 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 page's curvature is not a free parameter. The same trilateration, carried out on a sphere of stated radius instead of on a plane, with the error of the distances the construction decides rather than holds. At the radius the distances were measured on it is 5.1e-15 — exact, by construction, which is the refusal this figure exists to make. Two per cent either side of it the worst decided distance is already out by around 10%. Below 90 per cent of the Earth's radius the construction cannot be completed at all: two circles that must cross do not, and the page is simply too small to hold the places. The flat sheet is the right-hand limit, at 152%. The impossibility

The escape is not a dimension

Four places that will not lie in a plane surely lie in a space — and they do not. The double-centred matrix of their great-circle distances has a negative eigenvalue, so no Euclidean space of any dimension holds them, and buying a third dimension improves the picture by nothing whatever, to fifteen decimal places. What does work is a page with curvature, and the curvature is pinned to within two per cent.

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