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

What each extra dimension buys, which is nothing

Drawn here at the parameters it defaults to, with every essay that calls it.
What each extra dimension buys, which is nothing. The worst relative error of the best k-dimensional picture of London, New York, Tokyo, Sydney, for every k the set allows. Going from a line to a plane buys almost everything — from 180.2% to 1.27%. Going from a plane to a space buys nothing at all: the two numbers agree to every digit, because the third eigenvalue the extra dimension would spend is 2.85e-8 — zero — and the eigenvalue that would help is negative and cannot be spent in a Euclidean space of any dimension. The flat sheet is not the obstruction. Euclid is.

The worst relative error of the best k-dimensional picture of London, New York, Tokyo, Sydney, for every k the set allows. Going from a line to a plane buys almost everything — from 180.2% to 1.27%. Going from a plane to a space buys nothing at all: the two numbers agree to every digit, because the third eigenvalue the extra dimension would spend is 2.85e-8 — zero — and the eigenvalue that would help is negative and cannot be spent in a Euclidean space of any dimension. The flat sheet is not the obstruction. Euclid is.

It is drawn by curvature-figure with show: "distance-dimensions" — 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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