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

One axis for every page a picture could be drawn on

Drawn here at the parameters it defaults to, with every essay that calls it.
One axis for every page a picture could be drawn on. The three escapes of this rung are one parameter. A hyperbolic page of curvature radius k has Gaussian curvature −1/k², a flat sheet has zero, and a sphere of radius R has +1/R²; the axis runs through all of them in units of the Earth's own curvature, so the Earth sits at +1. The same construction runs at every point — hold 2n − 3 distances exactly, read the rest — and the curve has exactly one zero, at +1, where the error is 3.3e-16. The left branch never dips below the flat sheet's 7%; it approaches it from above as the curvature goes to zero, which is also a check that two constructions written separately with cosines and with hyperbolic cosines agree where they must. Below about +1.2 on the right the sphere is too small to hold the places and the construction refuses rather than fits.

The three escapes of this rung are one parameter. A hyperbolic page of curvature radius k has Gaussian curvature −1/k², a flat sheet has zero, and a sphere of radius R has +1/R²; the axis runs through all of them in units of the Earth's own curvature, so the Earth sits at +1. The same construction runs at every point — hold 2n − 3 distances exactly, read the rest — and the curve has exactly one zero, at +1, where the error is 3.3e-16. The left branch never dips below the flat sheet's 7%; it approaches it from above as the curvature goes to zero, which is also a check that two constructions written separately with cosines and with hyperbolic cosines agree where they must. Below about +1.2 on the right the sphere is too small to hold the places and the construction refuses rather than fits.

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

The whole library