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

The spectrum that decides whether a picture exists

Drawn here at the parameters it defaults to, with every essay that calls it.
The spectrum that decides whether a picture exists. The four eigenvalues of the double-centred matrix of squared distances, for the same four places measured two ways. Drawn straight through the rock, the chordal distances give three positive eigenvalues and no negative one, which is Schoenberg's statement that they are the distances of four points in three dimensions — and so they are, because the places are already there. Measured along the surface, one eigenvalue is negative: -2.165e+6 against a largest of 1.879e+8. A negative eigenvalue is not a large error. It is the statement that no Euclidean space of any dimension holds these distances.

The four eigenvalues of the double-centred matrix of squared distances, for the same four places measured two ways. Drawn straight through the rock, the chordal distances give three positive eigenvalues and no negative one, which is Schoenberg's statement that they are the distances of four points in three dimensions — and so they are, because the places are already there. Measured along the surface, one eigenvalue is negative: -2.165e+6 against a largest of 1.879e+8. A negative eigenvalue is not a large error. It is the statement that no Euclidean space of any dimension holds these distances.

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

2 essays call 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 four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes. The impossibility

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

 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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