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

Every three of them fit, and the four together do not

Drawn here at the parameters it defaults to, with every essay that calls it.
Every three of them fit, and the four together do not. Each subset of the four places, with the smallest eigenvalue of its own double-centred distance matrix — the quantity that is zero exactly when a flat picture exists and negative when none does. Every one of the four triangles sits at zero to rounding, which is the triangle inequality doing what it always does. The one four-place subset does not, and the bar it draws is the whole of the impossibility this ladder is about. Nothing about the places was chosen to make this happen; it happens to any four points on a sphere that are not on one great circle.

Each subset of the four places, with the smallest eigenvalue of its own double-centred distance matrix — the quantity that is zero exactly when a flat picture exists and negative when none does. Every one of the four triangles sits at zero to rounding, which is the triangle inequality doing what it always does. The one four-place subset does not, and the bar it draws is the whole of the impossibility this ladder is about. Nothing about the places was chosen to make this happen; it happens to any four points on a sphere that are not on one great circle.

It is drawn by curvature-figure with show: "distance-triples" — 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 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 whole library