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

Total curvature is a counting number in disguise

Drawn here at the parameters it defaults to, with every essay that calls it.
Total curvature is a counting number in disguise. The Gaussian curvature of each surface, integrated over the whole of it. Gauss–Bonnet fixes the answer at 2πχ, where χ is the Euler characteristic — vertices minus edges plus faces, a quantity with no geometry in it at all. The sphere gives 12.5664 against 4π = 12.5664. The torus gives -2.4e-11 against zero, and it does so by cancellation: its outer half is positively curved and its inner half negatively, in exactly equal measure.

The Gaussian curvature of each surface, integrated over the whole of it. Gauss–Bonnet fixes the answer at 2πχ, where χ is the Euler characteristic — vertices minus edges plus faces, a quantity with no geometry in it at all. The sphere gives 12.5664 against 4π = 12.5664. The torus gives -2.4e-11 against zero, and it does so by cancellation: its outer half is positively curved and its inner half negatively, in exactly equal measure.

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

3 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.

Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not. The impossibility

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

 The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside. The impossibility

Measuring curvature from inside

A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.

 One length, three corrugations. A straight segment shortened by a factor of 0.9, then wiggled across its own direction until its length is back to what it was. The wiggle's amplitude is what buys the length and the number of wiggles is free, so all three curves have exactly the target length while the third stays 16 times closer to the shortened segment than the first. That is the mechanism: the family converges to a map that is not isometric, while every member of it is. The impossibility

Impossible in two derivatives, possible in one

The impossibility this whole collection rests on computes a second derivative, so it is a statement about maps that have two. Take one away and it is false: a corrugation restores an exact length while converging to the map that does not, and iterating it gives a flattening whose derivative converges and whose curvature runs to half a million.

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