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

Gaussian curvature across a torus

Drawn here at the parameters it defaults to, with every essay that calls it.
Gaussian curvature across a torus. Curvature along a cross-section of a torus. curvature of both signs — positive outside, negative inside, zero on two circles. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.

Curvature along a cross-section of a torus. curvature of both signs — positive outside, negative inside, zero on two circles. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.

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

5 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.

 A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere. The impossibility

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

 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.

 Three surfaces with the same curvature, one of which is a sphere. Every one of these is a surface of revolution whose Gaussian curvature is 1 at every point, built by solving r″ + r = 0 for the meridian rather than by writing a shape down. The spindle closes to a point with an angle deficit, the sphere closes smoothly, and the bulge does not close at all — it ends in two circular edges. A surveyor confined to a patch of any of them, measuring angles and distances, cannot tell which one it is. The impossibility

Two surfaces with the same curvature

The ladder's base says curvature is the obstruction to a faithful map. It has never asked whether curvature is the whole obstruction — and it is, locally: there is a whole family of surfaces with the unit sphere's curvature at every point, none of them a sphere, and a geodesic circle drawn on one agrees with the same circle on another to one part in 10¹⁴.

 Everywhere Albers is exactly right, and the band round it. The set on which both principal scale factors are one — the only ground where a ruler on this map, at the map's own stated scale, measures the true distance in every direction. It is the parallels at 20.000° and 60.000°, drawn as a curve, with the band within 0.01 of true scale shaded round it. That band is 2.673 per cent of the sphere, and it narrows as ε as the tolerance tightens. The curve at its centre has no width at all, and no tolerance makes it have one. The impossibility

The places where a map is exactly right

Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.

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