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

Antimeridian — where it appears

The meridian at 180 degrees, where the longitude number wraps and a stored geometry that crosses it becomes two pieces or one enormous one. It is a discontinuity in the coordinates rather than in the ground, so every operation that treats longitude as a real number has to decide what to do at it.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

How each projection escapes being one to one. Eighteen projections, and the theorem allows no fourth column. A map of the whole sphere either leaves ground undrawn, or draws one place as a curve, or is cut so that one ground curve appears twice — and 11 of the eighteen do more than one of those. The three columns are a solid angle, a count of places and a page length, which is why they are three columns rather than one score: there is no rate at which a hemisphere converts into a pole.

No map of the whole sphere is one to one

Eleven essays establish that no map preserves distance, and every step of that argument needs a distance. There is a second impossibility underneath it that needs nothing at all: a sphere is compact and has no boundary, so a continuous map of it into the page cannot also be one to one. Eighteen library projections, eighteen escapes, and not one of them free.

impossibility · Topology A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres.

The antimeridian is a cut in the numbers

A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.

applied · Dataset The cut makes 20 faces out of 14 regions and needs no more colours. The same partition on a sheet cut at the antimeridian. six of the 14 regions are drawn in two pieces, one against each edge, and they are shown darker. The sheet has 20 faces where the globe had 14 regions, and it needs exactly the same four colours — because the two pieces of a split region between them touch exactly what the region touched, so identifying them gives back the sphere's own graph, edge for edge.

Four colours, and what a cut cannot do to them

Every projection removes a set, and a map cut at the antimeridian draws six of its fourteen countries in two pieces — twenty faces where the globe had fourteen regions. The obvious guess is that a map with split countries is the exclave problem and needs a fifth colour. It needs exactly four, and the reason is that the cut adds faces and adds no edges.

impossibility · Topology A ring round the pole at 80°, and the two pieces it makes. the boundary of a small polar cap — and of everything else. A closed curve divides a sphere into two pieces and neither of them is the outside: one is 4 thousand square kilometres and the other is 506 thousand, a ratio of 130.6 to one, and the coordinates are the same either way. The two colours are the two pieces, sampled at points rather than shaded, because shading one of them would already be the decision this figure is about.

A polygon on a sphere has no outside

Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.

applied · Dataset

Named alongside it

The objects these essays reach for when they reach for this one.

TopologyBoundaryConventionInterruptionSeamShoelaceVerificationAdjacencyAntipodeAreaBounding boxBranch cut

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