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

Adjacency — where it appears

Which regions of a partition share a boundary, as a graph rather than as a picture. It is what a colouring, a contiguity constraint or a neighbour query is really about, and it survives every projection because a cut adds faces and adds no edges.

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

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 ruler on a cartogram is not measuring anything. Twelve pairs of places, each measured on the ground and on the page of the triangular, x first cartogram of four cities, with the page scaled so that the median pair reads exactly right — the most generous calibration available. The line is that calibration. The worst pair, Tokyo to Quito, reads 21598 km for a ground distance of 14439 km, an error of 49.6%. The open marks are the same pairs on the equal-area map the cartogram was built from.

Everything else on the page pays for the areas

A cartogram gets one quantity exactly right and every other reading a page supports is collateral. A ruler on it is out by 35 per cent after the most generous calibration available, and ten of sixty triples of places change which one is in the middle.

distortion · Cartogram

Named alongside it

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

Angular deformationAntimeridianAreal factorBetweennessCartogramChromatic numberDensityExclaveFour-colour theoremGraphNon uniquenessPartition

All concepts