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

Pareto front — where it appears

The members of a set that nothing beats on every criterion at once. It says what is not ruled out rather than what a stated preference selects, and a weighted sum of the criteria can only ever choose a vertex of the front's convex hull.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Every library projection over Europe, on the two axes it can be wrong on. Each dot is one projection, scored over Europe on the two independent failures: how much it turns angles and how much it changes areas. The lower-left corner is the isometry that does not exist. The line joins the five projections nothing beats on both counts — the rest are inside it, and a reader who prefers either failure to the other should still not choose one of them, whatever weighting they hold.

The projections that are beaten on both counts

Two rungs of this ladder scored a rule of thumb over thirty regions and then forty-five. The same populations answer a harder question the ladder has never put: which library members are never the right answer at all. Two are beaten outright on both criteria everywhere, one is on no regional front in any population — and it is on the world's.

wrong · Audit Four of the seven on the front can never be first. Every library projection over the whole sphere, with both errors normalised to the table's own range. The seven filled circles are the Pareto front — nothing beats them on both counts. The line through three of them is the lower convex hull, and a weighted sum of the two errors is a straight line in this space, so only a hull vertex can ever come first. Four projections — Web Mercator, Miller cylindrical, Equirectangular, Winkel tripel — sit in the dents, undominated and unchoosable.

Which projection a weighting can make best

Rung eight finds the seven world projections nothing beats on both counts and tells a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.

wrong · Audit Two regions, three answers. Britain and New Zealand, 166° apart, with the pole of the best oblique conic under each of three objectives. Pooling the samples and taking an area-weighted score puts the pole in one place; refusing to let either region be worse than the other puts it somewhere else. The regions are drawn on Mollweide so that equal ground areas are equal page areas.

The pooled score abandons a region

Fifteen rungs optimise for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.

choosing · Choosing

Named alongside it

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

Trade-offOptimisationProjection selectionPurposeRobinsonWeightingWinkel tripelAggregationAngular deformationAreal factorAspectAudit

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