Concept

Contrast — where it appears

The ratio of the largest value of a density to the smallest, which is the single number every cartogram result here is reported against. It sets the areal range a map is being asked to produce, and the shape cost of producing it rises with it.

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

Also named here as diffusion cartogram — the same set of essays touches all of them, so they are one junction rather than several.

Four cartograms of one density. The same stated density — four cities — met four different ways, drawn on the same cells. Every panel is a correct cartogram of the same numbers: a region's page area is proportional to its mass in all four. They do not look alike, because the areal scale factor fixes one number per point and a map has four derivatives, so three degrees of freedom per point are left over and each construction spends them differently.

Every density can be met and none is free

Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.

distortion · Cartogram

The straight line between two correct maps passes through an incorrect one. Both ends of this sequence are exact cartograms of four cities: the triangular, x first construction at s = 0 and the triangular, y first at s = 1, each meeting the density to arithmetic noise. The panels between them are the straight-line blend of the two, which is what an animation between two maps computes. The cells drawn solid have turned inside out — their signed area is negative, so the map has folded over itself there and two places on the sphere are drawn at one place on the page.

A map that meets its target can fold

The flow a diffusion cartogram integrates is a diffeomorphism at every instant and cannot fold. Every discretisation of it can, and the point at which one does is a root of a quadratic — written down rather than searched for.

distortion · Cartogram

Named alongside it

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

Areal factor Cartogram Density Diffusion cartogram Non uniqueness Triangular map Angular deformation Diffeomorphism Discretisation Equal-area Fold Homotopy

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