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

Four cities, met — the triangular, x first construction

Drawn here at the parameters it defaults to, with every essay that calls it.
Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing.

Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing.

It is drawn by cartogram-figure with show: "cartogram-figure" — one member of a family of 0 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.

8 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.

Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing. Measuring distortion

A map drawn to a density it was handed

Two hundred and twenty-six essays measure distortion after the fact. This one specifies it: a density is handed to a map as a boundary condition, the areal scale factor comes out equal to it to five parts in a million, and every other invariant the site owns becomes the price.

 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. Measuring distortion

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.

 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. Measuring distortion

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.

 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. Measuring distortion

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.

 Two maps of one density: one costs nothing and one costs forty-four degrees. Both drawings meet the same areal request — an exponential ramp of contrast 79.8 to one, whose logarithm is harmonic — to arithmetic noise. The first is a conformal map written down in closed form, the conformal map that meets an exponential ramp, log-harmonic, whose angular deformation is 2.4e-8 degrees. The second is the triangular construction the previous rungs use, at 43.8° on the same request. A grid of squares is drawn through each: the first keeps every angle and the second does not. Measuring distortion

The cheapest map that meets its areas

An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.

 The density every real cartogram is handed. Two islands in an ocean that is not lightly populated but empty: the density is exactly zero over 83.7 per cent of the sphere. Four rungs of this anchor assume a positive density, because the construction divides by it — and the conditional cumulative of a column with no mass in it is zero over zero. Drawn on the equal-area base, so a cell's ink is a density and its area is ground. Measuring distortion

A density that asks for no room at all

Five rungs assume the density is positive everywhere, because the construction divides by it. Every cartogram anybody draws has an ocean, and an ocean is not sparsely populated but empty — 83.7 per cent of the sphere, exactly zero, and the construction returns nothing at all for a third of the probes.

 One number changed, and the whole map moved. Two cartograms of the same density differing in one bump's weight — London's, raised from 6 to 9 — with an arrow at each sample showing how far the second puts that place from where the first does. Every other density in the specification is identical. The largest movement is 0.255 of the rectangle's half-height and it is not at London: it is about forty-five degrees away, and places on the far side of the world move by a tenth of it. Measuring distortion

One number changed and the whole map moved

Raise one bump's weight in a density specification, leave every other number identical, and solve again. London's own value is what changed; London moves 0.054 and Delhi moves 0.226 — four times as far, with its own number untouched. The largest displacement anywhere is thirty degrees from the change, and the antipodal band still moves a fifth of the peak.

 Six regions, before and after. Six circular regions of unequal size on the equal-area rectangle, and the same six after the cartogram of four cities has been solved. Each drawn area is exactly its base area times the density it was asked for. The four that grow stay recognisably round; the two that shrink are drawn out into shapes that no longer resemble what they were, and nothing in the construction chose to treat them differently. Measuring distortion

A cartogram keeps the shapes it inflates

Seven rungs build cartograms and none reads one back. Reading means recognising a region and dividing its drawn area by its base area, and the construction is against the reader twice: the correlation between how much a region grows and how much of its shape it keeps is −0.996, and the base area a reader has to divide by is the map the cartogram replaced.

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