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

How many maps want to move their axis, and under which criterion

Drawn here at the parameters it defaults to, with every essay that calls it.
How many maps want to move their axis, and under which criterion. Every projection here was scored at twelve candidate axes and kept its best. Under the uniform weighting 5 of 13 symmetric maps and 1 of 7 asymmetric ones prefer an axis other than the poles — which is nearly none, as it should be, since a uniform criterion has no direction in it. Under the concentration, 8 of 13 and 2 of 7 move. That is the freedom the conjecture said the symmetric family could not use, and it uses it — and it still loses by 1.204 times.

Every projection here was scored at twelve candidate axes and kept its best. Under the uniform weighting 5 of 13 symmetric maps and 1 of 7 asymmetric ones prefer an axis other than the poles — which is nearly none, as it should be, since a uniform criterion has no direction in it. Under the concentration, 8 of 13 and 2 of 7 move. That is the freedom the conjecture said the symmetric family could not use, and it uses it — and it still loses by 1.204 times.

It is drawn by projection-figure with show: "symmetry-aspect" — one member of a family of 16 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.

1 essay calls 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.

The test the ladder asked for, and it refutes the conjecture. How much better the best asymmetric projection is than the best symmetric one, under weightings of four different symmetries, with every symmetric map allowed to re-aim its axis at twelve candidate poles. The conjecture rung eight recorded was that the seven earn their place by PLACING distortion where a symmetric map cannot, so their advantage should collapse under a criterion with no place preference. It does the opposite: the advantage is largest at 1.343 under the uniform weighting and smallest at 1.144 under a band, with the fully asymmetric concentration at 1.204 in between. The winner is named on each row and the map it beat is Equirectangular throughout. The seven are simply better maps. The families

The maps with no family are simply better

Rung eight found seven projections with no continuous symmetry and noticed they are almost exactly the set anybody would choose for a world map, then offered a conjecture with a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.

The whole library