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

Every projection in the library, ranked under the flat weighting

Drawn here at the parameters it defaults to, with every essay that calls it.
Every projection in the library, ranked under the flat weighting. Kavrayskiy's criterion — ½[(ln a)² + (ln b)²] in the two principal scale factors, so shape and area are charged together — weighted by the flat field, with every projection allowed its best of twelve axes. The seven with no continuous symmetry are marked. three of them come before the first symmetric map, and the best symmetric map is Equirectangular at 0.3539 against Winkel tripel's 0.2635. The ordering is not the one this collection's other criteria produce, and it is not meant to be — it is the one the conjecture had to be tested on.

Kavrayskiy's criterion — ½[(ln a)² + (ln b)²] in the two principal scale factors, so shape and area are charged together — weighted by the flat field, with every projection allowed its best of twelve axes. The seven with no continuous symmetry are marked. three of them come before the first symmetric map, and the best symmetric map is Equirectangular at 0.3539 against Winkel tripel's 0.2635. The ordering is not the one this collection's other criteria produce, and it is not meant to be — it is the one the conjecture had to be tested on.

It is one of 17 figures drawn by the same construction, and the drawing above is this one with nothing changed.

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 conjecture 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 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

Seven projections have no continuous symmetry, and they are almost exactly the set anybody would choose for a world map. That coincidence came with a conjecture and 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