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

What the picture gets wrong is everything it was not asked about

Drawn here at the parameters it defaults to, with every essay that calls it.
What the picture gets wrong is everything it was not asked about. The twelve constraints the six-constraint picture was not asked to hold, with what it makes of each — a relative error for a distance and an angle for a bearing, both drawn against the same bar so the two kinds can be seen together. The worst distance is Tokyo–Sydney at 6.9% and the worst bearing Tokyo → Sydney at 169°. Six exact constraints are not six-eighteenths of a correct picture: they are six statements that happen to be true about a drawing that is otherwise as wrong as any flat picture of four world cities has to be.

The twelve constraints the six-constraint picture was not asked to hold, with what it makes of each — a relative error for a distance and an angle for a bearing, both drawn against the same bar so the two kinds can be seen together. The worst distance is Tokyo–Sydney at 6.9% and the worst bearing Tokyo → Sydney at 169°. Six exact constraints are not six-eighteenths of a correct picture: they are six statements that happen to be true about a drawing that is otherwise as wrong as any flat picture of four world cities has to be.

It is one of 65 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.

Six of eighteen, where either kind alone can hold only five. London, New York, Tokyo, Sydney, drawn holding five distances exactly — the thick edges — and one bearing exactly, the arrow from London to New York. Six exact constraints, where five distances is the most any flat picture of four places can hold and five bearings is the most it can hold of those. All six are satisfied to 1.9 × 10⁻¹¹. The sixth is not free room found by luck: a picture is blind to where it sits, how it is turned and how large it is, distances see the last of those three and bearings see the second, and a picture holding both kinds is blind only to where it sits. Tokyo–Sydney, dashed, is the distance left out; the picture is wrong about it by 6.9%. The impossibility

A mixed picture holds one more, and what it is holding is north

A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.

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