Figure

The room runs out faster than the count of constraints suggests

Drawn here at the parameters it defaults to, with every essay that calls it.

$The room runs out faster than the count of constraints suggests. The share of choices of 2n − 3 pairs and a direction on each that give a picture with every held arrow pointing forwards, against the number of places, over 20,000 seeded choices at each size. It falls from 9.2% at four places to 2.75% at five, 0.68% at seven and 0.070% at eight. The count of bearings held rises only as 2n − 3 while the count available rises as n(n − 1)/2, so a picture is asked to satisfy a shrinking fraction of the pairs and finds it harder rather than easier — because the difficulty is in the directions, and every held pair adds one more arrow that has to come out forwards.$

The share of choices of 2n − 3 pairs and a direction on each that give a picture with every held arrow pointing forwards, against the number of places, over 20,000 seeded choices at each size. It falls from 9.2% at four places to 2.75% at five, 0.68% at seven and 0.070% at eight. The count of bearings held rises only as 2n − 3 while the count available rises as n(n − 1)/2, so a picture is asked to satisfy a shrinking fraction of the pairs and finds it harder rather than easier — because the difficulty is in the directions, and every held pair adds one more arrow that has to come out forwards.

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.

Two pairs of ten are in every picture that can be drawn. London, New York, Tokyo, Sydney, Cape Town, drawn holding seven of their ten pairs' bearings exactly — the most any flat picture of five places can hold. Of the 15,360 ways of choosing seven pairs and a direction on each, 152 give a picture whose held arrows all point forwards, and London–Tokyo and Sydney–Cape Town, drawn heavy, appear in every one of them. The three pairs this particular choice leaves out are dashed. The price is paid by the rest: the worst, New York to Sydney, is 142.1° from the truth.

The impossibility

The pair that cannot be held is not the one the geometry names

Four world cities have exactly one pair of the six that no drawable picture holds, and the obvious candidate — the pair whose bearing there and bearing back disagree most — is not it. That quantity predicts something else exactly: whether a held pair's arrow has a forced direction. It separates the two groups with a clean gap and nothing in between. And at five places there is no forbidden pair at all, so the four-place case was a statement about how little room five constraints leave rather than about the map.

The whole library