Origami & Folding
what one sheet can do
Concept

Determinism — where it appears

A procedure whose answer depends only on its input, with no coin anywhere in it. A deterministic search returns the same lettering however many times it is run, which removes every spread from its cost and leaves exactly one witness where a randomised one returns many.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

the dot is one run's cost, ranked; the rule is the constant order1001e+31e+4nodes visited40 seeds, ranked by cost80 nodes, every seed15 unfinished at 20,000same pattern, same conditions, same test at every node — the only difference is which letter is tried first

The difficulty was in the coin

One tessellation patch, one search, one test at every node — and a cost that runs from eighty-six steps to fifteen thousand depending on nothing but the starting seed. The heavy tail is real, it was measured carefully, and it was made by a single line of the search that nobody had thought of as a choice at all.

flat-folding · Forced order the bar is how many DIFFERENT letterings 20 runs returneda coin at every choice1414 of 20 runs found onea constant, with the coin only on the creases no vertex constrains120 of 20 runs found onea constant at every choice120 of 20 runs found oneon the rhombille patch, 157 panels and 282 creases

One witness or forty

Taking the randomness out of a search made it three orders of magnitude cheaper in the worst case and cost it thirty-nine of its forty answers. The compromise everybody reaches for — randomise only the choices that cannot matter — recovers four of the forty on two patches and none on the other three, because the diversity was never where it looked.

flat-folding · Layer multiplicity the curve is stop-and-restart; the rule is a constant letter order1001e+31e+41001e+31e+4563 at a cutoff of 10080 nodes, deterministic, nothing to restartexpected nodes in totalcutoff, in nodes

Restarting what cannot be restarted

Stopping a search early and starting it again with a fresh seed costs five hundred and twelve steps in expectation against sixteen thousand for patience. Every number in that is right. The distribution it is right about was made by the search's own coin, and taking the coin out costs eighty — with nothing left to reseed.

complexity · Hardness of folding

Named alongside it

The objects these essays reach for when they reach for this one.

SearchAssignmentLayer orderSearch costWitnessBacktrackingDistributionEnumerationExpected-valueLayer countMeasurementPatch

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