Origami & Folding
what one sheet can do
Concept

Variance — where it appears

How widely a quantity spreads across repeated measurements that ought to agree. For a backtracking search the spread across branch orders can be wider than any difference between the problems, so a single cost is one draw from a distribution rather than a property of the sheet.

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

what ten branch orders cost on the same four sheetsnodes of search; the sheets are the same drawings as the sweep abovethe square grid, 4×4, glued69 to 24636, 1 gave upthe square grid, 4×4, cut42 to 55the square grid, 3×3, glued20 to 731the square grid, 3×3, cut25 to 32each bar runs from the cheapest of 8 branch orders to the dearest, on a logarithmic scale; a dot is the middle one

The route, not the sheet

Every cost measured for a glued sheet has been one number from one branch order, and a backtracking search's cost belongs to the pair. Asked under eight orders instead of one, a cut cell's cost barely moves — 42 to 55 nodes — while the torus over the same drawing runs from 69 to 24,636, with one order giving up entirely. The glued sheet's best order costs less than twice the cut sheet's, so most of what a single order charged to the gluing belongs to the route through it.

complexity · Search order nodes per free letter, cheapest route against the middle onecheapest of eightmiddle of eight× where no route of that kind finished · periods along the bottom0.3110100×2345the square grid, glued×123the triangular grid, glued××1234the honeycomb, glued0.3110100×123the elongated triangular tiling, glued××12the rhombille tiling, glued×123the rhombille tiling, cut

The cheapest route crosses later

A search for a consistent lettering has a threshold: below it the letters propagate and the cost is a third of a node per crease, above it the search backtracks and the cost explodes. The threshold was measured with one branch order. Measured with eight, the cheapest route never starts searching before the typical one, and on most sheets it starts a period or two later — so part of every threshold on the record belongs to the route. And the one cut sheet past its threshold, the rhombille's, spreads across nearly three orders of magnitude of cost, which moves the spread off the gluing and onto the threshold.

complexity · Search order nodes of search under drawn and built routes, sheet by sheetlogarithmic, from 10 to the budget of 20,000; a cross at the right marks searches that did not finish101001,00010,000the rhombille tiling, 2×2, glueddrawn0 of 8joined first6 of 8crossed first0 of 8the rhombille tiling, 2×2, cutdrawn7 of 8joined first7 of 8crossed first7 of 8the rhombille tiling, 1×1, glueddrawn8 of 8joined first8 of 8crossed first8 of 8

The route built from the gluing

A search for a consistent lettering of a glued sheet costs a third of a node a crease on its cheap routes and thousands on its dear ones, and an account was offered of which routes are cheap: the ones that take first the creases whose choice propagates through the gluing. That account can be turned into a route rather than tested on samples, and there are two ways to read it. Built the first way, a route finishes nearly twice as many searches past the threshold as a drawn one does, and on the rhombille torus — where none of eight drawn routes finishes inside twenty thousand nodes — six of eight built routes do, the cheapest in 1,183. So the rhombille's difficulty belongs to the route. But which reading works changes with the tiling, and on the elongated tiling, where every drawn route finishes, the built ones lose searches.

complexity · Search order

Named alongside it

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

GluingMeasurementPatchSearchSearch costBoundary

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