Origami & Folding
what one sheet can do
Concept

Witness — where it appears

A single object that settles an existence question by being one. A lettering with no circle in its arcs proves the pattern admits one, whatever a sample of two thousand draws found, and it can be checked in one pass by machinery that had no part in producing it.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

a lettering of the patch that agrees with itselffound by testing the arcs while the letters were chosen, not after561 nodes · 246 backtracks · verified against a rebuilt folded sheet157 panels · 282 creasesits own lettering sends its panels round in a circle0 of 200 random letterings agree with themselvesthis one was found in 561 nodes and 246 backtracksit differs from the drawn lettering on 155 of 282 creasesthe drawing is the pattern; nothing here is a picture of the folded object

The lettering nobody could draw

Two hundred letterings drawn at random from the rhombille tessellation patch, and not one of them agrees with itself. Two thousand, and still not one. The patch was left as an open question — and it has an answer, found in five hundred and sixty-one steps by a search that tests the arcs while it is choosing the letters instead of after it has chosen them all.

flat-folding · Forced order the bar is how many creases the found lettering writes differentlymeasured against the lettering the pattern's own construction producedthe square patch4545 of 84 creases · 31 of them buriedthe elongated patch6666 of 106 creases · 42 of them buriedthe hexagonal patch6767 of 142 creases · 45 of them buriedthe triangular patch8787 of 142 creases · 65 of them buriedthe rhombille patch155155 of 282 creases · 117 of them burieda buried crease has an interior vertex at each end, and no legal move ever changes one

One solution of a search nobody ran

A crease pattern arrives with its letters already on it, and they look like part of the drawing. They are not. Every construction here ends in a propagation, a propagation ends wherever its first guess took it, and the lettering that comes out differs from the one a search finds on between a half and three-fifths of the creases — on patterns whose own letters are perfectly good.

flat-folding · As drawn 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 which blocks a single strand can routecolumns are the block's width and rows its height, both counted in helicessquare lattice1234567891012345678honeycomb lattice12345678910a route through every helixthe count allows it, and no route existsa column the rows never reach

A row the route cannot leave

Every rectangular block of helices up to ten by eight routes on the square lattice. On the honeycomb, the lattice a double helix's pitch prefers, twenty of the eighty do not — and every block odd in both directions fails for a reason visible along one row: every second helix on the top row has no neighbour off it, the two corners have one neighbour each, and a route forced through them runs the length of the row and ends. The colour count passes all of them, and a degree count along a single row refuses them.

biology · Molecular folding the bar is the size of the smallest shape the tests pass that no route reacheseach row adds one more cheap test to the ones above itsquares: the colour count, the ends and the cuts9 helices64 shapes of that size pass and have no routesquares: and the steps the ends force11 helices68 shapes of that size pass and have no routesquares: and the colour of every forced end11 helices32 shapes of that size pass and have no routesquares: and the colour count on a stretch a cut has fenced off12 helices12 shapes of that size pass and have no routehoneycomb: the colour count, the ends and the cuts12 helices18 shapes of that size pass and have no routehoneycomb: and the steps the ends force15 helices12 shapes of that size pass and have no routehoneycomb: and the colour of every forced end16 helices6 shapes of that size pass and have no routehoneycomb: and the colour count on a stretch a cut has fenced off16 helices6 shapes of that size pass and have no routeevery shape of every smaller size is either refused by the tests or routed by the search

A test that only knows one lattice

The cheapest argument that refuses the smallest shape no cheap test could refuse was read off that shape: cut at one helix, find the piece with no end in it, and count the colours of the stretch the route is then forced to cross. Added to the census it refuses every one of the square lattice's thirty-two eleven-helix survivors and pushes the smallest survivor to twelve, where twelve placements of two shapes survive out of half a million. On the honeycomb it refuses none of the six at sixteen. A test inherits the lattice of the witness it was read off, and the staircase is two staircases.

biology · Molecular folding

Named alongside it

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

AssignmentSearchLayer orderDeterminismDNA origamiLocalityNecessary conditionPatchRoutingSamplingBacktrackingBuried crease

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