Origami & Folding
what one sheet can do
Concept

Hamiltonian path — where it appears

A route through a network that visits every point exactly once. Deciding whether a shape of helices has one is hard in general, so the cheap tests that refuse shapes are necessary conditions and never the whole answer.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

a strand of 7249 bases at 64 to a helixthe bar is the largest member of the family that is still buildablea square blockstopped by the strand's length100 helices · 88% of the stranda single rowstopped by the strand's length113 helices · 100% of the stranda comb of teethstopped by the routing5 helices · 4% of the stranda plusstopped by the routingno size works at allan L with equal armsstopped by the strand's length113 helices · 100% of the strand

Two ceilings

A DNA origami is limited by the length of one viral strand and by whether its helices can be visited once each in a single pass. Grown one step at a time, a square block runs into the first at a hundred helices and a plus runs into the second at five — so which limit a shape meets is decided by the shape and not by the chemistry.

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 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 routeevery shape of every smaller size is either refused by the tests or routed by the search

Every cheap test misses a shape

A strand routed through a bundle of helices has to visit each once, and whether a shape allows that is hard to decide — so the cheap tests that refuse shapes are necessary and never sufficient, and for every set of them there is a smallest shape they pass and no route reaches. Listing every connected shape and searching the ones the tests let through finds it: nine helices on the square lattice for the colour count, the ends and the cuts, eleven once the steps a route's ends force are added, and still eleven once the ends' colours are checked. On the honeycomb the same three stages give twelve, fifteen and sixteen. Each test pushes the smallest unroutable shape out or leaves it where it is; none removes it.

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 the honeycomb lattice's 16-helix survivor, read by its corridorsa junction is a helix with other than two neighbours; a corridor runs between junctions through helices with twoeach corridor in its own colour; numbered circles are the junctions with three neighbours or more123end16 helices, 4 corridors, 3 junctionsjunction 1: 3 compulsory mouths, 1 to absorbjunction 2: 3 compulsory mouths, 1 to absorbjunction 3: 3 compulsory mouths, 1 to absorb1 helix with one neighbour spends an end1 end left to lie inside a corridorneeds 3, has 1: refused

A corridor has two mouths

Every cheap test for whether a strand can be routed through a bundle of helices had been built on knowing where the route ends, and the honeycomb's sixteen-helix survivor is a shape where the ends are mostly unknown. A test that needs no ends at all reads the shape by its corridors — chains of helices with two neighbours — and its junctions. A route passes a junction once, so a junction with three corridors leaves one of them to be entered only from its far mouth, which needs an end of the route inside it. The survivor has three such junctions and one free end. The count refuses all six of its placements, leaves the honeycomb with no survivor to sixteen helices, and on the square lattice refuses one of the two twelve-helix shapes and not the other.

biology · Molecular folding

Named alongside it

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

DNA origamiRoutingNecessary conditionLocalityParityWitnessDesign constraintScaffold

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