Origami & Folding
what one sheet can do
Concept

Routing — where it appears

Laying one continuous strand through every part of a target shape. It is folding with no sheet in it: what is designed is a path rather than a crease pattern, and the obstructions are combinatorial rather than metric.

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

a route the search found123456121110987131415161718242322212019helices 24colours 12 : 12a route is not forbiddenscaffold used 21%48 staples of 3224 helices · 1536 bases · 48 staples · colours 12 : 12

A sheet that routes itself

DNA origami folds one long strand into a shape by holding it against itself with a few hundred short ones. There is no sheet and no crease — what has to be designed is a route — and the first thing that can go wrong is a counting argument crease patterns already know under another name.

biology · Molecular folding 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 bases at which a crossover is possiblethe backbone turns 360° over a turn's worth of bases, and a crossover needs it facing the neighbourhoneycomb — three neighbours6 exact, 0 near071421283542square — four neighbours2 exact, 4 near4.3°4.3°4.3°4.3°07142128354210.5 bases to a turn · a mark is a base whose backbone faces a neighbour to within 5°, and the number under it is the miss

The helix chooses the lattice

A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.

biology · Molecular folding 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 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

Named alongside it

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

DNA origamiNecessary conditionParityHamiltonian pathLocalityWitnessDesign constraintError propagationGridPeriodicityScaffoldSelf assembly

All concepts