Origami & Folding
what one sheet can do
Concept

Locality — where it appears

The property of being decided by what is nearby. Every condition this subject checks is local to a vertex, and the questions that matter — layer order, rigid folding, global consistency — are not.

Named by 15 essays across 5 fields — each of them below, with the objects they name alongside it.

246810012345units (creases, or joints)log₁₀ statesa chain, 3 states per jointa sheet, two letters per creaseat one vertex, 4 of 16 assignments survive four local conditionsthe sheet's count has a local test that removes 75% of it · the chain's has none

Two things called folding

A protein folds and a sheet folds, and the word is the same word by accident. Both have exponentially many states and that is not the difference. The difference is that one of them can be filtered by four conditions checked at a single point, and the other cannot be filtered by anything local at all.

biology · Molecular folding a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

Even is not enough

Every vertex theorem in the subject is a statement about one point, and the two-colouring of the panels looks like the exception. It is not — on a square of paper it is a parity at each vertex and nothing more. Cut a hole and the two come apart: a loop of paper with three creases has no interior vertices at all, satisfies every theorem there is, and cannot be folded flat.

flat-folding · Two-colourability what the vertex conditions settle once one crease is chosenpatternsettled, against what is therepreliminary base1 of 81 vertices still choosingmiura 6×41 of 3815 vertices still choosingwaterbomb 4×41 of 7625 vertices still choosingyoshimura 6×51 of 8422 vertices still choosingsquare twist grid3 of 14464 vertices still choosingtriangular twist grid2 of 236104 vertices still choosingKawasaki was settled by the angles before a letter was written; the letters are what is left, and they are nearly all left

How little the conditions decide

Local is not global is a statement about sufficiency: every vertex can pass and the sheet still fail. There is a sharper complaint available, and it is about strength. Fix one crease of a tessellation and propagate every condition the subject has to a fixed point: three creases out of a hundred and fifty-eight follow, and sixty-six vertices are still holding more than one answer.

flat-folding · Flat-foldability a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

A cut that removes no paper

Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.

material · Kirigami the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false

The cost of asking the wrong sheet

A test written for a sheet with an edge, run on a sheet without one, does not fail. It exhausts — proving, at three, thirty-five and three thousand four hundred and fifty-five nodes, that no lettering exists — and the letterings it proved impossible fold, on the collection's own machinery, at every size they were tried at.

complexity · Hardness of folding a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

A theorem with an unstated hypothesis

Maekawa's and Kawasaki's conditions are quoted everywhere without saying which sheet they are about, and they do not need to be — they are conditions at a point and every point is the same. The two-colouring is quoted the same way and it is not a condition at a point, and the omission there is not harmless.

history · Attribution 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 two ways to come aparta crane left hanging, or every crane held and the piece still in islandsgridsubsetsnothing hangingholds togetherfailures localheld, of those passing3 × 34 joins1611100.0%100.0%4 × 49 joins512322197.8%65.6%5 × 516 joins65,5361,21578599.3%64.6%6 × 625 joins33,554,432260,625141,62199.6%54.3%a crane hangs from nothing when none of the joins at its four corners is kept — one crane, four points, no search

Nearly every cutting fails at one crane

Six by six connected cranes have twenty-five joins and thirty-three million ways to keep some of them, and an exhaustion over all of them takes a fifth of a second. Of the 33,412,811 that fail, 99.64 per cent fail at a single crane — one left holding none of the joins at its corners — which a maker can check by looking at each crane in turn. The arrangements that pass that check hold together less often as the grid grows: all of them at three by three, 54 per cent at six by six.

history · Folklore of folding the bar is the share of sound cuttings that hold the cranes in one piecesound means every crane keeps at least one join at its corners3 × 3100.0%2⁴ subsets of 4 joins · 1 states carried4 × 465.6%2⁹ subsets of 9 joins · 7 states carried5 × 564.6%2¹⁶ subsets of 16 joins · 23 states carried6 × 654.3%2²⁵ subsets of 25 joins · 52 states carried7 × 751.1%2³⁶ subsets of 36 joins · 123 states carried8 × 846.7%2⁴⁹ subsets of 49 joins · 294 states carried9 × 943.6%2⁶⁴ subsets of 64 joins · 714 states carried10 × 1040.5%2⁸¹ subsets of 81 joins · 1,758 states carried11 × 1137.7%2¹⁰⁰ subsets of 100 joins · 4,380 states carried12 × 1235.1%2¹²¹ subsets of 121 joins · 11,024 states carriedcounted one join at a time, carrying only which cranes on the frontier are already joined

The border is where the cranes come apart

Counted one join at a time rather than one subset at a time, the slit grid of connected cranes runs to twelve by twelve, where there are 2¹²¹ ways to keep some of the joins. Among the cuttings that hold every crane by something, the share that also hold together keeps falling — 54.3 per cent at six by six, 35.1 at twelve — and from eight by eight on it falls by the same factor at every size. A constant factor is the signature of the border: at six by six, 99.5 per cent of the sound cuttings that come apart do so through a stray piece touching the outermost ring of joins.

history · Folklore of 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 circle behind a refusal, on the smallest maps that need each onedash-dot: mountain; dashed: valley; heavy loop: the circle, each step to a panel the letters put above the last3 by 38 creases4 by 412 creases5 by 414 creases5 by 516 creases

The long circles go round the middle

A four-by-four map has 64 letterings the one-pass refusal catches with no three-by-three pinwheel anywhere in them, and they are not larger pinwheels. They are four shapes, sixteen versions of each, none unchanged by any turn or mirror, and every one is a circle of twelve creases running round the middle of the map. Read the same way on larger maps, every refusal is a circle round some rectangle of panels, and it needs the whole of that rectangle: one in 64 three-by-three rectangles closes one, one in 512 four-by-fours, and rectangles three cells across close nothing a pinwheel does not. Two window sizes explain all but 0.07 per cent of a twenty-by-twenty map's letterings.

complexity · Map folding what refuses a lettering of a tube 5 panels longone bar to a circumference; the whole bar is the share refused0%25%50%75%100%2 round49.0%4 round72.9%6 round59.5%8 round52.6%10 round52.7%12 round54.7%a three-by-three windowa four-by-four windowa circle round the tubea longer circle on the sheet1,500 letterings drawn at random for each tube; each is filed under the first test that refuses it

A tube refuses round itself

On a flat map the cheap refusal — a circle in the arcs a lettering forces — is almost entirely local: small windows of the map explain all but a sliver of it. Glue the map's left edge to its right and the windows still refuse at exactly the rate their number predicts, counted across the seam. But a new refusal appears that no window can contain, a circle that goes once round the tube. It refuses half of every lettering of a tube two panels round, over half of one four round, a quarter at six, and it falls by about half with every two panels of circumference while the tube's length hardly matters.

complexity · Map folding a 6 by 5 Miura with zigzag creases slit in pairsa slit ends at two vertices; two slits that meet share one10 of 10 slits20 of 20 interior vertices releasedshare multiplied by 2^20slitreleased, admits every letteringreleased, still refuses somenot released

An array is priced by its vertices

One slit along a Miura's zigzag crease multiplies the share of letterings that pass every local test by four, and one along a straight crease by three. An array of slits is not the product of those factors. Two slits that meet at a vertex release it once, not twice, and on a small Miura the slits' own factors price correctly exactly the 22 of its 128 arrays in which no two slits touch. What prices every array is the product of each vertex's own share — the ring conditions where nothing was cut, the strip rule wedge by wedge where something was. So the cheapest way to release every vertex is a matching: zigzag slits laid in pairs that never meet, half as many slits as vertices. And the count can be made before anything is cut only on a pattern whose vertices decide independently, which the Yoshimura's do not.

material · Kirigami a 6 by 5 Yoshimura and the loops of three that cost ita shaded triangle is a loop of three creases whose correlations multiplyrows 1.732 half-columns tall30 letterings at each vertexevery pair correlated -0.066727 triangles, (−1⁄15)³ eachcounted ⁄ product = 0.99252

What a loop of creases carries

Multiplying each vertex's own share of letterings gives the exact share for a whole Miura, both twists and a tapered corrugation, and three quarters of a per cent too much for the Yoshimura. The difference has an exact form. The count is the product times a sum over loops in the graph of creases, each loop weighted by the correlations its vertices put on the pairs of creases it uses. A Miura vertex correlates no pair at all, so no loop can carry anything. An equilateral Yoshimura vertex correlates every pair by −1⁄15, and its 27 small triangles take the product down by 0.8 per cent before the larger loops pay a little back. Draw the rows taller than √3 and the big-little-big lemma locks three pairs and frees the rest, no loop closes, and the product is exact.

material · Kirigami

Named alongside it

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

Necessary conditionBoundaryParityDNA origamiGluingKirigamiMaekawa's theoremRoutingTwo-colouringThe big-little-big lemmaConnectivityThe counting problem

All concepts