Origami & Folding
what one sheet can do
Concept

Measurement — where it appears

What a figure or a claim here is checked against, as opposed to what it asserts. Every quantity in this collection is produced by running a stated rule over a stated object, so a measurement carries its population, its resolution and its refusals with it.

Named by 23 essays across 7 fields — each of them below, with the objects they name alongside it.

populationpassfoldhave a gapbranchcut twice at random51 vertices, 13 kinds9.68.020%0%whole multiples of 45°60 vertices, 1 kinds30.019.3100%69%whole multiples of 30°60 vertices, 13 kinds19.713.177%26%a named vertex, jittered60 vertices, 4 kinds8.08.00%0%

Which vertices are the random ones

Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.

complexity · Typical instances each row is an exhaustive count over the patterns that construction producedon the edgedeepest piletimes smallercrease densitythe printed patterns8 patterns62%19.415.5×7.9twist tessellations12 patterns52%10.02.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2

Four ways to draw a pattern

Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.

complexity · Typical instances the bar is the shortest crease in the pattern, on a scale of powers of tenthe hexagonal patch at four turns of its polygons, everything else heldturn 0.21.4e-2130 creases · every one carries an arc · 2.25 mm on a 160 mm sheetturn 0.357.9e-6142 creases · 12 of them carry no arc · 1.3 µm on a 160 mm sheetturn 0.57.5e-3154 creases · every one carries an arc · 1.20 mm on a 160 mm sheetturn 0.72.8e-2154 creases · every one carries an arc · 4.53 mm on a 160 mm sheetone turn of one patch drops four orders of magnitude below the others, and it is the turn this collection prints

Twelve creases a micrometre long

A patch this collection has drawn for a long time carries a hundred and forty-two creases and a hundred and thirty arcs, and nobody had asked what the other twelve were. They are fragments left where the clip caught a pleat almost exactly at a corner — between one and nine micrometres long on a printed sheet, at one turn angle out of four, and it is the turn the collection prints.

flat-folding · Boundary the bar is the middle run of a hundred and twentysame pattern, same code — only the order the letters are tried in differsthe square patch2725 at best · 27 at the middle · 36 at worstthe elongated patch3432 at best · 34 at the middle · 39 at worstthe hexagonal patch4339 at best · 43 at the middle · 51 at worstthe triangular patch4539 at best · 45 at the middle · 53 at worstthe rhombille patch16684 at best · 166 at the middle · 48 of 120 unfinished at 20000an unfinished run is left out of the middle rather than counted as its budget

Four easy patches and one that is not

Run the same search a hundred and twenty times on each of five tessellation patches, changing nothing but the order the letters are tried in. Four of them answer in between twenty-five and fifty-three steps every single time. The fifth answers in eighty-four steps at best, a hundred and sixty-six in the middle, and does not answer at all in forty-eight runs of the hundred and twenty.

tessellation · Twists the bar is what the whole job costs if every attempt is stopped thereon the rhombille patch, read off 120 measured runsstop at 10051219% of runs finish by thenstop at 20053033% of runs finish by thenstop at 500105435% of runs finish by thenstop at 1000162442% of runs finish by thenstop at 2000263847% of runs finish by thenstop at 5000569749% of runs finish by thenstop at 100001060450% of runs finish by thenstop at 200001629160% of runs finish by thena run that never finished counts as above every cutoff, so the tail is read conservatively

Stopping is cheaper than finishing

A search whose cost varies by a factor of two hundred with nothing but the order of its guesses should not be waited out. Give up after a hundred steps, reseed and start again, and the whole job costs five hundred and twelve steps in expectation; run each attempt to twenty thousand and it costs sixteen thousand two hundred and ninety-one. Patience is thirty-two times more expensive than impatience.

complexity · Hardness of folding the bar is how many patterns the population holdseach one sampled forty times and then searched, to see whether the two methods ever disagreethe printed patterns80 never lettered by 40 draws · all 8 settled by search · worst 60 nodestwist tessellations70 never lettered by 40 draws · all 7 settled by search · worst 19 nodesquadrilateral meshes60 never lettered by 40 draws · all 6 settled by search · worst 6 nodesfold-and-cut patterns70 never lettered by 40 draws · all 7 settled by search · worst 14 nodesthey never do here — the patterns that separate them are not in any of these four

Four populations with nothing to separate

This collection keeps four standing populations of crease patterns to test its machinery against. Twenty-eight patterns, sampled forty times each for a lettering that agrees with itself and then searched for one — and on every single member the two methods return the same verdict in the same breath. The patterns that separate them are in none of the four, and the reason they are not is what the populations are for.

complexity · Typical instances the bar is how many nodes the search visitedone sheet crumpled deeper and deeper, its letters rechosen each time4 folds1716 panels · 34 of 40 random letterings agree · 1 backtracks5 folds1918 panels · 34 of 40 random letterings agree · 1 backtracks6 folds3435 panels · 15 of 40 random letterings agree · 0 backtracks7 folds3839 panels · 19 of 40 random letterings agree · 0 backtracks8 folds7271 panels · 11 of 40 random letterings agree · 2 backtracksthe share that agrees falls by more than half along this ladder; the search's cost tracks the panels and nothing else

Rare is not hard

Crumple a sheet deeper and the share of its labellings that agree with themselves falls from thirty-four in forty to eleven. The number of steps a search needs to find one of them does not move at all: it stays at about one per panel, with no backtracking, the whole way down. How often an answer turns up at random and how much work it takes to find one are different quantities, and a crumpled sheet is where they come apart.

material · Crumpling the bar is the largest gap between anywhere on the sheet and a referencea third fold specifies more folds than can be listed, so a sample of them is taken insteadnone of them0.089565 references, from two folds50 of them0.0763,498 references · 0.02% of the round100 of them0.0508,056 references · 0.04% of the round200 of them0.04723,480 references · 0.07% of the round400 of them0.02274,694 references · 0.15% of the round800 of them0.014270,882 references · 0.29% of the roundevery row is a lower bound on what the whole round would buy, because leaving folds out can only make the gap larger

The third fold cannot be listed

Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.

construction · Reference points each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all

A population nobody chose

Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon the collection had spent so long measuring belongs to the one tiling the grid leaves out.

complexity · Typical instances the curve is stop-and-restart; the rule is a constant letter order1001e+31e+41001e+31e+4563 at a cutoff of 10080 nodes, deterministic, nothing to restartexpected nodes in totalcutoff, in nodes

Restarting what cannot be restarted

Stopping a search early and starting it again with a fresh seed costs five hundred and twelve steps in expectation against sixteen thousand for patience. Every number in that is right. The distribution it is right about was made by the search's own coin, and taking the coin out costs eighty — with nothing left to reseed.

complexity · Hardness of folding each mark is one crease, ranked shortest to longest10⁻⁵10⁻⁴0.0010.010.1a factor of 498, and no crease in itlength, as a fraction of the sheet's side142 creases, rankedthe 12 in magenta are drawn, counted, lettered and put through every theorem, and none of them is visible

The shortest crease is not a crease

A crease pattern's density is usually quoted as total crease length over sheet area, which treats a metre of folding as a metre whether it arrives as one long line or ten thousand short ones. Reading the lengths individually instead finds twelve creases on a printed patch that are shorter than a wavelength of light.

material · Crease density the folded period of the Yoshimuradrawn periods across the top1234567the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyes240°yesnoa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

The period nobody measured

Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.

tessellation · Found patterns the bar is the share of each round's new references that lie on the sheet's own edgeafter one fold80%4 on the edge · 1 inside · 12 fold linesafter 2 folds9%48 on the edge · 508 inside · 92 fold linesan edge is a line a fold can cross twice, so it yields marks with the folds and not with their pairs

A reference on a sheet with no corner

Every construction in this subject begins from the sheet's own boundary: two edges meet at a corner, a corner is a point, and a point is what an axiom takes as input. A cylinder has two circles of edge and no corners at all, so a construction on one has nothing to start from and the seam is not a mark.

construction · Reference points 00.20.40.60.81-11234radius of the three marks, on the flat sheetmiss against flat (%)grown by the same factor everywhere · −0.00%grown more at the rim · +4.27%the growth of a spherical cap · −1.71%grown so that the curvature cancels · +3.52%a centre mark and three a third of a turn apart · on a flat sheet the three sit √3 times their radius apart

Three marks see nothing

The measurement a rim cannot make is an interior one, and the obvious interior measurement — two marks a known distance apart, measured again after growth — cannot detect curvature at all, because a uniformly enlarged sheet changes that distance and stays flat. Three marks cannot either: any three distances obeying the triangle inequality are the sides of a flat triangle. Four marks give six distances, and six distances are not free on a flat sheet. The growth profile a rim measurement reads as flat misses by three and a half per cent with four marks at the rim.

biology · Growth as metric 00.20.40.60.81-3-2-11radius of the cut, on the flat sheet∫K dA inside the cutR ⁄ √2grown by the same factor everywheregrown more at the rimthe growth of a spherical capgrown so that the curvature cancelsa cut at radius r reads the total curvature of the disc inside it, −2πr (ln Ω)′(r) — the rim measurement moved inward

A cut reads a slope

A rim measurement returns one number for a whole grown disc, and a family of growth patterns share it. Cut the disc in a circle and the piece inside has a rim of its own, and its reading is minus two pi r times the slope of the log of the growth at the cut — so a cut reads a slope, a set of cuts reads the slope at each radius, and the slopes add up to the growth profile itself. The one thing no cut can recover is how much the whole sheet was enlarged, which is the one kind of growth that curves nothing.

biology · Growth as metric the length a pattern reports, and the length a sheet of paper hassix of the eight are built on a unit square, so on those the distinction does not arisepatternown widthraw lengthreportedon paperThe Miura fold6.3739.36,6791,049out by 6.37xThe tapered corrugation1.187.81,2431,057out by 1.18xThe preliminary base1.004.8724724the sameThe square twist1.004.7704704the sameThe hexagon twist1.006.1916916the sameThe Yoshimura pattern1.0014.02,3802,380the sameFold and cut — the triangle1.001.7258258the sameThe waterbomb tessellation1.0014.32,2902,290the samea builder working in cells returns a pattern several units across, and not dividing by that is the whole of the error

A length needs a scale

These essays measure crease length, and a crease length is a length in the pattern's own coordinates. Six of the eight printed patterns are built on a unit square, so their coordinates are sheet widths and the distinction never arises. Two are not — a Miura laid out as six cells of unit width spans 6.37 — and on those two the shelf multiplied by the printed size without dividing by the width. The Miura's folding length was reported as 6,679 millimetres and is 1,049, and the same pattern's printable sheet has carried the right number all along.

material · Crease density metres of crease a square metre: what each pattern asks for, and what each paper allowsthe pale bars are patterns and the dark ones are paperswhat foil-backed tissue allows641026 µm, a crease 0.16 mm acrosswhat washi allows416740 µm, a crease 0.24 mm acrosswhat kami allows238170 µm, a crease 0.42 mm acrosswhat copier paper allows1667100 µm, a crease 0.60 mm acrossThe waterbomb tessellation89printed at 160 mmThe Yoshimura pattern82printed at 170 mmThe tapered corrugation41printed at 160 mmThe hexagon twist41printed at 150 mmThe Miura fold36printed at 170 mmThe preliminary base32printed at 150 mmThe square twist31printed at 150 mmFold and cut — the triangle11printed at 150 mma crease occupies about 6 sheet thicknesses, so the closest two creases can be laid is that, and the ceiling is its reciprocal

The density a paper allows

Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.

material · Crease density each family's dial, run, in the plane of the two directional factorsmeasured on the folded state of every pattern, not predicted from its rule1234561234along the patternacross itthe accordionthe square twistthe Miura foldthe Yoshimuraa point on the lower edge leaves one direction alone; a point on the dashed diagonal draws in equally both ways

The plane the five points were in

Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.

tessellation · Shrinkage the cross factor of a leaf corrugation against its zigzag anglemeasured off the folded state's own extent, at every angle drawnno change0.600.680.760.840.921.00zigzag angle, radians0.981.001.02below one between 0.7 and 0.88 radians, at worst 0.98555 — and the areal factor never falls below 1.360

The direction that gets longer

A shrink factor below one is a direction in which the folded sheet is bigger than the flat one, and the leaf corrugation has one. The cause is not the taper and not the angle: a corrugation's folded extent across its own creases is a constant of the cell, the same number at two rows and at ten, so the cross factor is the sheet's height divided by a fixed length — a straight line through the origin that crosses one at a definite row count.

tessellation · Shrinkage 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 the other fault a drawing can have, counted and measureda stub is a crease with a free end; the distance is how far from the rim it stoppedpatcheswith a crossingwith a stubstubsdistinct depthsshallowestdeepest120761566270.27 mm35.1 mm0.27 mm35.09 mm66 stubs27 depthsdistances at the 150 mm these patterns print at; the scale is logarithmic because the range is a factor of 128

A stub is never alone

A crossing is a crease running past another and it has a depth. A stub is a crease that simply stops, and it has one too — how far from the rim it stopped, which is also how much shorter than a crease it is. Measured across a hundred and twenty drawings: sixty-six stubs, from 0.27 mm to 35 mm at printed size, every one of them paired with another at exactly the same distance, and not one on a drawing that did not already have a crossing.

rigid · Tolerance drawings carrying each fault, of those drawna junction is split in two: the fault, and the rim ending the reading also calls a junctionextended, of 120clipped, of 120a crossing760two creases pass through each othera stub150a crease stops in the middle of the papera crease ending on a crease00the junction as a faulta crease ending on the rim120120the junction as the reading counts ita fragment25a crease too short to seeevery stub is on a drawing with a crossing; every clipped fragment is on a drawing with nothing else wrong

Two faults, not four

A drawing departs from its crease list in four named ways — a crossing, a stub, a junction and a fragment — and a checker tests for all four. Counted side by side on two hundred and forty drawings, two of the tests find nothing the others do not. No crease anywhere ends on another crease: every junction the reading finds is a crease meeting the rim, which is where creases are meant to end. Every stub is on a drawing that already has a crossing. What is left is two independent faults, and the clipped construction, which makes no crossings, still makes the second — creases a thousandth of a millimetre long, in pairs.

rigid · Tolerance 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

Named alongside it

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

SearchSearch costCrease patternPatchTessellationBoundaryIdealisationSamplingGluingTypical instancesWorst-case analysisCrease density

All concepts