Origami & Folding
what one sheet can do
Concept

Crossing — where it appears

Two creases drawn across one another at a point neither of them has as an endpoint. It is a vertex of the drawing rather than of the crease list, and it can never fold flat: Maekawa's count refuses it at every angle, and Kawasaki's condition refuses every crossing but a right-angled one.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

The vertex the list does not have

Every condition this collection checks is asked at a vertex of a crease pattern, and a crease pattern is handed to the checker as a list of points and segments. A reader is handed ink. Read the same patterns the second way and eight printed sheets gain nothing at all — while four tessellation patches gain 12, 18, 12 and 5 vertices that nobody wrote down, every one of them a place where two creases were drawn across each other.

flat-folding · As drawn the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge

Two creases that cross

A crossing is four creases at a point, so the four conditions of the subject apply to it — and three of them can be satisfied. It is developable at every angle, it satisfies the big-little-big lemma whenever its two lines carry different letters, and it satisfies Kawasaki's condition when the lines meet squarely. Maekawa's refuses it always, at every angle and under every lettering, because a crossing's four spokes belong to two creases and can only be four and none, two and two, or none and four.

flat-folding · As drawn the same tessellation on the same square, cut out of the plane two waysassembled from whole unitsclipped from the plane12 crossings · panels 1.73 apart0 crossings · panels closemountainvalleyraw edge

Cutting a patch out of a plane

A tessellation is infinite and a sheet is not, so every picture of one is a decision about where the paper stops. Assembling whole twist units on a square and running the outstanding pleats to the rim puts 12, 18, 12 and 5 creases across other creases on four of five tilings; generating the pattern over a larger region and clipping it puts none. The panels then place exactly — and what is waiting behind the repair is a different refusal that could not be asked about before.

tessellation · Twists the bar is the share of the twists on the paper that the paper's edge cuts0.5 of the sheet86%1 whole · 6 cut by the edge0.42 of the sheet55%5 whole · 6 cut by the edge0.34 of the sheet59%7 whole · 10 cut by the edge0.28 of the sheet70%7 whole · 16 cut by the edge0.22 of the sheet37%17 whole · 10 cut by the edge0.18 of the sheet49%23 whole · 22 cut by the edgea patch is a picture of a tessellation, and the smaller the unit the less of the picture is edge

Most of a patch is edge

Between 34% and 91% of the vertices in the crease patterns drawn here sit on the edge of the paper rather than inside it, and on the tessellation patches — the figures that are meant to show what a repeating pattern looks like — it never falls below a third. A boundary is one unit deep whatever the unit is, so the share falls like one over the number of units across and reaches nothing at any size a page can carry.

flat-folding · Boundary the sectors are 40°, 140°, 40°, 140° — each row is one way the vertex can start to foldcrease 1crease 2crease 3crease 4mode 10.7100.7102 of the four creases movemode 200.7100.712 of the four creases movethe numbers are the four fold angles' ratios to one another as the vertex leaves the flat state

Two mechanisms at one point

Two creases drawn across each other cannot fold flat — Maekawa's count refuses them at every angle. They move perfectly well as rigid panels, and they move in two ways: bend along one line while the other stays flat, or the reverse. Every other developable vertex of degree four has two ways too, and in both of them all four creases move together at a fixed ratio. The crossing is the case where the two motions have nothing to do with each other.

rigid · Rigid folding the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

How deep is a crossing

A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

rigid · Tolerance the bar is how many of the 33 patterns each refusal is the first to catchtwo creases cross5one sweep over pairs of creasesa vertex condition fails0one pass over the verticesthe panels do not place0one walk over the panelsthe letters force a loop0one pass over the crease listno ordering exists6every ordering of the panels22 of the 33 are refused by none of these and are folded, undecided, or waiting on a search too large to run

The order the refusals come in

This collection can say no to a crease pattern in five ways, and they cost wildly different amounts: a sweep over pairs of creases, a pass over the vertices, a walk over the panels, a pass over the crease list, and an enumeration of every ordering of the panels. Run all five over the thirty-three patterns in the four test populations and the cheapest refuses five, the most expensive refuses six, and the three in between refuse nothing at all.

complexity · Hardness of folding the bar is the share of random drawings with at least one crossing in them2 segments23.1%0.23 crossings on average3 segments51.2%0.69 crossings on average4 segments73.5%1.36 crossings on average6 segments95.2%3.48 crossings on average8 segments99.4%6.53 crossings on average12 segments100.0%15.30 crossings on average20 segments100.0%43.76 crossings on averageevery crease pattern in this collection has none, and none of them was drawn at random

Drawn by the same hand

Two straight segments dropped on a square cross about 23% of the time; four of them cross 74% of the time; twelve cross with certainty, about fifteen times over. Every crease pattern in this collection's four test populations has none — not because the checkers were catching them, but because the same rules that drew the patterns were incapable of producing one, and nothing looked until a construction finally did.

complexity · Typical instances 34560%20%40%60%80%100%creases in the stripreachable by simple folds72%30%17%13%the basic symbolsa dashed line — valleya dotted line — mountainan arrow — fold it nowand what they missreverse, squash, sink,petal — every one of thema move no dashed linecan ask forevery assignment of 68 seeded spacings

A file has no paper

The field's interchange format is three arrays — where the vertices are, which pairs of them an edge joins, and a letter for each edge — and that is exactly the object every computation on a crease pattern starts from. A list of edges cannot say that two of them must not cross, because crossing is a property of the drawing and the list has no drawing in it. So a pattern that no paper could carry is a perfectly well-formed file, and four of this collection's own were.

history · Notation the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge

The reader decides the junction

Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

history · Pedagogy each circle holds a crease shorter than a thousandth of the sheet12 fragments7.9·10⁻⁶ of a sheet · M5.9·10⁻⁵ of a sheet · M7.9·10⁻⁶ of a sheet · M5.9·10⁻⁵ of a sheet · V7.9·10⁻⁶ of a sheet · V5.9·10⁻⁵ of a sheet · Mand 6 more1018× to draw the longest142 creases and 60 interior vertices, 12 of the first and six of the second invisible

The crease the drawing cannot show

Twelve creases on a printed crease pattern are eight millionths of a sheet long. They are in every count the collection takes of that patch, they pass every theorem, and no printer resolves them and no hand folds them. They are also the only thing holding the folded sheet together.

flat-folding · As drawn 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

Named alongside it

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

Crease patternInterior vertexBoundary vertexNecessary conditionFlat-foldabilityIdealisationDecision procedureDocumentary recordFace graphFolded stateLayer orderMeasurement

All concepts