The vertices a crease list does not have
as-drawn is one function. Everything below was drawn by it when the site was
last published, at settings taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
With nothing specified
view: "placement", show: "paper-map", width: 0.6
view: "placement", show: "paper-drawing", width: 0.6
view: "placement", show: "paper-law"
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- composing the reflections around every vertex of the drawing: 30 of 32 come back to the identity, and the 2 that do not are exactly the 2 crossings ×4
- a crossing's four sectors are two pairs of equal angles, so Kawasaki asks the two lines to meet squarely — here they meet at 22.9° and the two alternating sums differ by 268.3° ×3
- a stub never occurs alone: on every one of the 15 patches that carry any, the count is even — a pleat has two creases and they stop together ×3
- and no patch here has a stub without also having a crossing — 15 of 120 carry stubs, 76 carry crossings, and the first set is inside the second — so a test for crossings already refuses every drawing a test for stubs would ×3
- and the two stop at the same distance: every depth measured occurs an even number of times, so the 66 stubs are 33 pairs rather than 66 separate faults ×3
- 4 of 13 patterns drawn here have creases crossing one another, and every one of those is a patch cut out of a tessellation ×2
- a drawing moved by one whole period of its square or triangular tiling is the centred drawing again, segment for segment, so the moved drawings are the tessellation's own ×1
- a pattern is read twice — as the crease list its generator returns, and as the ink a reader would fold — and the second reading adds a vertex wherever two creases cross ×1
- a placement chosen at random puts a fragment on 11.2% of drawings with fragments counted below a thousandth of the sheet, and 1.8% below a ten-thousandth; the centred placement puts one on 4.2% ×1
- and its four spokes are two creases, so a crossing has four mountains, four valleys or two of each — never the difference of two Maekawa asks for (here 2 and 2) ×1
- at a crease 0.6 mm wide 63% of placements are clean, far more than the 30% independent pieces would leave at the same mean, because the pieces arrive together ×1
- each of the 5 centred drawings that carry fragments has, for every fragment, a partner of the same length on the opposite edge of the sheet; on 4 the partner is the fragment's image under the half-turn about the sheet's centre, and on the the elongated triangular tiling patch, whose drawing has no such symmetry, under a half-turn about a point off it ×1
- each of the 5 drawings loses every fragment once the pattern is moved 0.127 mm or less across the rim they sit on, and part way the count can be odd — the pairs exist only while the sheet's centre is a centre of the pattern ×1
- every drawing falls in one of five groups, and in none of the others the four faults could make — no stub without a crossing, no fragment beside a crossing without a stub ×1
- every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
- every one of the 120 drawings has a placement clean at the width of a crease in copier paper; the tightest reaches 1.57 mm, and every drawing that stops short of 2 mm (1 of them) is rhombille ×1
- every way of lettering a crossing as two lines, at every angle: 32 tried and none of them satisfies both conditions ×1
- fragments come in pairs of equal length, as stubs do: on every drawing that has any, each length occurs an even number of times ×1
- lettering the four spokes independently instead, 8 of 128 satisfy both — and every one of those is the right angle ×1
- moved 0.13 mm either way across their rim, all 10 drawings are free of fragments, pass every vertex condition and close round every vertex; 1 of them comes back from the lettering search with letters that force a loop where the centred drawing's did not ×1
- of 24 placements of the sheet chosen at random, none leaves all 120 drawings free of fragments — the fewest is 3 and the most 24 — against 5 at the centred placement ×1
- on 240 drawings, no crease ends on the interior of another crease; every junction the reading finds is a crease meeting the rim, which every drawing has; and every stub is on a drawing that also has a crossing ×1
- on every one of them the two readings agree: the patterns whose panels cannot be placed consistently are exactly the patterns drawn with a crossing ×1
- on this drawing, at the width of the crease on this frame, 29% of random placements leave no piece too short to form, and the grid drawn agrees to within four points at 27% ×1
- over 120 drawings the measured mean count of pieces shorter than 0.42 mm is 1.004 against 0.995 from the derived rate, and no one drawing misses by more than a factor of 1.25 ×1
- the clipped construction makes no crossing and no stub, and makes fragments on 5 of its 120 drawings — where they are the only fault the drawing has ×1
- the pieces drawn as fragments are exactly the creases the reading counts as fragments at this corner — 1 with the pattern moved 0 mm ×1
- the rate per crease lies between 2.42 and 2.74 on every drawing, around the 2.55 that creases pointing evenly in all directions would give, eight over pi, so the fragment count follows the number of creases and not their length ×1
- the turn at a crossing is twice the amount Kawasaki is out by there — 240.0° to 240.0° on this patch ×1
- this placement leaves 6 pieces of crease shorter than a crease 0.6 mm wide, on a drawing that is otherwise whole ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
A fragment counts creases, not length
A fragment was a piece of crease under a thousandth of the sheet, a threshold chosen for being smaller than a pencil line. Set it instead at the width a crease occupies in real paper, which is the first threshold a folder would recognise, and the rate at which a sheet carries one can be derived from the drawing without placing it: the crease width, times about 2.5, times the number of creases on the sheet. Measured over 1,024 placements on each of 120 drawings it comes out at 1.00 against 0.99 at the width of a crease in kami, with no drawing off by more than a quarter. It reads how many creases there are, and hardly how long they are. It also says what it cannot, which is the clean share: 63% of placements at the width of copier paper, against the 30% independent pieces would leave, because a rim that catches one corner catches its whole row.
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.
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.
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.
No placement cleans every drawing
Five of a hundred and twenty clipped twist patches carry fragments — creases a fraction of a millimetre long, left where the sheet's edge cut through the pattern — and the proposed repair was to nudge the sheet. A fragment turns out to be a corner of a twist polygon lying within a seventh of a millimetre of the rim, and moving the pattern 0.127 mm clears any one drawing. No single movement clears them all. Laid at random, a sheet puts a fragment on one drawing in nine; of twenty-four random placements the cleanest still leaves three drawings with one, and the centred placement every drawing was made at leaves five. A fragment is not a fault of the construction. It is the threshold meeting a placement, and the only clean set of drawings is one placed drawing by drawing.
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.
The drawing does not say what is glued
One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.
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.
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.
The refusal that reads the list once
There are five ways of saying no to a crease pattern here, and their costs are two hundred and eighty-two, a hundred and twenty-six, a hundred and fifty-seven, thirty-nine thousand six hundred and twenty-one — and a search that is refused outright. On the largest patch the four cheap tests together do less work than one of them looks like it should, and the fifth cannot be started. A refusal that reads the crease list once is the only kind that scales.
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.
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.
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.
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.
All the figures · The flat-folding field · The patterns a reader can fold