How much of a folded sheet lies over the rest of it
stack-orders is one function. Everything below came out of it during this
build, at arguments 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.
At its defaults
view: "populations"
view: "constraints"
view: "shelf"
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×3
- 112 letterings of the preliminary base fold, in 336 states, and show 2 different faces between them ×1
- 13 of 33 patterns in the populations this site tests its checkers on are known to have a folded state ×1
- 2 of 8 patterns generate no taco-tortilla rule at all and are held entirely by the other one ×1
- 26 of 73 crumples measured have more than one folded state, against none of the printed patterns ×1
- 3 of 4 patterns have letterings that force a loop of panels, and every one of them has letterings that do not ×1
- 3 of 8 printed patterns have a point of their footprint with every panel of the sheet over it ×1
- 4 of 4 patterns keep panels out of view from above entirely — the preliminary base shows two of its eight ×1
- 4 of 8 printed patterns have their ordering decided; the rest have too many panels for the search to finish ×1
- 6 strips give the same stacking count and the same state count by two routes that share no code ×1
- 8 letterings of the square twist fold, in 8 states, and show 4 different faces between them ×1
- 9 maps of stamps fold in the same number of ways counted by ordering their panels and counted by a rule that never places one ×1
- a panel is counted at the rim when one of its own sides is a raw edge of the sheet, taken from the planarised pattern rather than from the drawing ×1
- all 32 letterings whose central ring reads as one letter satisfy every condition at every vertex, and none of them has a folded state ×1
- every crumple here is a sheet that folded itself, so a search that failed to order one would be reporting on its own machinery rather than on the paper ×1
- every face any lettering of these patterns shows is the printed face, the printed face turned round, or the whole face changed side at once — none recolours part of it ×1
- every lettering of fold and cut — the triangle with a folded state was searched to the end ×1
- every lettering of the preliminary base with a folded state was searched to the end ×1
- every lettering of the square twist with a folded state was searched to the end ×1
- every one of the 112 letterings of the preliminary base with a folded state was searched to the end ×1
- every one of the 8 letterings of the square twist with a folded state was searched to the end ×1
- every pair of panels is tested for shared ground, and the constraints are read off the folded state rather than off the drawing ×1
- every pattern in these populations satisfies the conditions at each of its interior vertices, which is what put it in the population ×1
- every pattern with one folded state keeps the other side one layer down, and every pattern with more than one puts it two down somewhere: 4 of the first kind, 4 of the second ×1
- Fold and cut — the triangle: 64 letterings, 30 pass every vertex condition, 18 have a folded state ×1
- on every orderable pattern the other side, wherever it is under a point at all, is one or two layers down, never deeper — whatever the pile's depth ×1
- on none of the 6 patterns with panels away from the edge does a rim panel lie over more than a middle one, and on 5 it lies over fewer ×1
- the deepest point of a folded leaf is 1.93 times the average depth, and the multiple moves by less than 0.000 across the patches measured ×1
- the depth is read off the ordered pile at every sample of the footprint, over every folded state the ordering search returns, from above and from below ×1
- the ladder grows one patch of one tessellation, so the only thing changing between rows is how much paper there is ×1
- the layer count is sampled on a grid over the folded footprint and integrates back to the area of the sheet, which is the check the sampling gets rather than an assumption ×1
- the layer count is sampled over the folded footprint and integrates back to the area of the sheet, which is the check the sampling gets ×1
- the letter fold puts two panels showing the same side on top of each other, so one layer down is a fact about these patterns and not a law of folding ×1
- the loop is read off the crease list in one pass, so it is available on patches whose orderings could never be enumerated ×1
- the pile is one ordering the rules admit, drawn panel by panel from the bottom up ×1
- The preliminary base: 256 letterings, 112 pass every vertex condition, 112 have a folded state ×1
- the search finishes at 12 panels and is refused past it, on a pattern whose orderings are 12! at that size ×1
- the side showing at a point is read off the highest panel covering it, so it is a consequence of the ordering rather than of the drawing ×1
- The square twist: 4096 letterings, 256 pass every vertex condition, 8 have a folded state ×1
- the three counts are nested by construction: a lettering with a folded state passes every vertex condition, and one that fails a condition is never asked for an ordering ×1
- the two non-crossing rules are counted separately, because which of them holds a pattern down is a property of the pattern ×1
- where it is two layers down it is so over at most 5.7 per cent of a pattern's samples, counted over every state and both faces ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
A collision is an order
Paper passing through paper is treated here as a thing that happens during a motion and is caught by watching for it. At the flat state it is not an event at all: it is the absence of an ordering, and it can be proved rather than observed. Four of the six quadrilateral meshes this site solves for rigid folding place perfectly and admit no ordering of their nine panels — so every one of them must pass through itself, and none of them was ever driven to find out.
A proof in one pass
Deciding whether a crease pattern has a flat folded state is hard, and the search that decides it gives up at twenty-four panels. One line of the same machinery does not search at all: each crease says which of the two panels it joins lies above the other, and a circle in what those statements demand is a proof that no folded state exists. It costs one pass over the crease list, and on a tessellation patch of a hundred and fifty-seven panels it answers in milliseconds.
Consistent is not foldable
The square twist has 4,096 mountain-valley labellings. Two hundred and fifty-six satisfy every condition at every vertex; two hundred and fifty-two of those have letters that do not contradict themselves; and eight have a folded state. So the cheap proof that reads the letters in one pass accounts for four of the two hundred and forty-eight failures, and the other two hundred and forty-four are refused by a search over orderings that nothing shorter replaces.
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.
No height to swap
A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.
One sheet down
A colour change has been priced by how deep the pile is — eight layers over every point of the preliminary base, six on a small Miura. Ordered, the piles say something else: wherever the other side of the paper lies under a point, it is the next sheet down on every pattern with one folded state, and never more than two down on any. And relettering the same creases cannot reach it. Of 112 letterings of the preliminary base that fold, every one shows either the printed face or the whole face turned the other colour; the square twist's eight only turn its face round.
Taught with a wrong reason
Four mountains and four valleys is what the preliminary base's symmetry suggests and Maekawa forbids it; a twist looks like a twist when its central ring reads as one letter, and no such lettering folds; a tessellation is verified because its unit is, and a forty-nine-panel patch of one had no folded state at all. In each case the conclusion taught is right and the reason offered for it is not, and the site that repeats them is this one.
The crumple keeps its options
Every crease pattern this site prints has exactly one folded state and not one of its thirty-nine available rearrangements is legal. A sheet creased by folding it at random four times has an average of 2.16 folded states, one of them has nine, and six of three hundred and thirty-five rearrangements are legal. The sheet nobody designed is the one with room left in it.
The deepest point pays for the paper
A folded design uses its sheet according to its mean layer count and its paper according to its deepest point, and the ratio of the two is a property of the crease pattern. Measured on every printed pattern it runs from exactly one to nearly six — and it does not split tessellations from bases, as expected. It splits patterns whose every panel lies over every point from patterns that keep a footprint with structure in it. The ratio moves the corner of the substrate map by its square root, so the fold-and-cut triangle can reach the paper's eighty layers at 326 millimetres where the preliminary base must stop at 134.
The file records no verdict
A crease pattern file records vertices, edges and letters. Every one of the square twist's two hundred and fifty-six admissible letterings makes a perfectly valid file, and two hundred and forty-eight of them describe an object that does not exist. The format has a field for the layer order — the one thing that would settle it — and nothing fills it in, so a file is a drawing rather than a claim, and the field exchanges them as though they were claims.
The grid a division makes
Dividing a square into thirds in both directions is a construction: four creases, each exact, each landing on a rational the ladder can name. The object it leaves behind is a three-by-three map of stamps, and how many ways that folds is the oldest open problem in the subject — 1,368 at three, 300,608 at four, and unknown at five.
The lettering that folds nowhere
The conditions at a vertex admit 256 letterings of the square twist. Eight of them have a folded state. The other 248 satisfy developability, Kawasaki, Maekawa and the big-little-big lemma at every vertex of the pattern and cannot be folded by anyone — and this site printed one of them for years, at true scale, with instructions to fold it first.
The map counted from the layers
The classical map-folding counts are computed from a rule that never places a panel: work out which edge of the folded square each fold wraps around, and refuse the orderings that interleave two folds at one edge. Place the panels instead and order them by the general non-crossing rules, and the same numbers come out — 2, 6, 16, 50, 144, 8, 60, 320, 1368 — on nine sizes, by machinery that shares no line of code with the first.
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 patterns a checker is tested on
This site keeps four populations of crease patterns and runs its checkers over them, which is what makes a claim about typical instances measurable rather than rhetorical. Asked whether the members actually fold, the populations answer: thirteen of thirty-three do, six place and cannot be ordered, five cannot be placed at all, and nine are past what the search will finish.
The rim lies over less
A folded sheet's boundary is usually discussed as the place the theorems stop applying. It is also visible in the pile: a panel carrying a raw edge of the paper lies over fewer of the other panels than one that does not, on every printed pattern that has both kinds — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb tessellation, and never once the other way round.
The tiling the unit could not promise
Every twist on this site carries the same caveat: the unit is verified and the plane is not, because deciding a whole pattern is intractable. There is one thing about a whole pattern that costs a single pass over its crease list, and it says no. The square twist tiling was drawn with a lettering that contains a loop of twenty-eight panels, so the patch on this site had no flat folded state at all — and only seven of forty independent redraws avoid one.
Fourth of eight, and still not chosen for it
A deployable is sold on compaction: large in use, small in transit. Measured, the pattern that actually gets built converts folding into compaction at 0.67 sheet-widths of crease per layer, which is fourth of the eight printed patterns — nearly three times worse than the Yoshimura, which nobody deploys, and nearly three times better than the hexagon twist, which nobody deploys either. The ranking does not pick out the pattern that flew from anywhere on the shelf, and that is the finding.
Twice as thick where it is thickest
A folded leaf's thickness is quoted as an area calculation: so much lamina, so much footprint, so many layers on average. The average is not what has to fit in the bud. Sampling the folded state of corrugated leaf patterns of two to five rows gives a deepest point of eight, twelve, sixteen and twenty layers against averages of 4.15, 6.23, 8.31 and 10.39 — a ratio of 1.926 that does not move at all.
Two panels that are one panel
Paper cannot pass through paper, and every test for it compares pairs of panels. On a glued sheet two pieces of the drawing can be the same piece of paper — so a test that does not know the identification either reports a collision between a panel and itself, or misses one where the sheet meets itself round the loop.
Two refusals that refuse differently
Four of the six developable quadrilateral meshes this collection solves have no ordering of their nine panels — they must pass through themselves, and a search over every ordering proves it. On all four, the letters agree with themselves perfectly. The linear proof and the exponential search are not a fast test and a slow one: they answer different questions, and neither contains the other.
Which side arrives
A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.
Every generator · The flat-folding field · The patterns a reader can fold