Folded state — where it appears
Named by 50 essays across 8 fields — each of them below, with the objects they name alongside it.
The paper is all still there
A folded sheet is smaller than it was and none of it has gone anywhere. How much smaller it is and how many layers deep it is are not two properties of a pattern — they are one number, and their product is the sheet.
The lengths are free
Kawasaki reads angles, Maekawa counts letters, and the big-little-big lemma compares one sector with its neighbours. Not one condition in the subject mentions how long a crease is — so a single vertex is not a pattern but a whole family of them, every member folding, no two folding into the same shape.
More than one way to lie flat
A crease pattern with its mountains and valleys marked is spoken of as though it named a folded object. It does not. The legal stackings can be counted exactly in one dimension, the count is routinely more than one, and its size is a property of the pattern that nobody quotes.
The outline is mostly crease
The edge of a folded model is what a reader looks at, and almost none of it is the edge of the paper. Measured across five patterns, the sheet's own boundary accounts for between nothing and a third of the exposed edge; the rest is fold, and on a waterbomb tessellation the raw edge does not reach the outside at all.
The shadow does not name the pattern
A photograph of a folded model carries an outline and a thickness at every point of it, and that is the whole of what it carries. It is not enough. Crease patterns in genuinely different places fold to identical outlines with identical layer counts, and nearly a third of the folded objects a short strip can reach are reached by more than one pattern.
The order does not name it either
A photograph of a folded model carries an outline and a layer count, and that is not enough to recover the pattern. Hand the observer the layer order as well — everything the object physically is — and most of the ambiguity goes. Most. What is left are pairs of genuinely different crease patterns that fold to the same object, which no better photograph reaches.
Decided before the design
A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.
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.
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 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 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.
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.
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.
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.
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.
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.
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.
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.
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.
The property a patch does not have
A folded corrugation is described as a material — a packing ratio, a stiffness, a Poisson's ratio — and every one of those is a statement about an unbounded medium. Fold the same tiling at six sizes on the same square and the compaction climbs from 3.89 layers to 4.79 as the share of units the rim cuts falls from nine tenths to four, and it has not settled at the fine end. The number a patch gives is the material's number minus its own boundary.
A cut that reaches the edge
A ring of paper with three creases running from its hole to its rim satisfies every condition the subject has — vacuously, because it has no interior vertex at all — and cannot be folded: its panels take no two colours and the two routes to one of them end up 1.75 sheet widths apart. One cut from the hole to the edge, crossing no crease and changing no letter, and it folds exactly. The cut removes an adjacency, which is the one thing neither a fold nor an edge can do.
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 part of the same method 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.
The loop is not the tangle
A search that finds a contradiction in a pattern's letters reports the first circle it meets, and on a tessellation patch that is eight to twelve panels of forty-nine. It reads as a local fault. Decompose the same arrows a second way and the set of panels that lie on some circle is thirty-five of forty-nine on the square patch and ninety-nine of a hundred and fifty-seven on the rhombille — which is why the smallest available repair does not reach it, and cannot be tried on most of the creases at all.
A contradiction is even
A crease pattern's letters can demand a circle of panels each of which lies below the next, which is a proof that the sheet has no folded state. Every such circle found here — one thousand one hundred and forty-nine of them, across every family of patterns this collection draws — has an even number of panels in it, and none has four. Both facts are theorems rather than observations, and they come from opposite ends of the subject.
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.
The ring is the loop
The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself.
The letters a crumple was given
A sheet creased by folding it and folding it again arrives with a mountain-valley labelling that cannot be wrong, because a folding produced it. Nothing about the pattern protects it: reletter the same creases and the share of labellings whose letters agree falls from every one of forty at eight panels to eleven of forty at forty-one. The foldability of a crumple is a fact about its history, not about its drawing.
One cut removes one arc
A crease pattern whose letters contradict themselves has, in principle, an obvious smallest repair: cut one crease and the statement it was making goes away. Cut every crease of four tessellation patches in turn — four hundred and seventy-four cuts — and sixteen of them leave a sheet whose panels still land anywhere at all. A cut gives the paper a freedom, and a sheet with a freedom in it has no folded state to order.
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 first thing about layers
A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.
Closure is not the identity
Walk a folded state from panel to panel, composing a reflection at every crease, and come back to where the walk started: the composition has to be the identity. That is the rule everybody states, and it is a special case. On a sheet whose edges are glued the walk does not come back to where it started, and what the composition has to equal is the gluing map.
The turn a column costs
The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.
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.
The field is empty where it would say nothing
The interchange format for crease patterns has a field for the layer order and nothing ever fills it in. Filling it in where the folded states can be listed — four of the eight printed patterns, and Miura patches to twelve panels — finds that the preliminary base and both twists have exactly one folded state, so every one of the field's signs follows from the crease pattern and the field would record nothing a reader could not compute. The fold-and-cut triangle has two states. The Miura is different: every patch with three or more columns has several — three, six and eleven on the three-by-two, three-by-three and four-by-three — so on the pattern that gets built the field carries information from six panels up, and the field's size had been measured as log₂ of the panels' orderings, which on the preliminary base is fifteen bits for an object that has zero.
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.
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.
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.
One choice with eleven answers
A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.
The folded strip lies at its own slant
The two directional factors read a folded state against the axes the pattern was drawn on, with its first panel held where it was drawn. A flat-folded Miura is not lying along those axes: it is a strip of fixed width laid at exactly its own slant to them, and the leaf corrugation's strip lies at exactly its zigzag. So the Miura's cross factor changes with the number of columns — 1.748 on four, 0.948 on ten, at one slant — and on the leaf, on every Miura with an odd number of columns and on the twists of three and six sides, it changes with which panel is held. Read without axes, both families compact least at 0.80 radians, not at the 1.05 the box on the axes reports.
The rows are free
A Miura's hinges per layer of compaction were found to climb without a ceiling as the pattern is refined, and the footprint was blamed for growing with the sheet. Measured one direction at a time it grows in only one. Every band between two straight creases folds exactly onto the band before it, as an accordion's pleats do, and the folded box does not change at all as they are added. Every band between two zigzags walks the stack a step along itself, the sine of the slant each. So the hinges a layer costs settle as rows are added and climb as columns are, and the same sixty-four cells cost 1.94 hinges a layer laid out long in their rows and 16.14 laid out long in their columns.
Every panel holds a frame
A flat-folded pattern can be put down on any of its panels, and on some patterns the pair of shrink factors changes with the choice. Holding a panel does not move the paper so much as turn the frame it is measured in, by twice the alternating sum of the crease directions crossed to reach that panel — a number Kawasaki's condition makes independent of the route. So the frames are read off the crease pattern before anything is folded. A pattern with one frame gives one answer whatever its folded shape; the five-sided twist has five frames and five answers; and the four-column Miura agrees on every panel although its folded strip is not its own mirror image, because a box cannot tell a shape from its half-turn.
The slant that stacks in one depth
A folded Miura's rows lie exactly on one another, and its columns each lie one step along the folded strip from the last, so the pile at any point is the rows times the number of column images over it — on average two over the sine of twice the slant. When that number is an even whole number the pile is one depth everywhere but the ends: two columns deep at 45 degrees, four at exactly 15. When it is odd the pile stays mixed, because neighbouring columns lean opposite ways. The printed Miura at 20.05 degrees sits beside the odd count three, and its graded pile is that and nothing more; at 15 degrees the same single-freedom mechanism stacks every panel away from its ends over one depth.
A sheet that turns with the twist
Holding a different panel of a folded twist can give a different pair of shrink factors: on a square sheet the three- and six-sided twists gave two pairs each and the five-sided twist five. Cut each twist from a regular sheet of its own number of sides and every panel gives the same pair. The reason is a count of rotations. A K-sided twist's panels hold its folded state in frames that are multiples of 180⁄K or 360⁄K degrees; a sheet that shares g of the twist's rotations makes the widths agree in every frame those rotations reach, with the half-turn a width cannot see thrown in. On twenty-two pairs of twist and regular sheet, that count gives the exact number of answers every time — a hexagon twist on a triangle gives one, a pentagon twist on a square five.
A line in the columns, a binomial in the rows
A Miura strip two rows high was found to have 2c − 3 folded states for c columns, from three columns to seven, and the search that found it ran out at eight. Counted instead from the pairs of panels that actually overlap, the count runs to twenty columns and nine rows, and 2c − 3 turns out to be one row of a formula: a patch c columns by r rows has 1 + (c − 2)(B − 1) folded states, with B the central binomial coefficient of r + 1. The line in the columns has a reason that can be drawn. The binomial in the rows is measured and not explained, and it holds only while the folded columns two apart still overlap, which is the same as the pile being more than two deep.
A stack is a shared fate
Hinges that fold onto one segment of a folded stack share whatever happens to that segment. On a Miura every zigzag line puts one hinge from every row in the same place, so a sheet cut between its rows into five modules leaves every place in every module, and one jam anywhere stops all five. Cut between its columns, each place stays inside one module — except at the slants where two over the sine of twice the slant is an even whole number, the very slants that pile the stack evenly, where columns that far apart share places across the cut.
A corner turned shows the sheet beneath
Relettering the preliminary base turns its whole face one colour or the other and nothing between. One added crease is the cheapest change that does better, and the angle condition leaves it a single direction: across a diagonal at right angles, cutting a corner off. Placed s along each edge from the corner, it divides the face into three regions — where the corner lay and where it lands, each twice s squared of the face, and the rest — and the letterings of the result colour those three regions in all eight possible ways.
Two stacks shuffled in glued pairs
A Miura patch three columns wide and r rows high has a central binomial number of folded states — 2, 3, 6, 10, 20 and on to 252 at nine rows — and the count was measured without a reason. The reason is a shuffle. In every state the middle column lies at the bottom and the two end columns lie above it, interleaved like two halves of a deck. Some pairs of an end column's panels can never be parted, because the row crease joining them lies inside the other column's folded footprint, and the two columns take those welds in turn, one at every row crease. That leaves r + 1 unbreakable blocks, and the number of ways to interleave them is the binomial.
What refused the mirror was the rim
A twist tessellation on a sheared square grid folds, for a window of pleat angles, to a mirror image of its own arrangement, and whether its layers can be stacked was open. Asked of the drawn patches, the answer is no at every pleat angle, mirrored or not. Every refusal is a ring of flaps — flaps folded onto one face of a panel, each lying across the crease of the next — and on this grid every ring sits at the rim. Cut two creases in and every patch stacks, mirrored ones included.
Two corners compete across an edge
One crease across a corner of the preliminary base cuts its face into three regions, and the letterings colour them all eight ways. Two creases across two corners cut it into five, and the question was whether the corners compete for the paper beneath. It depends on which two. Opposite corners colour the five regions in all thirty-two ways on either face. Adjacent corners, joined by an edge the base folds in half, reach twenty-nine on each face and lose three; the three one face cannot show are the other face's three with every side exchanged. And two corners cut at the same distance land on each other, so the face is back to three regions however many states the letters make.
The proportion everyone draws is the hard one
The Yoshimura is drawn equilateral, and every cost and count of it has been measured there. Moved a quarter of a step along its own height dial, a search for a stacking that took 268,650 steps takes 808, and a sheet with 9,790 folded states has 92. The equilateral pattern is the most expensive member of its family on every sheet measured, because exact coincidence removes the conditions that a panel lying across a crease would impose: from 1,024 conditions to none. A drawing chosen for being easy to construct is the one instance where almost nothing constrains the layers, and a population of one chosen drawing had been reporting that freedom as the difficulty of the pattern.
Named alongside it
The objects these essays reach for when they reach for this one.
Layer orderingCrease patternNecessary conditionLayer countLayer orderAssignmentFlat-foldabilityFootprintNon-crossing conditionShrinkageUnit cellMiura