Local rules, global behaviour

Flat-folding

Two conditions at a point

Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.

Flat-folding

Why the difference is two

Maekawa's theorem says mountains and valleys differ by exactly two at every flat-foldable vertex. The constant is not empirical — it is a full turn, and the theorem is about winding rather than about paper.

Flat-folding

The smallest sector decides

Two assignments can satisfy both flat-folding theorems and only one of them folds. What separates them is a condition about the smallest angle, and it is the first rule in the subject that is not about counting.

Flat-folding

Local is not global

Every vertex can satisfy every condition and the sheet still not fold. Deciding whether a whole crease pattern folds flat is NP-hard, which means no figure will settle it and no algorithm will scale.

Flat-folding

Which layer goes on top

The mountain-valley assignment says which way each crease turns. It says nothing at all about which sheet ends up above which, and that second question is a different object with its own rules — and all of the difficulty.

Flat-folding

A strip is decidable

Take the same problem down one dimension and it stops being hard. The reason is not that strips are small — it is that overlaps on a line form a chain, and chains cannot contain the cycles that make the two-dimensional question intractable.

Tessellations

One vertex, repeated

Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.

Tessellations

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

Tessellations

A square that turns

A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.

Tessellations

A material made of creases

A folded sheet has a Poisson's ratio, a stiffness and a packing behaviour that the flat sheet did not. None of them belong to the paper — they belong to the pattern, and changing the pattern changes them without changing the material at all.

Tessellations

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

Rigid folding

What the vertex does on the way

A four-crease vertex is a linkage on a sphere. Solving its closure gives the fold angles at every moment, and two theorems that are usually proved about the flat state turn up in the answer without being put there.

Flat-folding

How many assignments fold

The local conditions throw away most of the ways a pattern could be creased. They throw away a smaller and smaller fraction as the pattern grows, and what survives grows faster than what is discarded — which is why a strong filter is not a decision procedure.

Designing a base

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

Tessellations

A property you can dial

Steel has one Poisson's ratio. A Miura-folded sheet has a surface of them, and where on that surface it sits is set by the panel shape and by how far it happens to be folded — which is why it is a mechanism rather than a material.

Rigid folding

A state no motion reaches

Flat-foldability asks whether a folded state exists. Rigid-foldability asks whether there is a path to it. The two sets are different, and the difference can be counted on a single vertex.

What it costs to know

A machine that can only crimp

Change the atom and the whole picture changes. A machine whose single move folds two adjacent creases at once reaches strips no simple-fold machine reaches, is defeated by strips they handle easily, and cannot fold an odd number of creases at all — for reasons that are pure arithmetic.

What it costs to know

Four questions about one sheet

Deciding, counting, listing and optimising are not four difficulties of one problem. They are four problems, and folding is the subject that proves it: a ruled map is trivial to decide and unsolved to count, while a general crease pattern is the other way round.

What it costs to know

Hardness is about the worst one

Flat-foldability is NP-hard, and every crease pattern on this site is decided in under a second. Both are true, and holding them together is the difference between using the result and repeating it: hardness is a statement about the worst instance a family contains, and nobody folds the worst one.

What it costs to know

Where the exponent comes from

The number of ways a strip of stamps folds grows exponentially, and the base of the exponential is a number nobody has proved exists. The ratio of one term to the last climbs past three and is still climbing where the computation stops — which is the only structural handle anybody has on the sequence.

What it costs to know

What a checker cannot check

Every crease pattern on this site is run past four theorems before it is allowed onto a page, and passing all four proves nothing. The gap is not a bug to be closed: it is the NP-hardness result, arriving as a property of a hundred lines of code.

Tessellations

Which polygons twist

Twist tessellations come in three kinds — triangle, square, hexagon — and it is natural to read that as a fact about twists. It is not. A twist can be built around any regular polygon and every one of them folds; what stops at three is the tiling, and the tiling is a fact about the plane.

Who found it, and when

The name is not the date

Kawasaki's theorem is in Husimi's book ten years before Kawasaki's paper. Maekawa's is Justin's too. The mean gap between a result in this field and the name it is known by is twenty-two years, and it runs in one direction.

Who found it, and when

Found before it was designed

Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.

Who found it, and when

The same vertex, found four times

A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

Folding nobody designed

A sheet that routes itself

DNA origami folds one long strand into a shape by holding it against itself with a few hundred short ones. There is no sheet and no crease — what has to be designed is a route — and the first thing that can go wrong is a counting argument crease patterns already know under another name.

Folding nobody designed

Two things called folding

A protein folds and a sheet folds, and the word is the same word by accident. Both have exponentially many states and that is not the difference. The difference is that one of them can be filtered by four conditions checked at a single point, and the other cannot be filtered by anything local at all.

Flat-folding

The sheet has two sides

Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing.

Flat-folding

Almost every pattern fails

Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.

Flat-folding

Where the paper stops

Every flat-folding theorem is a statement about a full turn of paper, so a vertex at the edge of the sheet is subject to none of them. Cutting a patch out of a pattern removes conditions rather than preserving them, and a small enough patch has almost none left.

Tessellations

A unit that folds is not a tessellation

Of the 512 repeating rules for the waterbomb tessellation, 56 pass every condition on a two-by-two patch and 32 pass on every larger one. The twenty-four that die were never foldable — the small patch simply contained one of the four kinds of vertex the pattern makes, and the failures were at the other three.

Flat-folding

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.

Flat-folding

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.

Flat-folding

Nothing meets at three

Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars.

Rigid folding

Paper through paper

Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.

Tessellations

Any tiling makes a twist

A twist tessellation is usually drawn, admired and copied. It can be derived instead: hand the construction any tiling of the plane and it returns a crease pattern that folds flat, with the twist polygons' shapes forced by the tiling's own angles and nothing left to choose but how large and how turned.

Tessellations

Where two twists share a pleat

Every twist tessellation the tradition draws has one size of twist, because every tiling it is drawn on has one kind of vertex. Hand the construction a tiling with two, and the pleat between a large twist and a small one turns out to fix their sizes exactly — three to one, and nothing else folds.

Tessellations

Fenced at both ends

The twist angle of a tessellation looks like a free dial, and it is fenced twice. Turn too far and the pleats have no paper left. Turn too little and something stranger happens: every angle condition in the subject goes on holding and the pattern loses its mountain-valley assignment entirely.

What it costs to know

Two directions that will not separate

A map has rows and columns, and a strip of stamps is a map with one row. The obvious hope is that the two-dimensional count is built from the one-dimensional one — fold the rows, then fold the columns. It is not: a two-by-three map folds 60 ways against a product of 12, and the discrepancy grows from a factor of two to a factor of thirty-eight over the counts anybody has.

Rigid folding

The vertex is geared

A rigid four-crease vertex has one degree of freedom, which says that one number decides everything and not how. The how is a fixed ratio: the tangents of the half fold angles at two creases stay in constant proportion for the whole of the motion, and the proportion is a function of the sector angles and nothing else.

Flat-folding

Even is not enough

Every vertex theorem in the subject is a statement about one point, and the two-colouring of the panels looks like the exception. It is not — on a square of paper it is a parity at each vertex and nothing more. Cut a hole and the two come apart: a loop of paper with three creases has no interior vertices at all, satisfies every theorem there is, and cannot be folded flat.

Flat-folding

Where the lemma says nothing

The big-little-big lemma asks for a sector strictly smaller than both its neighbours, and the word doing the work is strictly. At a vertex whose two smallest sectors are equal the lemma has no opinion at all — and those are the vertices origami actually uses. The count of markings the conditions admit doubles, discontinuously, at exactly the angles everybody folds.

Flat-folding

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.

Flat-folding

How little the conditions decide

Local is not global is a statement about sufficiency: every vertex can pass and the sheet still fail. There is a sharper complaint available, and it is about strength. Fix one crease of a tessellation and propagate every condition the subject has to a fixed point: three creases out of a hundred and fifty-eight follow, and sixty-six vertices are still holding more than one answer.

Curves and material

A cut that removes no paper

Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.

Rigid folding

The family the Miura belongs to

Move one vertex of a Miura and the sheet has no rigid folded position at all — which leaves the obvious question unanswered. What else moves? A row of paper reflected in each of a fan of lines is flat-foldable for nothing at all, and whether it also folds rigidly turns out to be a condition on a table of cosines: it has to be a column of numbers times a row of numbers.

Rigid folding

Where an error goes

A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.

Designing a base

The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.

Tessellations

The corrugation that curves

A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.

What it costs to know

The map that is not a rectangle

Take one square out of a three-by-three map and the number of ways it folds does not go down by an eighth. It goes up — to 848 if the square came from a corner, and to 8,016 if it came from the middle. Two maps of eight squares in the same box, differing by nearly a factor of ten, and no function of the box tells them apart.

Flat-folding

Crimp it away and ask again

Four conditions decide whether a vertex folds flat, and they decide it exactly at a vertex whose sectors are all different sizes. Everywhere else they over-count: two markings of every tied four-crease vertex, twelve of the degree-six vertex this site prints nine of on one sheet. What decides the case is not a fifth condition but a procedure — fold the smallest sector away and ask the smaller vertex.

Flat-folding

One marking, many objects

A crease pattern with every mountain and valley written on it is spoken of as though it named a folded model. At four creases it does. At six it need not, and at the eight-crease vertex in the middle of the first base anybody folds, a single marking can be folded into four genuinely different objects — same creases, same letters, four answers.

Flat-folding

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.

Rigid folding

The condition that is not flat-foldability

Take away the assumption that one crease family runs straight through every vertex and ask what makes a quadrilateral mesh move. It is not flat-foldability. There is a one-parameter family of meshes, every one of them developable and flat-foldable at every vertex to machine precision, and exactly one member of it folds — the Miura. Slide a single vertex along the ray that keeps every condition exact and the sheet stops moving, first order in the displacement.

Rigid folding

One crease decides the sheet

Fix one crease of a flat-folding problem, propagate every condition the subject has, and three creases out of a hundred and fifty-eight follow. Fix one fold angle of a rigid one and every crease on the sheet follows, with a single consistent answer. The same experiment, two questions, opposite answers — and it is why a self-folding sheet needs one biased vertex rather than one per vertex.

Tessellations

The propagation that never had to work

The twist construction carries one equation per edge of its tiling and propagates the twist sizes outward from a seed. On every tiling anybody has drawn a twist on, every one of those equations is satisfied trivially — both ends of an edge read the same two numbers, because a regular polygon has one interior angle. The construction has been running and doing nothing, and the one tiling where it did something is the one whose tiles are not regular.

Tessellations

The dial that decides nothing

Turn a twist tessellation's angle from one fence to the other and every measurable thing about it changes: the smallest sector goes from 88 degrees to under one, the pleats swallow a quarter of the sheet and then almost none of it, the folded footprint changes by a third. The number of ways it can be creased does not change at all — sixteen, at every angle tested — because the lemma reads which sector is smallest and never how small.

Designing a base

The whole alphabet of a grid

Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.

What it costs to know

A short reason to say no

When a folding question comes back yes it brings an object anybody can check. When it comes back no it usually brings nothing but the assurance that a search looked everywhere. At one vertex that is false: a refusal comes with a witness one or two steps long, out of a search space of a hundred and twelve, and the witness is a vertex the crease pattern does not contain.

Flat-folding

A tie is not a decision

The crimp reduction decides a vertex by folding its smallest sector away, and where two sectors tie for smallest it has no forced move and must try each of them. That search is not rare — on the vertex at the centre of the first base anybody folds it happens for fourteen of the sixteen letterings — and it has never once changed the answer.

Flat-folding

The order decides the count

Ask how many mountain-and-valley letterings a vertex admits and the answer looks as though it should depend on the angles. It does not. Three of the four conditions never see an angle at all, and the fourth asks only which sector is smallest — so the count is a function of a combinatorial arrangement, and a walk round the cycle that never looks at a vertex reproduces it exactly.

Flat-folding

Which side is showing

Two earlier rungs asked what a folded object records about the pattern that made it, first from its outline and then from its complete layer order. Neither observation is one anybody can make. A photograph of duo paper carries the outline, the thickness and the colour showing at every point — and over the whole census the colour separates nothing at all.

Rigid folding

Solving every face at once

A quadrilateral mesh that folds rigidly has to close round every one of its faces, and the rung that built the general mesh could close one. Four of them at once resisted a descent that drove each free length to its own root, because closing a loop is a condition on several lengths together — and solving them jointly finds a sheet with no two vertices alike that folds, and a surface of them sixteen dimensions wide.

Rigid folding

Which crease to push

Deciding one fold angle settles every other one on a quadrilateral mesh, which is what makes a self-folding sheet buildable with a single actuator. It leaves a question that sounds like an afterthought: which crease. Driving each of a mesh's twenty-four in turn gives twenty-four different answers to how far an error in it travels — and on the sheet that repeats one vertex, it gives several answers to what shape the sheet takes.

Designing a base

The shapes the optimum has

Requiring a circle packing to be its own mirror image halves the number of coordinates a search has to find, so the same effort covers a much smaller space. Whether that helps depends on something the search cannot know in advance — whether the best packing was symmetric — and measured flap count by flap count the answer alternates without a pattern anybody could use.

What it costs to know

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.

Flat-folding

The vertices nobody checks

Every figure on this site is gated on four conditions evaluated at every interior vertex, and the word interior has been carrying the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, not one of them has ever been examined, and the condition that decides them has been available since the second phase of the collection.

Flat-folding

Walking between two foldings

The letterings a vertex folds in are always counted and never navigated. Counting says a generic degree-six vertex has eight of them; navigating says that changing any two creases turns any one into any other, and that changing two neighbouring creases does not — and that the vertices which come apart are the ones with no coincidences in them, which is the opposite of what every other measurement here would suggest.

Flat-folding

The creases that cannot move

One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.

Rigid folding

Closing is not building

A quadrilateral mesh solved so that every loop closes to within a millionth of a radian is a mesh whose fold angles are consistent. It is not necessarily an object. One of the six solved here drives a panel through another at every angle of its motion — there is no part of the fold at which it could be made of solid panels — and the pair that crosses is two steps apart in the sheet, where nothing evaluated at a vertex could see it.

Curves and material

A cut is a licence

What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.

What it costs to know

The cost is in the coincidences

How big an instance is, is what a hardness statement is about, and it is the weaker predictor of what deciding one costs. Hold the degree fixed and vary only how many of a vertex's sectors are equal: the work of deciding it rises by a factor of nearly three, against a factor of two for doubling the number of creases. The expensive instances are the ones a designer draws on a grid.

Flat-folding

The pieces without the list

The letterings a pattern folds in fall into pieces no folder can cross, and the count was found by writing every lettering down — which stops at eighteen creases. The Miura has thirty-eight, the Yoshimura eighty-six, and the number of pieces can be read off the drawing without listing anything: four million and two hundred and eighty-one million million.

Flat-folding

Nothing slides past anything

A marked strip has several legal stackings and this site has counted them at length. Nobody asked whether a folder holding one can reach another by lifting a flap over its neighbour: five hundred and sixty of six hundred and seventy-two stackings have no such move at all, and whether any exists depends on the parity of the segment count.

Flat-folding

A ring and a line

A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.

What it costs to know

The count counts labels

One, two, six, sixteen, fifty, a hundred and forty-four: the oldest sequence in the subject counts foldings of a strip of numbered stamps. A folded strip of blank paper has no first stamp and no top side, and neither of those operations ever leaves a folding alone — so the count of objects is 1, 2, 5, 14, 38, 120, and it is not the count over four.

What it costs to know

Where the machine catches up

The weakest machine in the subject folds every layer at once and is stopped by a strip with two creases in it. On a strip of equal stamps it is stopped by almost nothing: every one of the 288 folded states a six-stamp strip has is reachable by a sequence of all-layers folds, and on every unevenly creased strip tried it reaches none of them. At seven stamps the completeness ends, and finding out where it ended is what checking it past six was for.

Tessellations

Thirty-two rules, thirty-two pieces

The waterbomb tessellation's surviving repeating rules fold to one object — same panels, same places, same areas. Put them in the space of letterings the patch admits and they occupy thirty-two different pieces of a quarter of a million, so no two of them can be reached from one another without unfolding the sheet.

Folding nobody designed

Two ceilings

A DNA origami is limited by the length of one viral strand and by whether its helices can be visited once each in a single pass. Grown one step at a time, a square block runs into the first at a hundred helices and a plus runs into the second at five — so which limit a shape meets is decided by the shape and not by the chemistry.

Flat-folding

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.

Flat-folding

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.

Tessellations

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.

What it costs to know

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.

What it costs to know

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.

Rigid folding

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.

Axioms and construction

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.

Who found it, and when

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.

Flat-folding

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.

Tessellations

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.

Tessellations

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.

Rigid folding

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.

Designing a base

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

What it costs to know

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.

Curves and material

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.

Who found it, and when

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.

Flat-folding

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.

Flat-folding

The loop a vertex cannot close

A crease pattern's letters can contradict themselves, and the contradiction is never local. Enumerate every mountain-valley labelling of a single interior vertex at degree four, six and eight — a hundred and fifty pass every condition the subject has — and not one of them sends its panels round in a circle. The one labelling that would is refused by Maekawa, alone: Kawasaki holds on it and so does the big-little-big lemma.

Flat-folding

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.

Flat-folding

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.

Tessellations

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.

Tessellations

The rule that breaks the count

The waterbomb tessellation has five hundred and twelve repeating rules for its letters and thirty-two of them fold. A hundred and twenty of the other four hundred and eighty send four panels round in a circle — the shortest circle a crease pattern can have — and every single one of those hundred and twenty has broken Maekawa's count at the very vertex the circle goes round. The theorem that closes the shortest circle, caught doing it, a hundred and twenty times.

What it costs to know

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.

What it costs to know

A population that cannot fail

Thirty-three crease patterns are kept here to run the checkers over, and every one of them has letters that agree with themselves. That is not a property of the patterns. It is a property of how they were made: each came from a construction that returns a lettering, so a test looking for letters that contradict themselves has nothing to fire on. Reletter the same thirty-three and the failure is available at once — on one member, four of sixty redraws.

Rigid folding

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.

Who found it, and when

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.

Folding nobody designed

The taper decides nothing

A leaf's corrugation narrows toward its margin, and the taper is what the pattern is for. It has no effect whatever on how often the pattern's letters agree with themselves: four width profiles from perfectly even to strongly tapered give a hundred and seventy-four consistent letterings of two hundred, identically. What moves the number is the count of rows, and on that measure a leaf tracks a Miura rather than the corrugation it most resembles.

Flat-folding

Which condition does the refusing

A search for a lettering carries five conditions: developability, Kawasaki, Maekawa, the big-little-big lemma, and the demand that the arcs the letters force have no circle in them. Run it on five tessellation patches and count what makes it take a letter back. The four everybody checks refuse nothing at all. Every single backtrack is the fifth.

Flat-folding

Every move leaves the verdict

The only change a folder can make to a lettering without breaking it is to push one vertex through, flipping two creases at once. Try every such move on five tessellation patches, from two different letterings each: nineteen of two thousand nine hundred and sixty-four survive the conditions, and not one of the nineteen turns a lettering that agrees with itself into one that does not, or the other way about.

Tessellations

Sixty-four rules, sixteen fold

The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.

Tessellations

The loop is in the rule

Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.

Tessellations

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.

Tessellations

Where a rule can close a loop

Three corrugation families have repeating rules whose letters send four panels round in a circle, and one has none at all. The one that has none is the one whose vertices are all of degree six — and the reason is that a straight line through a point carries a single letter under any repeating rule, while a strict alternation round six creases needs the two halves of that line to differ.

Flat-folding

Where a sector crosses sixty

Turn the twist polygons of a tessellation patch a hundredth of a radian further and the pattern goes from having no mountain-valley labelling at all to having one immediately. Nothing about its graph changes across the transition — the same eighty-three panels, the same hundred and forty-two creases, the same four labellings at every one of its sixty vertices. What changes is which sector at a vertex is the smallest one.

What it costs to know

The test that never fires on a map

The cheapest refusal this collection has reads a crease list once and reports that no arrangement of the layers exists. Enumerate every labelling of every map from two panels to nine and it fires on four of the four hundred and fifty-four — all four on the largest map, none at all below it. On the oldest open problem in the subject, the cheap test has essentially nothing to say.

Who found it, and when

Half the recipe is decoration

Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.

Folding nobody designed

The leaf's rules are the Miura's

A plicate leaf packs into a corrugation that is broad in the middle and narrow at both ends. Enumerate every repeating mountain-valley rule it admits and the table is the Miura fold's table, rule for rule, number for number — and it stays that table when the leaf is redrawn with even columns, a violent taper, taller rows or a steeper zigzag. The plant's geometry cannot reach its own letters.

Flat-folding

The order that is its own mirror

Trying a mountain first and trying a valley first are two different searches, and on a hundred and forty-two crease patterns they cost the same number of steps — not on average, not nearly, but identically, pattern for pattern. The reason is a symmetry of every condition the subject has, and it is four lines long.

Flat-folding

A region with no lettering

One turn angle at which a tessellation patch has no consistent lettering was found by sweeping a dial. Sweeping two dials finds nine patches with none, across three tilings, filling a corner of the parameter space — and never touching the square tiling, whose sectors have no sixty degrees to cross.

Tessellations

The dial and the tiling that is not alike

Four of the five tilings a twist tessellation can be built on behave identically under every dial the construction has. The fifth has two kinds of vertex, and everything about it is different: it is the only one whose search has a tail, the only one whose shallow patches take minutes to draw, and the only one where a distance has to be solved rather than assumed.

Tessellations

A corrugation never backtracks

As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.

Rigid folding

A search with nothing to reorder

One search on a crease pattern costs eighty steps or fifteen thousand depending on the order it takes its decisions in. The other search on the same crease pattern costs 1,188,571 steps whatever order it is given — twelve permutations of the panels, twelve identical counts. The difference between them is one line of code that neither has and one has.

Designing a base

Ninety-nine in a hundred pass

A designer checks a box-pleated pattern the way every text teaches: vertex by vertex, counting mountains and valleys, watching the smallest sector. At sixteen divisions that check passes a hundred letterings in a hundred, and one of them folds. The check that separates them costs a single sweep over the crease list and is in no recipe anywhere.

Designing a base

The edge is what makes it hard

Grids, crumples, leaves, corrugations and fold-and-cut patterns all give up a consistent lettering at one step per panel with no wrong guess anywhere. The one family that does not is a tessellation clipped to a square, and what separates it from the others is not disorder, not size and not irregularity. It is having a rim.

Folding nobody designed

The plant's pattern is not a hard case

A hornbeam leaf packs into its bud by corrugating, and the pattern it uses gives up a consistent lettering at nine, twelve, fifteen, eighteen, twenty and twenty-four steps on nine, twelve, fifteen, eighteen, twenty and twenty-four panels. Nothing about the plant's problem is combinatorially difficult, and saying so is worth as much as finding a case that is.

What it costs to know

What the rim was doing

One rectangle of a twist tessellation, cut out of the plane in the ordinary way, gives up a consistent lettering in forty-eight steps. Join its opposite edges so that no crease is divided and the same drawing, at the same vertices, under the same conditions, takes fifty-six thousand seven hundred and seventy-two. The edge of the paper was never the difficulty. It was the slack.

Flat-folding

A loop that goes somewhere

Every crease says which of its two panels lies above the other, and a loop in those statements is a proof that the pattern has no flat folded state. On a sheet with no edge that sentence is false. The loops of a periodic pattern carry a lattice step each, and a loop that ends one cell to the right is not a contradiction — it is a stack of paper with no bottom layer.

Flat-folding

The lettering that was proved impossible

A search closed its whole tree on a glued square tessellation and reported that no mountain-and-valley assignment of it is consistent. Written onto ordinary patches of one, four and nine periods and handed to the four vertex theorems and a folded sheet rebuilt from scratch, the assignment it says cannot exist passes every check, on four tilings, up to fifteen hundred creases.

Flat-folding

The rim is four letters a cell

Cut a rectangle out of a tessellation and it asks exactly the vertices the tessellation asks, exactly the same questions. What it adds is four free letters for every period of edge — the creases the cut divides, which become two independently answerable creases instead of one. Eight letters on a two-period square, sixteen on a four-period one, and nothing else about the two objects differs at all.

Flat-folding

The bottom layer is at the rim

A hundred and sixty-nine panels of folded tessellation, and three of them have nothing underneath. All three touch the paper's edge, and the same is true on every tiling at every size measured. Which panel is at the bottom of a stack turns out to be a fact about where the sheet was cut rather than about the pattern, and the pattern itself has no bottom at all.

Flat-folding

One step per panel is a table size

Four families of crease pattern search at exactly one step per panel — a grid at nine sizes, a leaf, a Miura, six crumples — and it was read as a law about patterns that fill their own sheet. It is a number: the conditions at each of their vertices admit eight labellings. Where the conditions admit four, the cost is half. Where they admit thirty, it moves again, and the same pattern at two proportions demonstrates it with everything else held still.

Flat-folding

Six creases and the same straight line

The one family here whose vertices are degree six was said to break the arithmetic that every other family obeys, on the strength of a single pattern. Built as a family — six sizes from twenty-one panels to a hundred and nineteen — the Yoshimura is exactly as linear as a grid, with no decision ever withdrawn. What degree changes is the constant, and it changes it in both directions depending on one angle.

Flat-folding

A knife edge nine decimals wide

Draw the Yoshimura with its rows 1.7320508 half-columns tall and each vertex admits thirty labellings and the pattern costs fifty-seven steps. Draw it at 1.7320509 and each admits eight and it costs nineteen. The number between them is √3, which is the proportion everybody draws — and below it the sectors are unequal and the lemma is still silent, because the small ones sit next to each other.

What it costs to know

Pruning on proofs alone

A search that discards a branch it cannot prove wrong is not a search. Deciding whether a periodic pattern's layer relations really contradict themselves is far dearer than the disc's one-pass test, so the cheap test is asked first — it is sufficient, so it settles almost everything — and the expensive one runs only on what the cheap one rejects. Five of nine steps on a small cell, fifty thousand of fifty-seven on a large one.

What it costs to know

The cost of proving something false

A search closing its whole tree is the strongest result this collection can produce, and on a glued tessellation it produces one that is wrong. What it costs to reach is three steps at one period, thirty-five at four, three thousand four hundred and fifty-five at nine, and more than two hundred thousand at sixteen — growing far faster than the cost of finding the lettering it says does not exist.

Tessellations

The most decided vertex here

Sixteen ways to letter four creases; Maekawa allows eight; the big-little-big lemma allows four. A twist polygon's corner is one of the few vertices in this collection where the second cut applies, so it keeps four labellings where a grid, a leaf, a Miura and a crumple all keep eight — and the family the collection long called difficult turns out to be the one whose conditions decide the most.

Designing a base

The edge was not what made it hard

Five families of pattern searched at one step per panel and a tessellation patch did not, and the property left standing after four alternatives were killed was having a rim. Measured under a fixed letter order the patches cost between a half and two-thirds of a step per panel, at every tiling and every size — below the line rather than above it, and the rim is why.

Designing a base

The designer's grid is the dearest thing here

Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.

Rigid folding

An order with no least element

Enumerating every way a folded pattern can be stacked works by building upward from a panel with nothing below it. The smallest square twist patch has exactly one such stacking and takes eleven thousand steps to find it. The pattern that patch was cut from has no panel with nothing below it at all, so the enumeration has nothing to start from — and the sheet is perfectly well stacked anyway.

Who found it, and when

A test imported without its hypothesis

The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.

Folding nobody designed

Nothing grown was cut out of anything

A leaf's corrugation costs twelve steps on twelve panels, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four — exactly one per panel at every geometry, which is the most any pattern here costs. A tessellation patch costs half that, and the reason is that somebody cut it out of something. A leaf's creases stop at the margin because the plant stopped there.

Flat-folding

The seam carries a sign

A loop of paper folds flat when it has an even number of creases round it. A Möbius band folds flat when it has an odd number. The drawing is the same in both cases, the creases are the same creases, and what changed is a factor of minus one contributed by the sheet rather than by anything drawn on it.

Flat-folding

The band that needs an odd number

A Möbius band is the first sheet in this collection with one side, and the consequence is sharper than a reversed parity. Mountain and valley are defined relative to a side, so on a sheet with no consistent side a crease has no letter — and Maekawa's condition survives the loss while the assignment it is about does not.

Flat-folding

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.

Flat-folding

Parity is not enough

A Möbius band needs an odd number of creases round it. Give it three, square across the strip, and it does not fold — nor does five, nor seven, nor any odd number at all. The counting argument is necessary and it is not close to sufficient, and the thing it cannot see is which way the creases point.

Flat-folding

An alternating sum of angles

Kawasaki's condition says the sectors round a vertex alternate to a straight angle. A glued band has no vertices and obeys a condition of exactly the same shape: the crease angles have to alternate to a multiple of a straight angle. Two different quantities, two different sheets, one arithmetic — and in both cases what is being said is that a product of reflections came back the right way.

Flat-folding

Half a rim

A rectangle of tessellation cut out of the plane has four edges; glued into a torus it has none. Gluing one pair and leaving the other gives the middle of the scale — the same drawing, the same vertices, the same conditions asked of them, and exactly half the rim. What the rim costs turns out to be measurable per edge rather than only at the ends.

Flat-folding

The rim adds up

What one glued pair of a cell's edges saves in free letters is what the other pair saves, and gluing both saves the sum. That is a rate rather than an observation, it is the form of the claim two objects could never support, and it is what makes 'the rim costs four letters a cell' a statement about tessellations rather than about one drawing.

Flat-folding

Euler counts the gluing

Vertices minus creases plus panels comes to one on a rectangle of paper and nought on any gluing of it. That is the cheapest check that an identification did what it says, it costs three counts already being made, and it is what found a crease running exactly through the corner of a cell — a case the corner search could not see and no other check would have noticed.

Flat-folding

A grid that will not close

Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.

Tessellations

One node per panel, with the rim gone

A rectangle of repeating pattern cut out of the plane costs exactly one node of search per panel, on every family and at every size. Take the rim away and the total falls and the cost per panel rises, because the letters that were removed were the ones that could not be wrong.

Tessellations

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.

Tessellations

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.

What it costs to know

Half the slack

Gluing one pair of a cell's edges removes half the free letters and costs almost nothing. Gluing the second pair removes the other half and costs three orders of magnitude. The letters go linearly and the search does not, and the reason is that the last free letter is worth more than all the others.

What it costs to know

Which pair is glued

A cell's two cylinders have the same Euler number, the same amount of rim and the same name. On a symmetric drawing they have identical counts of letters, panels and vertices — and searching them costs twenty-four nodes one way and eighty-five the other. Half the rim is a description of the topology and not of the object.

Curves and material

A cut is surgery

Two cuts that look identical on the paper do completely different things to the sheet. A slit run inward from the rim changes nothing at all; a closed cut in the middle removes a disc and leaves a sheet carrying a condition it did not have before. What separates them is not the length of the cut or how much paper it removes.

Curves and material

Two holes are two conditions

One hole in a sheet of paper gives one loop that cannot be shrunk and one parity to satisfy. Two holes give two, and they are independent: an arrangement of creases can satisfy the condition round one hole and fail the condition round the other, and the sheet refuses on the strength of the one it failed.

What it costs to know

A proof in no nodes at all

A parity refuses a sheet before any search begins. It costs one addition, it is certain, and it says nothing about why — while a search that exhausts on the same sheet costs thousands of nodes and produces a proof of the same fact. Two proofs of one thing, and the cheap one is available only where somebody has noticed the invariant.

What it costs to know

The cost of asking the wrong sheet

A test written for a sheet with an edge, run on a sheet without one, does not fail. It exhausts — proving, at three, thirty-five and three thousand four hundred and fifty-five nodes, that no lettering exists — and the letterings it proved impossible fold, on the collection's own machinery, at every size they were tried at.

What it costs to know

A map with no edges

Counting the ways a rectangular map folds is the oldest open problem in the subject, and every version of it assumes the map has an edge. Join the map's opposite edges and the question changes shape: half the sizes have no folded state at all, and the ones that do have no bottom layer to count from.

Tessellations

The seam that is not a symmetry

Gluing a cell's edges looks like a symmetry of the drawing and is not. It is an instruction about which points of the paper are the same point, the drawing has to agree with it along the whole of a glued edge, and a rectangle that is not a period of the pattern does not glue at all — which turns out to be the only real restriction on which cylinders exist.

Flat-folding

A bottom layer on half a rim

The bottom of a folded stack lives at the paper's edge, which is why a sheet with no edge has an order with no least element. A cylinder has half a rim, so it has a bottom — and the count of panels that could be it falls with the rim, which makes the claim a measurement rather than a boundary case.

Flat-folding

The arc that arrived twice

Which of two panels a crease calls its near one is decided by the order a face walk happened to number them, and the mirrored record is the same relation. Except on one sheet, where it is not — and that sheet turned out to be the one whose folded state comes back the other way up, which is how a duplicate in a graph became a diagnosis.

Flat-folding

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.

Designing a base

The symmetry a gluing adds

A patch of a tessellation has whatever symmetry its outline allows — a few reflections, a rotation or two. Glue its edges and it acquires translations, and a lettering of the glued sheet has to be invariant under them. That is a much stronger requirement than a lettering of the patch, and it is why one answer covers every patch at once.

Curves and material

A crease with no vertex to belong to

Crease density is measured as length of line per area of paper, and everything else about a crease is measured at the vertex it runs into. A band of paper has creases that run from one edge to the other and meet nothing, so it has density and no vertices at all — and it still refuses to fold.

Rigid folding

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.

Rigid folding

A mechanism that closes on itself

A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.

Who found it, and when

A theorem with an unstated hypothesis

Maekawa's and Kawasaki's conditions are quoted everywhere without saying which sheet they are about, and they do not need to be — they are conditions at a point and every point is the same. The two-colouring is quoted the same way and it is not a condition at a point, and the omission there is not harmless.

Folding nobody designed

Nothing grown has a seam

A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.

Folding nobody designed

The census returns one

The rung below this one asked for a census: every pattern reaching a stated packing ratio while opening from a single input, with its crease density. The census is makeable for the corrugations and it comes back with one member. A corrugation piles its panels over a footprint as wide as its longest panel, so its ratio is the total length divided by that longest one — and that equals the panel count only when every panel is the same.

Folding nobody designed

The surface has to be supplied

The curve that turns over does so because the sheet's own thickness fills the box it is folding into. A surface in a body has to be reached as well as fitted, and the channel that reaches it takes depth out of the same box on exactly the same terms — so the best fold count and the surface it delivers both fall by the ratio of the sheet's thickness to the sheet and its supply together.

Folding nobody designed

Two surfaces in one box

A body folds several surfaces into one volume and each of them does a different job. Dividing the depth between them looks like a fair split costing nothing overall, and it is not: the area a single surface can reach goes as the square of the depth it has, so m surfaces sharing a depth reach a total of exactly one m-th of what one of them would have reached alone.

Folding nobody designed

The container picks the member

Two large waves and eight small ones are the same metric, and geometry has nothing to say about which. Bending has nothing to say either — it rises at every step of the family, so a least-bending rule always answers the fewest waves and never anything else. What decides is the container: amplitude times wave count is constant across the family, so a ceiling on the height is a floor on the number, exactly inversely.

Folding nobody designed

The test measures the rim

Flattening the specimen is the right test and the measurement anybody actually makes on a flattened specimen is a boundary one — how far the margin overruns its chord. Total curvature is a boundary quantity too: it equals minus two pi R times the growth profile's slope at the rim, and nothing else about the interior survives into it. So a sheet can be curved everywhere and integrate to nothing, and the test reports it flat.

Who found it, and when

Which cranes can stay joined

The 1797 book slits a square into a grid and leaves the cranes attached at the interior lattice points, each of which holds four of them at once. Of the sixteen ways to choose which of a three-by-three's four points to leave joined, exactly one leaves the piece in a single object — and it is the one that uses all four. The cutting is very nearly forced rather than chosen.

Who found it, and when

One lost source and the story changes

Five of the record's fifteen claims rest on exactly one surviving document. Take those away and the field's most-quoted statistic — how far ahead of its evidence a popular date runs — falls from three hundred and fifty-seven years to a hundred and ninety-three, while the median hardly moves at all. The effect is not a tendency spread through the record; it is two documents, and both of them are single-witness.

Folding nobody designed

A nest pays four a level

A corrugation folded inside the panels of another looks like the arrangement that multiplies surface rather than dividing it. It multiplies the factors and divides the depths, and the depths cancel: a packed level can hold at most its depth over four times what it folds, the thing it folds is as thick as the depth the level below was given, and so a nest of L levels reaches at most the box over 4ᴸ sheet thicknesses. One level with the whole box beats any nest of two by exactly four.

Folding nobody designed

A corrugation has one resting state

A folded wing held short of shut stores energy in its hinges, and a wing that could stay both open and folded with nothing holding it would need that energy to have two bottoms. A corrugation cannot provide them: every crease in it folds by one angle, so the energy of any set of crease springs is a parabola in that angle and has exactly one resting state, however much the springs disagree. A single degree-four vertex has two branches through the flat state, and the same springs give it two resting states on almost every setting tried.

Folding nobody designed

A row the route cannot leave

Every rectangular block of helices up to ten by eight routes on the square lattice. On the honeycomb, the lattice a double helix's pitch prefers, twenty of the eighty do not — and every block odd in both directions fails for a reason visible along one row: every second helix on the top row has no neighbour off it, the two corners have one neighbour each, and a route forced through them runs the length of the row and ends. The colour count passes all of them, and a degree count along a single row refuses them.

Folding nobody designed

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.

Folding nobody designed

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.

Who found it, and when

Nearly every cutting fails at one crane

Six by six connected cranes have twenty-five joins and thirty-three million ways to keep some of them, and an exhaustion over all of them takes a fifth of a second. Of the 33,412,811 that fail, 99.64 per cent fail at a single crane — one left holding none of the joins at its corners — which a maker can check by looking at each crane in turn. The arrangements that pass that check hold together less often as the grid grows: all of them at three by three, 54 per cent at six by six.

Who found it, and when

A recipe needs degree four

The Miura's letters are taught as a recipe of same-and-differ clauses, and the recipe is exact: the sixteen repeating rules that fold are precisely the rules two clauses allow. The Yoshimura's twenty-six cannot have such a recipe, because twenty-six is not a power of two. The waterbomb's thirty-two is a power of two and still has none — its clauses allow sixty-four rules and half of them fail. The difference is one vertex: at degree four the counting theorem leaves a parity, and at degree six it leaves 'not all alike', which no clause of that kind can say.

Who found it, and when

The border is where the cranes come apart

Counted one join at a time rather than one subset at a time, the slit grid of connected cranes runs to twelve by twelve, where there are 2¹²¹ ways to keep some of the joins. Among the cuttings that hold every crane by something, the share that also hold together keeps falling — 54.3 per cent at six by six, 35.1 at twelve — and from eight by eight on it falls by the same factor at every size. A constant factor is the signature of the border: at six by six, 99.5 per cent of the sound cuttings that come apart do so through a stray piece touching the outermost ring of joins.

Folding nobody designed

The wall is the flat sheet

A sprung degree-four vertex usually has two resting states, one on each branch of its motion, and the branches meet in one place a sheet can pass through: the flat state. So the only road from one resting state to the other crosses the flat sheet, and on every setting of the springs tried on two vertices the flat sheet is the highest point of that road. Its energy is each spring's stiffness times its rest angle squared, summed, which does not contain the vertex's sector angles at all — the same springs put on four different vertices give a wall of exactly the same height. The geometry decides only how far below the wall each state sits, and the shallower one sits a median of five per cent below it.

Who found it, and when

Two sentences for the Yoshimura

No recipe made only of same-and-differ clauses picks out the Yoshimura pattern's twenty-six folding rules, because its vertices have six creases. A recipe allowed one other kind of sentence does, and it is short: an even number of the four zigzag classes are mountains, and no course carries the letter all four zigzags share. That is three clauses, one parity and a prohibition for each of the pattern's two kinds of vertex, and it allows exactly the twenty-six. The waterbomb tessellation, with four kinds of six-crease vertex, needs three parities and only two prohibitions, because its parities do half the prohibiting.

Folding nobody designed

Every cheap test misses a shape

A strand routed through a bundle of helices has to visit each once, and whether a shape allows that is hard to decide — so the cheap tests that refuse shapes are necessary and never sufficient, and for every set of them there is a smallest shape they pass and no route reaches. Listing every connected shape and searching the ones the tests let through finds it: nine helices on the square lattice for the colour count, the ends and the cuts, eleven once the steps a route's ends force are added, and still eleven once the ends' colours are checked. On the honeycomb the same three stages give twelve, fifteen and sixteen. Each test pushes the smallest unroutable shape out or leaves it where it is; none removes it.

Folding nobody designed

A chain of vertices switches all at once

One sprung degree-four vertex rests in two states, one on each branch of its motion, and switches between them only by passing through the flat sheet. Chain vertices together by sharing a crease between each and the next, and a branch can be chosen at every vertex: two, four, eight and sixteen combinations for chains of one to four. The median spring setting rests once on every combination. And every combination's curve of configurations passes through the same single point — the whole chain flat at once — and meets no other anywhere else, so every switch, even of one vertex's branch, takes the whole chain back to flat. The wall that switch climbs is every spring's flat energy added up, growing by a crease's worth for every crease, and a chain's second state sits several times further below it than a single vertex's does.

Folding nobody designed

Springs that disagree do not offer a choice

A corrugation of hinges that remember different angles was expected to have more than one position in which nothing pushes. It has exactly one, at the stiffness-weighted mean of what they remember, because a sum of parabolas in one variable is a parabola. The second resting place comes from somewhere else entirely — the mirror pattern — and the flat sheet is the barrier between them, which is also why a creased sheet cannot be pulled flat at all.

Folding nobody designed

Standing up beats lying down by eight

A level that fills its clearance with plies lying flat makes every ply share the clearance, and the sharing costs a factor of four. A level that fills the same clearance with walls standing on the base gives every member the whole height and charges them only for footing. The ratio is exactly eight, at every clearance and every thickness — and it is the difference between a cost charged against depth and a cost charged against the space beside it.

Designing a base

What the condition does not decide

A tree of seven leaves imposes twenty-one separations and eight of them bind. The rest are slack, the eight pin six of the seven leaves against the sheet's own edges, and the seventh can be moved half a per cent of the sheet for nothing. The requirement that looks quadratic is doing linear work, and what it leaves undecided is the part a designer is actually choosing.

Curves and material

Where a ring of divisions belongs

A pattern that divides the circle everywhere as finely as its rim requires is over-divided for most of its radius, because the excess grows from nothing. Putting a ring of new divisions in wherever the residual would otherwise pass what the material takes gives seven rings on a hemisphere at five per cent of stretch, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out — and the first of those sits where a completely different criterion put its first tuck start.

Rigid folding

Only four creases decide a Miura

Driving one crease of a rigid quadrilateral mesh settles every other one — except that on the pattern everybody builds it often does not. Enumerated properly, four of a four-by-four Miura's twenty-four creases leave exactly one folded state and the other twenty leave two, four or eight. A mesh whose vertices all differ leaves one from every crease. The ambiguity is not a property of quadrilateral meshes; it belongs to the symmetry.

Folding nobody designed

A domain too short to be unique

A staple holds the scaffold by pairing with a stretch of it, and two arguments decide how long that stretch has to be. One is combinatorics — a stretch of seven bases has about four hundred other places in a 7,249-base strand it would also match. The other is thermodynamics, and it is the one that binds: a duplex that is unique at eleven base pairs still comes apart at the temperature the design is held at, and staying paired takes seventeen. Rounded up to the crossover period, that is three periods on both lattices, and the lattice the helix prefers is the one whose three-period domain leaves some of its staples unattached.

Folding nobody designed

A test that only knows one lattice

The cheapest argument that refuses the smallest shape no cheap test could refuse was read off that shape: cut at one helix, find the piece with no end in it, and count the colours of the stretch the route is then forced to cross. Added to the census it refuses every one of the square lattice's thirty-two eleven-helix survivors and pushes the smallest survivor to twelve, where twelve placements of two shapes survive out of half a million. On the honeycomb it refuses none of the six at sixteen. A test inherits the lattice of the witness it was read off, and the staircase is two staircases.

What it costs to know

Deciding is not making

Four earlier essays here ask which machines can flatten a strip at all, and the answer sorts them into a lattice with one column full and three with holes in it. Asked instead what each machine can produce, the three sort completely differently: the machine that may choose its block reaches every folded state of every strip tried, the machine that takes one layer reaches exactly four whatever the strip is and however long, and the machine that takes the whole pile is the only one whose answer depends on the spacing at all.

Designing a base

A price holds until the arrangement moves

Every edge of a subject's tree has a price — the scale lost per unit of extra length — and the obvious use of a price list is to spend a fixed total of limb where it is cheapest. Done a tenth of a unit at a time, re-pricing at every step, it works and then stops: the bird's scale rises 6.3 per cent in four steps and no further. But the prices do not hold while it happens. The bird's free tail stops being free after the first tenth, and its legs nearly treble in price without being touched. The lizard's prices hold for four steps, because its arrangement keeps the same three pairs at their limit for four steps. A price is a statement about which pairs are at or near their limit, and it lasts as long as they stay there.

Designing a base

Rounding in the cheap direction

A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.

Who found it, and when

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.

Rigid folding

The deciding set does not move

A driven Miura leaves several folded states from most of its creases and exactly one from a few, and those few are where an actuator belongs. It was reported that the few change along the motion — four of twenty-four at 0.6 radians, fourteen at 0.8 — and that a five-by-five sheet had a crease leaving fifteen states where every other count was a power of two. Mapped at twenty angles from 0.1 to 3.0 radians on three sizes of sheet, neither survives. Every crease leaves the same number of states at every angle, every number is a power of two, and the same creases decide the sheet throughout. The changes were the vertex solver losing one of a vertex's two configurations on 138 of 8,640 solves, and the configurations it lost can be carried exactly from an angle where it finds both.

Folding nobody designed

Holding a fold moves the force

A creased hinge held at an angle slowly comes to remember that angle, so a leaf or a wing packed in a bud for a season is gradually holding itself and the bud has less to do. The force is not used up in the process. The torque the container must supply falls as e^(−T⁄τ), the torque later needed to open the structure rises by exactly the same amount, and the two sum to the same total at every moment of the holding. Held for three relaxation times, a corrugation creased to 0.8 radians and packed to 2.4 needs 4 per cent of the total from its container and 96 per cent from whatever opens it — and the barrier to its mirror image has grown more than eightfold. A packing that lasts buys independence from its container with a harder unfolding.

Folding nobody designed

A loop takes choices away

A chain of four sprung degree-four vertices has sixteen combinations of branches, each a resting state, and switches between them only through the flat sheet. Close the chain into a loop round one panel and the combinations must agree when the fold angles come back round. On a face whose four vertices all differ, one combination survives; on a Miura face, four. The count is the same at every angle the face is driven to, and two surviving assignments at the same driven angle are never closer than one and a half times that angle — so they separate as the face folds and meet only when it is flat. A loop does not create the junction a region would need to switch on its own. It removes choices and leaves the switch as global as before.

Tessellations

Closing the loops is not folding

The twist construction propagates one equation along every edge of a tiling, and it can only work where the equations agree round every loop. Asked which irregular tilings pass, a linear map gives a clean answer: the square grid, the triangular grid and the honeycomb pass under every shear and stretch tried, because each edge has a half-turn symmetry that makes its equation exactly one at both ends, and a half-turn survives any linear map. The rhombille passes only as drawn. But passing is not folding. On every one of those images — including the ones whose loops close exactly — the construction produces a pattern that fails the angle condition at every turn tried. The loops were a necessary condition all along, and the construction needs something the tilings' images do not give it.

Tessellations

One number where the corners wanted four

The twist construction gives a vertex a single side distance, and every account of these patterns does the same — it is what rotate-and-shrink means. The conditions never asked for it. Written out, the corner condition is one linear equation per pleat crease in the distances taken one per edge, so a degree-four vertex carries four unknowns against two independent equations. Given them back, the twelve sheared and stretched tilings that refused to fold all fold.

Tessellations

Every twist writes an equilibrium

Divide each side of a twist polygon by the length of the edge it faces. The polygon closing says those numbers, weighted onto the edges, balance at the vertex; the pleat matching says the two ends of an edge agree on the number. Together they are a positive equilibrium stress — the thing a tiling has when it is the plan of a spider web — and the construction has been writing one at every vertex without being asked for it.

Curves and material

Only one side can run out

A curved fold has two surfaces and every reach ever computed here has been one of them. The closed form's denominator is the crease's curvature plus the rate the ruling angle turns at, and crossing to the other surface negates both — so at any point of any crease at most one of the two surfaces can have its rulings converge. A crease that never changes the way it bends therefore has a surface with no such boundary at all, anywhere along it.

What it costs to know

Each drawing has its own threshold

Gluing a cell's edges was measured once, at one size, and found to cost three orders of magnitude — which cannot tell a threshold from a slope, nor say whether a cut sheet has one further out. Swept from one period to five on four tilings, every sheet starts at about a third of a node per free letter and every drawing leaves that behaviour at a size of its own: four periods on the square grid, three on the honeycomb, two on the triangular grid and two on the rhombille, where even the cut sheet crosses.

What it costs to know

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.

Curves and material

An angle that turns faster than the crease

Which of a curved fold's two surfaces runs out is decided by a sum of two rates — how fast the crease bends and how fast the ruling angle turns — and every measurement so far has set the second to zero. Let it turn and it carries the sign on its own: past a rate of exactly one, a crease of unchanging curvature bounds both of its surfaces, which no constant angle on that crease can do. Below that rate the turning costs reach without changing anything else.

Who found it, and when

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.

What it costs to know

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.