Every essay — page 3
Axioms and construction · Flat-folding · Designing a base · Tessellations · Rigid folding · Curves and material · What it costs to know · Who found it, and when · Folding nobody designed
What it costs to know
Deciding, counting, listing and optimising are four different questions about the same sheet, and folding answers them at four wildly different prices.
The oldest open problem
In how many ways can a map be folded? The question needs no notation to state, the answer is a small integer for small maps, and after sixty years there is still no formula — only a list of numbers, each one found by searching every possibility.
The fold a machine can make
A theorem that says a folded state exists says nothing about getting there. A machine that folds every layer at once is stopped by a strip with two creases in it — one that folds flat perfectly well, and that a pair of hands folds in about four seconds.
The patient machine is the weak one
A machine that folds one layer at a time sounds like a machine with more freedom, not less. It has less, and the reason is the most ordinary fact about paper there is: it is joined, so whatever a machine declines to hold it also cannot move.
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.
The machine that may choose
Three restricted machines lose patterns that fold perfectly well. Give one of them a choice — any block of layers, top or bottom — and the loss vanishes: over a hundred and seventeen spacings, every flat folding of every strip became reachable. Being forced was the whole problem.
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.
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.
The answer is bigger than the question
A twelve-square strip of stamps is twelve numbers of input and 146,376 objects of output. No algorithm writes that faster than it can be written, so 'efficient' has to be measured against the answer rather than against the question — and in folding that is the normal case.
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 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.
What universality costs
The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.
Getting close instead of getting it right
When the best answer is out of reach the question stops being what it is and becomes how much is lost. For packing discs into a square the loss is measurable: a seeded search in this repository comes within a fifth of a percent of the best radius anybody has proved, and proves nothing.
A no costs more than a yes
When a folding question comes back yes, it comes back with an object: a labelling, a stacking, a folded state that anybody can check in one pass. When it comes back no, it comes back with nothing but the assurance that a search looked everywhere — and that assurance is the first thing to break.
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.
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.
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.
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.
Four ways to draw a pattern
Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.
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.
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.
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.
The map counted from the layers
The classical map-folding counts are computed from a rule that never places a panel: work out which edge of the folded square each fold wraps around, and refuse the orderings that interleave two folds at one edge. Place the panels instead and order them by the general non-crossing rules, and the same numbers come out — 2, 6, 16, 50, 144, 8, 60, 320, 1368 — on nine sizes, by machinery that shares no line of code with the first.
The patterns a checker is tested on
This site keeps four populations of crease patterns and runs its checkers over them, which is what makes a claim about typical instances measurable rather than rhetorical. Asked whether the members actually fold, the populations answer: thirteen of thirty-three do, six place and cannot be ordered, five cannot be placed at all, and nine are past what the search will finish.
The 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.
Drawn by the same hand
Two straight segments dropped on a square cross about 23% of the time; four of them cross 74% of the time; twelve cross with certainty, about fifteen times over. Every crease pattern in this collection's four test populations has none — not because the checkers were catching them, but because the same rules that drew the patterns were incapable of producing one, and nothing looked until a construction finally did.
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.
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.
Stopping is cheaper than finishing
A search whose cost varies by a factor of two hundred with nothing but the order of its guesses should not be waited out. Give up after a hundred steps, reseed and start again, and the whole job costs five hundred and twelve steps in expectation; run each attempt to twenty thousand and it costs sixteen thousand two hundred and ninety-one. Patience is thirty-two times more expensive than impatience.
Four populations with nothing to separate
This collection keeps four standing populations of crease patterns to test its machinery against. Twenty-eight patterns, sampled forty times each for a lettering that agrees with itself and then searched for one — and on every single member the two methods return the same verdict in the same breath. The patterns that separate them are in none of the four, and the reason they are not is what the populations are for.
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.
Which choice the cost lives in
A backtracking search takes two decisions at every step — which thing to decide, and what to decide about it. The literature is almost entirely about the first. On these crease patterns the whole of the cost was in the second, and the structural improvement everybody reaches for first makes matters worse on fifty-two patterns out of eighty-seven.
The order that proves nothing exists
Twelve crease patterns with no consistent lettering at all. Proving it takes fifteen steps under one rule and half a million under another — and on three of the twelve the two rules swap places, so neither is the good one. The cost of a negative is two to the power of how many free choices sit above the contradiction.
A population nobody chose
Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon the collection had spent so long measuring belongs to the one tiling the grid leaves out.
Restarting what cannot be restarted
Stopping a search early and starting it again with a fresh seed costs five hundred and twelve steps in expectation against sixteen thousand for patience. Every number in that is right. The distribution it is right about was made by the search's own coin, and taking the coin out costs eighty — with nothing left to reseed.
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.
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.
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.
Where you cut hardly matters
Slide the same rectangle across one whole period of the same tessellation and every position gives a different patch: different creases divided, different half-panels round the edge, panel counts from forty-nine to sixty-one. The cost of lettering them runs from twenty-five steps to thirty-three. Whether a cut is made changes the answer by three orders of magnitude; where it falls changes it by a third.
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.
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.
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.
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.
One population, four sheets
A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.
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.
The tube a map makes
Join one pair of a map's edges and the result is a tube — a real object, foldable in the hand, and neither the strip's problem nor the torus's. It has one loop that cannot be shrunk instead of two, it keeps its bottom layer because it keeps half its rim, and half its sizes are refused by a parity the flat map does not have.
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.
The easiest strip needs the deepest reach
The patient machine and the machine that may choose are the two ends of one number: how many layers of the pile a machine is allowed to hold. At one it reaches four states whatever the strip; at the pile's full depth it reaches everything. In between it is a machine nobody has defined, and measuring where completeness arrives inverts these essays' own ordering — the evenly creased strip, which the machine that takes everything folds perfectly, needs the deepest reach of all, and one uneven strip is complete at two.
A shallow machine pays in states, not folds
A machine allowed to take only a few layers of the pile at a time reaches fewer folded states, and the natural fear is that it also reaches the ones it does by much longer sequences. Walked breadth first, so that every state's shortest sequence is found, it does not. On unevenly creased strips every state takes exactly one fold per crease at every depth, because no two creases ever lie on one line. On strips of equal stamps a shallower machine needs one fold more for a minority of states and two more for eight of the 924 states at seven stamps — and never more than the crease count, which no machine can exceed.
Fourteen states are one pile
A machine that folds every layer at once reaches every folded state of a strip of six equal stamps and misses fourteen piles at seven. The fourteen are not fourteen things. Taking a pile's bottom stamp and putting it on top maps foldings to foldings, so the 462 piles of seven stamps fall into 33 classes of exactly fourteen, and the missed piles are one whole class: the pile 0 6 1 2 3 4 5 — an accordion of five stamps with the last stamp wrapped round it and slid into the fold that holds the first — seen from each of its seven stamps. At eight stamps the machine misses 64 piles, and they are exactly the piles that leave that one when an end stamp is removed.
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.
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.
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.
Who found it, and when
Almost everything repeated about where folding comes from is dated too early, attributed to the wrong person, or both. This field checks the claims against the record — and is honest that a record is not a proof.
Nothing here is as old as it sounds
Paper folding is described everywhere as an ancient art. The oldest surviving book of it was printed in 1797, the oldest reference to folding for amusement is from 1680, and the median claim in this subject is dated two centuries before anything that attests it.
The oldest book cuts the paper
The Hiden Senbazuru Orikata of 1797 is the earliest surviving book of recreational paper folding, and its famous connected cranes are made by slitting one sheet into a grid. The founding rule of the modern subject is younger than the tradition it claims to describe.
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.
Fifty years in the wrong language
Margherita Beloch showed in 1936 that one fold solves a general cubic. The result was correct, published, and in a mathematics journal — and the subject that needed it did not find it until 1991. The cost of a paper nobody reads is measurable, and it is most of a century.
A record is not a proof
Every other claim here can be re-derived from the figure that makes it, and a wrong one shows. A date cannot: it is checked once, by hand, against a record that is itself a survivor. This field is the one most likely to be wrong and least likely to be caught, and saying so is the only defence it has.
The kindergarten was a geometry class
Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.
A schoolteacher's theorem
Kazuo Haga folded a corner of a square to the midpoint of the far side and found exact thirds. The construction needs one fold, no measurement and no compass, the numbers that come out are exactly 3/8, 7/8 and 2/3, and it was found by a biology teacher looking for something to do with a classroom.
What a dashed line can say
Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.
Publishing the pattern instead of the sequence
A diagram sequence is one picture per step and a crease pattern is one picture. When designers began releasing patterns rather than diagrams, the cost of publishing a model fell by two orders of magnitude and the difficulty moved onto the reader — which is what made the complex era possible and what made most of it unfoldable.
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.
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.
Two traditions and a merge
Ceremonial wrapping, recreational folding and the kindergarten syllabus are three separate lineages with three separate purposes, and they were independent until the late nineteenth century. Told as one continuous tradition, the oldest date in any of them becomes the age of all three.
The paper had to arrive first
A model with sixty-four layers at its thickest point, folded in ordinary copier paper, is six and a half millimetres of stack. The layer count a design can reach is fixed by the substrate, not by the folder — so the elaborate tradition is downstream of a manufacturing achievement with its own dates.
The star that was cut before it was proved
Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.
From a shell to a solar array
The Miura fold was published in 1970 and flew on a satellite in 1995. The gap is not ignorance — the pattern was known, understood and available the whole time — and the same twenty-five year lag appears between every folding result and the hardware that uses it.
The patterns nobody owns
This site prints crease patterns at true scale and prints no designer's work, and that has always been stated as a rule applied at the end. Read the printed shelf as a documentary record instead and the rule turns out to be a property of the record: every pattern that carries a date was published as mathematics, every undated one belongs to nobody, and the two silences are one silence.
The half no notation records
Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.
Two kinds of claim
This site has published a median overrun of two centuries and a mean of twenty-two years in the opposite direction, and both are right. Split the record by what each claim is about and the reason appears: every claim about a practice is dated earlier than its evidence, most claims about a result are dated later, and the two scatter by 979 years and 131.
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.
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.
The reader decides the junction
Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.
The file records no verdict
A crease pattern file records vertices, edges and letters. Every one of the square twist's two hundred and fifty-six admissible letterings makes a perfectly valid file, and two hundred and forty-eight of them describe an object that does not exist. The format has a field for the layer order — the one thing that would settle it — and nothing fills it in, so a file is a drawing rather than a claim, and the field exchanges them as though they were claims.
The first thing about layers
A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.
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.
The tail was named somewhere else
The search for a mountain-valley labelling of a tessellation patch costs eighty-four steps at best and does not finish at all two runs in five, and the cure is to stop and start again rather than to wait. None of that was discovered here. The distribution was described in the study of satisfiability solvers in the nineteen-nineties, the restart arithmetic is older still, and what a crease pattern contributes is one more instance.
The cure was named first
A heavy-tailed search runtime, the arithmetic for cutting it off and restarting, and the reason restarts work at all were established in the study of search between 1993 and 1998. This collection imported all three, and inherited with them the phenomenon they answer — which is that randomising a search's choices is what makes the tail.
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.
Found by people not folding paper
The shortest strip that makes a Möbius band has a literature, and it is in differential geometry rather than in origami. The two subjects have the same number, they reached it by completely different routes, and neither of them cites the other — which is the fourth time this collection has found that shape.
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.
No format has a gluing
A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.
A sheet has a size as well
The layer count a design reaches is fixed by how thin the paper is. What size the finished thing comes out at is fixed by how large the sheet is, through a factor the pattern decides: the folded footprint times the mean layer count is the area of the paper, so the linear shrink is the square root of the layer count and a sixty-four-layer model finished at a hand's width wants more than a metre of sheet.
Which ceiling is binding
Two constraints hold a design's layer count down and both are ceilings on the same number. The stack gets better as the paper thins; the grid gets better as the sheet grows, because piling layers needs divisions and a division cannot be finer than a folder can place it. They cross at a sheet size that rises as the paper thins — so on the papers a classical folder had, the substrate really is the limit, and only at tissue weights does the hand take over.
How many wedges the paper allows
A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.
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.
When two of them are first attested together
Every single date in this field is argued about by centuries, and a pairwise one is not. The earliest year at which two claims are both attested is simply the later of their two surviving sources, so it inherits the better-attested half of each pair rather than the worse — and the year at which all five lineages of practice are jointly on the record is 1838, one year from where the same claims are popularly dated together.
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.
What the hindsight was worth
The best restart cutoff for the one tessellation search with a heavy tail was read off a hundred and twenty measured runs, which nobody running the search could have done in advance. The universal schedule needs no such knowledge, and on the same runs it costs 3,222 nodes in expectation against 512 for the cutoff chosen by looking — a factor of 6.3, which is close to the base-two logarithm of that cutoff, as the theory of the schedule says it should be. A larger unit brings the schedule within a few per cent of the hindsight, and choosing the unit is choosing the scale the schedule was meant not to need.
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.
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.
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.
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.
The prediction held at eight and broke at ten
A join in a slit grid of cranes merges at most four pieces, so n² cranes need at least ⌈(n² − 1)⁄3⌉ joins, and exhaustion found four and six by six meeting that floor in exactly one way. The guess was that eight by eight would too, with twenty-one joins. Counted a join at a time, it does — twenty-one, one way. Ten by ten does not: it needs thirty-four against a floor of thirty-three, and has 7,076 ways to spend them. The grids that meet the floor with no merge to spare are four and eight by eight among every size to fourteen, and sixteen by sixteen by construction, because a perfect tree of joins on a grid twice as wide is four perfect trees and one join in the middle. The even grids were never the pattern; the doublings are.
The interval is wider than the number
The field's most-quoted statistic — how far ahead of its evidence a popular date runs — is the mean of eight positive gaps, and it is quoted as three hundred and fifty-seven years. Resampling the fifteen entries puts ninety per cent of its weight between a hundred and fifty and five hundred and ninety-one. The interval is wider than the number, the middle gap's interval is a fifth as wide, and the difference is the same two documents the leave-one-out found.
A question the record is too small to answer
Five of the fifteen claims rest on one surviving source and five on two, and those two counts are exactly what an estimate of the claims that left no source at all is made of. Applied, it says two and a half are missing. Its ninety-five per cent interval runs from fifteen to thirty-one, re-reading a single entry's source count moves it by a fifth, and its independence assumption is false in the one way documents actually fail — which is what makes computing it worth more than declining to.
Some discoveries would make it worse
The field's headline statistic averages the claims dated ahead of their evidence. So a document found for a claim that was nearly right removes a small number from a mean of large ones and the average overrun goes up: finding a source two centuries earlier for the kindergarten entry would take the figure from 357 to 408. The four claims where a discovery helps can take twenty-five years off each, and no single document at any date can bring the number to 250.
A sheet is as large as two arms
A model's finished size is its sheet divided by the square root of its layer count, so a sixty-four-layer model at a hand's width wants more than a metre of paper. A hand-made sheet is formed on a mould somebody lifts out of a vat and shakes, and what that bounds is an area rather than a thickness: over the whole plausible range of mould weights and what arms can do repeatedly, the largest square sheet runs from about half a metre to about two. The demand and the bound are the same sizes, which is the one thing about them nobody has to know the constants to see.
The paper that will not hold a crease
Every constraint these essays have found improves as the paper gets thinner: the stack, the size, the layer count. A crease does not. A crease is a plastic hinge in the fibres at the fold, and a sheet one fibre thick has nothing through its thickness to hinge — so there is a floor under the thickness that no manufacturing skill moves, because the fibre diameter is a constant of the plant. The papers a tradition folds sit between one and a half fibres and four, and the two below that in this collection's own shelf are tissues, which are backed with foil before anybody creases them.
Eighty layers and the sheet decides the rest
Five of these essays each bound one thing and none of them bounds a design. Put together they close. The crease floor fixes the thinnest usable paper at about a fibre and a half; that paper's stack runs out at eighty layers; the largest sheet two arms can make falls away as the square of the finished size. The region under both is every model anybody can fold, and it has a ceiling at eighty layers and a corner at about a hand's width — above which the paper is no longer the limit and the vat is.
A failure teaches a schedule nothing
The universal restart schedule costs 6.3 times the cutoff chosen by hindsight on the one folding search with a heavy tail, and the obvious repair is a schedule that learns its scale from the attempts it has already made. It cannot. A failed attempt costs exactly its cutoff and reports only that the run needed more, so every rule that chooses the next cutoff from its own failures writes down the same list whatever happens — a fixed schedule in disguise. On the measured runs, doubling after every failure costs at least 3.6 times the hindsight, and growing by half at least 2.4. What does come near is information from outside the run: the universal schedule given the longest search on four other patches as its unit costs 1.7 times the hindsight. The field that supplied the schedule reached the same conclusion, and answered it by watching runs from the inside.
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.
One choice with eleven answers
A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.
The deepest point pays for the paper
A folded design uses its sheet according to its mean layer count and its paper according to its deepest point, and the ratio of the two is a property of the crease pattern. Measured on every printed pattern it runs from exactly one to nearly six — and it does not split tessellations from bases, as expected. It splits patterns whose every panel lies over every point from patterns that keep a footprint with structure in it. The ratio moves the corner of the substrate map by its square root, so the fold-and-cut triangle can reach the paper's eighty layers at 326 millimetres where the preliminary base must stop at 134.
Folding nobody designed
Leaves, wings and single strands of DNA all fold, and none of them were folded by anybody. Growth changes a sheet's own metric where a crease changes only its shape — the same geometry read in the opposite direction, and every figure here draws a fold this repository computed rather than an organism it did not measure.
A sheet that grows cannot lie flat
A leaf does not decide to buckle. Growth changes the distances between a sheet's own material points, a set of distances determines a curvature, and a curvature that is not zero cannot be laid in a plane by anything — whatever the sheet is made of and however slowly it grew.
Which way the disc curves
A growing disc either domes or ruffles, and which one it does is not a matter of how much it grew. It is decided by where the growth was — more at the rim opens the sheet, more at the middle closes it — and one number in one formula takes it through both.
The excess does not choose its waves
A rim that has grown longer than its span has to put the extra length somewhere, and two large waves and eight small ones are the same metric — identical arc length, identical excess. Geometry fixes the family and is completely indifferent about the member.
A crease carries no curvature
A fold looks like the sharpest curvature a sheet could have, and intrinsically it has none at all. Developability — the first of the four conditions this site's checker runs — is exactly the statement that folding an uncut sheet creates no curvature anywhere, including at the creases.
A leaf packs by corrugating
A corrugation is the cheapest fold there is — parallel creases, no interior vertex to think about — and a leaf that uses one has to taper it, because a leaf is broad in the middle. Which direction the taper is allowed to run turns out not to be a matter of taste.
The bud chooses the pattern
Nothing in the geometry of a corrugation says how many folds it should have. The container does: too few folds is a strip too wide to fit, too many is a stack too thick to fit, and the window that fits at all is narrow and has a best point in the middle of it.
Opening with nothing to pull
A leaf is not opened by a hand, a hinge or a motor. It opens because it keeps growing — which puts a condition on the pattern that no folder ever has to satisfy, because a person can always push.
A wing that folds into nothing
A beetle stows a wing longer than its body under a case a fraction of that length, and the ratio is the whole engineering problem. What a fold achieves is computable from the pattern alone, and the four geometries available are not close to each other.
No motor in the fold
An insect's wing has muscles at its base and nothing out along its length, so the pattern has to carry the deployment by itself. The condition that makes that possible is a count: one degree of freedom means one number determines every panel, which means one thing has to pull.
Nothing in a body folds on a line
A crease in an organism is not a crease. It is a compliant region — a patch of thinner material that bends — and a region has a width. The width consumes surface in exact proportion to the number of folds, which puts a ceiling on how fine a pattern can usefully get.
How much surface fits in a body
An organ whose whole job is to have area — a gut, a gill, a cortex — is solving a packing problem in reverse. Folding buys surface inside a fixed volume, and with a sheet of zero thickness it buys an unlimited amount. With a thickness the curve turns over and then falls to nothing.
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.
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.
The same corrugation in four places
A leaf, a wing, a crushed cylinder and a solar array arrive at nearly the same fold, and none of them copied any of the others. Convergence stories are cheap; this one is checkable, because the constraint that forces it can be computed rather than admired.
The organism is not the model
Every figure in this field draws a fold this repository computed. Not one of them measures a leaf, a wing or a gut. That is the rule the field was built to, and it is worth stating as a table rather than as a preamble — because a field about living things is where a computed geometry is most likely to be read as an observation.
Four finders, one option
Four unrelated lineages arriving at the same corrugation is read as evidence that the corrugation is good. It is at least as much evidence that there was nothing else to arrive at: how far a folded sheet shrinks is exactly its average layer count, so a lineage choosing a packing ratio is choosing a number of layers and nothing else — and the quantity that is genuinely free turns out to be almost uncorrelated with it.
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.
Twice as thick where it is thickest
A folded leaf's thickness is quoted as an area calculation: so much lamina, so much footprint, so many layers on average. The average is not what has to fit in the bud. Sampling the folded state of corrugated leaf patterns of two to five rows gives a deepest point of eight, twelve, sixteen and twenty layers against averages of 4.15, 6.23, 8.31 and 10.39 — a ratio of 1.926 that does not move at all.
A leaf ends its pattern
A hornbeam leaf's corrugation does not stop at the margin by being cut off: the pleats narrow until there is nothing left of them, and the margin is where the pattern reaches zero rather than where it was interrupted. The same is available to a drawn pattern and costs nothing — a corrugation tapered by a factor of eighty-two across its columns folds with a Kawasaki residual of 4×10⁻¹⁶, exactly as an untapered one does, because the column widths never enter the condition.
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.
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.
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.
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.
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.
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.
Four materials, four optima
The convergence argument gets its pattern and stops there. A hinge has a radius, the radius takes a fixed length of surface out of every fold, and the fold count that gets the most packing out of a sheet is inversely proportional to it — so a leaf, a wing, a gut lining and a metal array agreeing on a corrugation still disagree by an order of magnitude about how many creases to put in one.
The number is the angle
Every packing ratio worked out so far is computed at a fold closed all the way, and a folded wing is not closed all the way. At zero thickness the ratio runs away as the fold shuts, so the size of a quoted number says how far the fold got and not what the pattern is — and the pattern contributes only an exponent, which makes the same quoted ratio mean two quite different angles depending on which geometry produced it.
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.
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.
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.
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.
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.
The fold count sets the spring
A corrugation sweeps the same span at every fold count, and the count decides only how much room the zigzag needs while it does it — which was counted as a gain with nothing pushing back. Something does push back. Every hinge is a spring, a finer corrugation has proportionally more of them, and the force to hold a given span rises exactly as the clearance falls: the product of the two is the same number whatever the count, and at each material's own best count the spring goes as one over the square of the hinge radius.
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.
The helix chooses the lattice
A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.
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.
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.
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.
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.
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.
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.
How far open is a question about the grip
A corrugation of hinges that rest flat is loaded when it is shut, so it opens by itself and the force in the held-state calculation is a restraint rather than a drive. Followed from shut to flat that restraint only ever falls, and by exactly π over two — so every partly open state a structure can occupy is squeezed into a band a third wide, and a grip that weakens by a third leaves the sheet nine tenths open.
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.
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.
The channel grows with what it feeds
A comb of standing walls beats a stack of plies by eight because its members do not share the clearance that pays them. Supply takes that back, and asymmetrically: a wall's channel has to be sized for the surface the wall carries, so it grows with the wall's height and is charged against the pitch, while a ply's channel is a constant charged against the clearance. The comb then saturates at twice the reciprocal of the channel's share, the stack does not saturate at all, and the two cross at a clearance the model gives in closed form.
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.
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.
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.
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.
The angle the eight does not know
A comb's members are always drawn standing square to the base, and nothing has asked why. Lean one to an angle and it must be longer to reach the same clearance, which is more surface; it also takes more of the base to stand on, which is fewer members. The two are reciprocal and cancel exactly — the surface a comb holds is the same number from a right angle down to one degree, where each member is fifty-seven times the clearance long and there are two of them where there were a hundred.
In a tube the standing members lose
Members standing across a clearance beat layers lying along it by eight, and every drawing of that argument has a flat base under it. Curve the base into a tube and the ranking inverts: radial fins converge, so the room they need is the room at their tips, and their best arrangement fills exactly half the cross-section. Concentric layers fill all of it. The eight becomes a half, and the half is exact.
The wedge belongs to one length
Radial fins inside a tube reach at best half of what any lining of sheet could hold, because converging fins leave empty wedges behind their tips. Tapering the fins cannot help: the tip already sets the count, and a fin cannot be thinner there than the sheet it is made of. Fins of several lengths can. Counted along the radius they are a staircase under a straight line, and the staircase with m steps is best with its steps equally spaced, where it holds exactly m ⁄ (m + 1) of the ceiling. The factor of two belonged to fins of one length, not to fins.