The oldest book cuts the paper

The rule everybody knows about origami is that the paper is not cut. One square, no scissors, no glue; if something is cut it is kirigami, which is a different thing with a different name and, by implication, a lesser one.

The oldest surviving book of recreational paper folding in the world breaks that rule on almost every plate.

What the book is #

Hiden Senbazuru Orikata — roughly, the secret method of folding one thousand cranes — was printed in 1797. It is a slim thing, it circulated as a novelty, and it is the earliest surviving publication anybody has that treats paper folding as a recreation rather than as ceremony.

Its content is forty-nine designs for connected cranes: groups of birds joined at wingtip or beak, folded from a single sheet that has been cut into a grid of squares which remain attached at their corners. The engineering is in the cutting pattern and in the order of folding, because a bird whose neighbour is attached at the wingtip cannot be folded independently of it.

That is a remarkable object, and it is not the object the modern rule describes.

The arithmetic of the slitting #

The cutting is regular enough to be arithmetic, and the arithmetic says something the plates do not.

Lay an n by n grid on the sheet. To free n² squares while leaving them joined, every internal grid line is slit except at the lattice points where four squares meet. There are 2n(n−1) internal line segments to cut, in units of the small square’s side, and (n−1)² lattice points left intact.

So a 2×2 arrangement is four cranes on four sides of slit, held by one join. A 6×6 is thirty-six cranes on sixty sides of slit, held by twenty-five joins. The slitting per bird rises from 1.00 to 1.67 sides as the piece grows — the more ambitious the work, the further it travels from the rule.

Why the difference is odd #

There is a small identity hiding in those numbers and it is the kind worth pulling out, because it explains why the arrangements look the way they do.

Cranes minus joins is n² − (n−1)², which is 2n − 1. Always odd, always exactly the number of squares along two adjacent edges of the grid counted once each. The generator asserts it at every size, because an off-by-one in either count would otherwise produce a plausible table.

What it means physically takes a little more care than the difference alone suggests, because the joins per bird are actually rising: a quarter at two by two, and $(n-1)^2/n^2$ in general, which climbs toward one. A bigger arrangement is better connected per bird, not worse.

Where the fragility actually is #

So the flimsiness has to be somewhere else, and counting incidences finds it exactly.

A lattice point is not a link between two squares. It is where four squares meet, so one join carries four birds, and the right question is not how many joins there are but how many each bird has.

A corner cell of the grid touches exactly one interior lattice point. An edge cell touches two. An interior cell touches four. Adding those up: $4 + 8(n-2) + 4(n-2)^2 = 4(n-1)^2$, which is four times the join count, as it must be.

So four birds in every arrangement, at every size, hang by a single point of paper each — and there is no size at which that improves. The corners of the piece are held by one fibre bundle apiece whether the arrangement is two by two or eight by eight, and a corner bird lost is a corner bird lost.

That is a sharper statement than a general flimsiness and it matches what a folder meets. The interior of a large arrangement is well tied — four joins per bird — and the failures are at the rim, where a bird pinned by one corner has to be rotated through a full crane’s worth of reversals with nothing else holding it.

Which explains the choice of join #

It also explains the wingtip-and-beak choice the plates make and never state.

A bird with four joins is constrained on all sides and can be worked gently; a bird with one join takes the whole torque of its own folding through that point. The mechanically demanding join is therefore the corner one, and there are exactly four of those in any arrangement.

The dense arrangements — where most birds have three or four joins — can afford the wingtip, which is the thicker and stiffer corner, because no single join is carrying much. The sparse ones, where most birds have one or two, need the join that survives being loaded alone, which is the beak.

The book’s choice tracks the incidence count rather than the size of the piece, which is what the plates show and is not what a reader would predict from the arrangement’s overall dimensions. It is a design rule derived from the adjacency graph by people who had no adjacency graph, and arrived at by folding the things.

What a corner join can bear #

A join at a lattice point is a mathematical idealisation with a physical cost, and this subject has an essay’s worth of practice at naming that gap.

In the idealisation the join is a point: zero width, infinite strength, no bending stiffness. In paper it is a few fibres. The forces at it are not small, because folding a crane requires rotating the whole square through a series of reversals while its neighbour holds one corner fixed, and the fibre bundle at that corner takes the torque.

That is a real constraint, and it is the reason the tradition is on thin, strong, long-fibred paper rather than on anything to hand. The join is the weakest structure in the piece and the fibre length is what holds it.

Folding a bird that is pinned at the corner #

The interesting difficulty in the book is not the cutting. It is what the cutting leaves behind.

An ordinary crane is folded from a free square: every stage can be turned over, flattened, and worked from either face. A crane in a connected arrangement cannot. One or two of its corners are attached to a neighbour, so the square cannot be turned over independently, cannot be lifted clear of the sheet, and has to reach the same folded state through a sequence that never asks for the moves the free square takes for granted.

That is a reachability problem, and it is the same one this site spends a whole field on. A theorem that says a folded state exists says nothing about whether anything can get there, and a machine that can only make one kind of move reaches a strictly smaller set of states than folds flat at all. A folder pinned at two corners is exactly such a restricted machine, and the book’s forty-nine plates are, read charitably, forty-nine solved instances of a constrained reachability problem worked out by hand in the eighteenth century.

What the order of folding costs #

The constraint has a second consequence which the plates record and no modern account mentions: the arrangements have a folding order, and it is not free.

In a grid of connected squares, folding one bird pulls on its neighbours. Fold the middle one first and the surrounding paper is dragged inward and can no longer be laid flat to work on; fold the outermost ones first and the interior stays accessible for longer. The book’s plates number their steps, and the numbering is not decorative.

That makes the connected crane a piece of sequenced origami in a stronger sense than an ordinary model, where the sequence is a convenience of exposition. Here the sequence is part of the design, and a wrong order does not produce an ugly result but an unreachable one.

Where the rule actually comes from #

If the oldest book cuts, the natural question is when the prohibition arrived and what it is for.

The honest answer is that it arrived with the twentieth-century codification of the subject as an art with standards, and that it is a generative constraint rather than a historical one. That distinction is worth the paragraph, because it changes what the rule is owed.

A historical rule is a claim about the past and is answerable to the record. A generative constraint is a rule adopted because of what it produces. One sheet and no cuts produces a subject: the sheet cannot stretch, so it can only take shapes with zero Gaussian curvature; material for a flap has to come from somewhere, so a flap costs a circle; every theorem about a vertex depends on the sectors summing to 360°, which is the statement that the paper was not cut there. Remove the constraint and every one of those results goes.

So the rule earns its place by consequences, not by ancestry. Claiming ancestry for it is unnecessary and happens to be false.

The cut as a first-class operation #

Once cutting stops being a lapse, it becomes a subject, and it turns out to be a deep one.

One straight cut releases any straight-line drawing whatever, given the right folding. That theorem is of 1998 and it is one of the most striking results in the field, and it is a result about cutting. The 1797 book is doing something in the same family two centuries earlier: using cuts to change what folding can produce, rather than using them to escape a difficulty.

The difference between the two is instructive. The senbazuru cuts are made first and are a grid; they partition the sheet and the folding happens afterwards, independently in each cell except at the joins. The fold-and-cut construction folds first and cuts once, and the cut is the last operation rather than the first. Both are one sheet. Only one of them is uncut.

The join as a design variable #

Reading the arrangements as engineering rather than as decoration turns up a choice the book makes consistently and never explains.

The birds can be joined at a wingtip or at a beak, and those are different problems. A wingtip join attaches the corner of the square that ends up at the extremity of a long flap, so the join is loaded in tension along a line of paper that has been folded to a point and is at its thinnest. A beak join attaches a corner that ends up at the head, where fewer layers meet and the paper is under less tension but is harder to reach with the fingers while folding.

So there is a trade between mechanical survival and manual access, and it is decided per arrangement. The plates that join at the wingtip are the sparse ones, where each bird has one or two neighbours; the dense arrangements, where a bird may be attached on three sides, tend to use the beak. That is a sensible engineering choice and it is nowhere stated in the book — it is visible only by looking at which arrangements use which, which is a form of evidence the documentary record does not usually offer.

A grid is not the only adjacency #

One more thing the arithmetic makes visible, and it is the reason the square grid was worth singling out rather than assumed.

The counts above — n² birds, (n−1)² joins, 2n(n−1) sides of slit — are facts about a square grid and not about connected cranes in general. What determines them for any arrangement is the adjacency graph of the squares: how many cells there are, how many places two cells touch at a corner, and how much internal boundary has to be released. Change the arrangement to a strip and the numbers change; the identity 2n − 1 is a property of the square and disappears with it.

The book contains both. Its strips and its irregular clusters obey the general rule and not the tidy one, which is a small piece of evidence that the arrangements were designed as objects rather than derived from a formula — nobody working from the square-grid identity would have produced the strips.

What the plates cannot tell #

There is a limit on how much the book can settle, and it is the standard limit in this field.

The plates show finished arrangements and cutting diagrams. They do not show who was doing this before 1797, or how long, or where. A printed book is evidence that a practice existed and was worth publishing, which usually means it had been around for a while — but “a while” is not a number, and the book itself makes no claim about its own antiquity beyond the word hiden, secret, which is a marketing term as much as a historical one.

So the 1797 date is a floor and nothing more. What it does settle, because the plates are unambiguous, is the character of the practice at that date: connected, cut, and comfortable about it.

Two centuries with nothing in between #

The most uncomfortable thing about the 1797 book is not what it contains. It is what surrounds it.

A publication of that sophistication does not appear out of nothing — forty-nine worked connected designs is the output of a practice, not of an afternoon — and yet the record on either side of it is close to empty. There is Saikaku’s line about folded butterflies from 1680, a century earlier and about ceremony rather than recreation. There is the Kayaragusa around 1845, half a century later. Between and around those, essentially nothing survives that shows the practice at work.

That sparseness is itself evidence, though not of the thing it is usually taken as evidence of. It does not show the practice was young. It shows the practice was not written down, which is what one expects of something transmitted between people who are in the same room — and which means every date in this field is a date of documentation and none is a date of practice.

The idealisation, named #

Every essay here names the assumption its figure rests on, and this one rests on two.

The first is that the cutting pattern was the regular grid drawn above. The plates show several arrangements and not all of them are square grids; some are strips, some are irregular clusters. The arithmetic generalises — for any arrangement, cuts and joins are determined by the adjacency graph of the squares — but the tidy 2n − 1 identity is a fact about the square grid specifically.

The second is that a join is a point. It is not, and the whole thickness field of this site exists because that kind of idealisation fails in ways that matter.

Where this goes next #

The ladder from here goes two ways. The star that was cut before it was proved takes the other famous piece of folk cutting and finds a theorem two centuries late behind it. And nothing here is as old as it sounds is the rung below this one — the record as a whole, of which this book is the single most important row.

The surprising connection is worth stating last. The rule this site works under is not inherited; it was chosen, comparatively recently, and it was chosen well. A subject defined by a prohibition that its own oldest document ignores is in a stronger position than one defined by tradition, because the prohibition can then be justified by what it generates rather than defended by how old it is.

There is a version of that argument with teeth. Suppose the prohibition really were ancient, and suppose somebody produced a document proving it. Nothing about the mathematics would change — Kawasaki’s condition would still follow from the sectors summing to 360°, a flap would still cost a circle, and the general flat-foldability problem would still be NP-hard. The rule’s entire value is downstream of its consequences and none of it is downstream of its age, which is exactly the test that separates a working constraint from a piece of heritage. The 1797 book, by ignoring the rule, makes that test easy to run.