Which cranes can stay joined

The oldest book cuts the paper counts the arithmetic of the slitting. For an n by n grid the piece has n² cranes, (n − 1)² corner joins holding them together, and 2n(n − 1) sides of slit — and the slitting grows faster than the crane count, so the violation of the one-sheet rule gets larger as the piece gets more impressive rather than smaller.

That rung takes the arrangement as given and prices it. This one asks how much choice there was in it.

The joins are hyperedges, not edges #

The first thing to get right is what a join is, because it decides the whole count and the obvious model is wrong.

A grid of squares has neighbours, and the natural picture is a graph: cranes as vertices, adjacent pairs as edges, and the joins as a subset of those edges — the kind of object a crease pattern’s face graph is. On that picture the piece is connected exactly when the chosen edges contain a spanning tree, which needs n² − 1 of them.

The book’s arrangement does not have n² − 1 joins. It has (n − 1)², which for a three by three is four rather than eight, and for a five by five is sixteen rather than twenty-four. So the edge picture is not the arrangement being drawn, and the identity the ladder below verifies — cranes minus joins is exactly 2n − 1 at every size — is a check that it is not.

What the book draws is a join at each interior lattice point, and a lattice point in the interior of a grid has four squares meeting at it. One join therefore holds four cranes at once: the wing tips of all four squares are left uncut at that point.

That is a hyperedge rather than an edge, and it is why the piece needs so few joins. Connectivity is cheap when one join connects four things.

Counting the subsets that work #

With the model right, the question is a finite one and it is small enough to answer by exhaustion.

For an n by n grid there are (n − 1)² join points. Each is either left joined or cut, so there are 2^((n−1)²) arrangements — two for a two-by-two, sixteen for a three-by-three, five hundred and twelve for a four-by-four, and sixty-five thousand five hundred and thirty-six for a five-by-five.

For each subset, union the four cranes at every chosen point and ask whether everything ends in one component. The answers:

• Two by two. One join point, four cranes. One of the two subsets works and it is the one that keeps the join.
• Three by three. Four join points, nine cranes. Exactly one of the sixteen subsets works, and it is the full set. Every join is load-bearing; there is no slack whatever.
• Four by four. Nine join points, sixteen cranes. Twenty-one of the five hundred and twelve subsets work — four per cent — with a minimum of five joins, achieved in exactly one way.
• Five by five. Sixteen join points, twenty-five cranes. Seven hundred and eighty-five of sixty-five thousand five hundred and thirty-six — one and two tenths of a per cent — with a minimum of ten joins, achieved fifty ways.

The share that work collapses as the piece grows, from a half at two-by-two to one in eighty at five-by-five, and even the largest piece needs ten of its sixteen joins.

Why the three-by-three has no slack at all #

The three-by-three result deserves its own explanation because “one of sixteen” invites the thought that something has been mismodelled.

Nine cranes in a three-by-three grid. The four interior lattice points sit at the centres of the four two-by-two blocks, and each holds the four cranes of its block.

Now look at the corner cranes. The top-left crane touches exactly one interior lattice point — the one at the centre of the top-left block — because the other three lattice points are not on its boundary. So each corner crane is held by exactly one join, and dropping that join detaches it.

Four corners, four joins, one each: every join is the sole attachment of a corner. Drop any of them and a corner falls off. That is the whole argument and it explains why the count is one rather than a few.

It also says what happens at larger sizes. The corners are still held by one join each, so the four corner joins are compulsory at every size — which is why the minimum never falls below four and why the counts stay so small. What varies is how much freedom the interior has, and the interior grows as (n − 3)² while the corners stay at four.

What this does to reading the plates #

The 1797 plates show particular arrangements and the natural reading is that they are designs — chosen from many, for their appearance or their difficulty.

The count says they are very nearly not choices at all. At the sizes a plate can show, the arrangements that hold together are a handful, and at three by three there is one. The book is drawing something close to the only thing there is.

That is a different kind of claim about a historical object than this field usually gets to make, and it is worth being clear about which part is computed. Computed: that a slit grid of a given size, joined at interior lattice points, is connected for very few subsets of those points. Not computed: which arrangements the plates actually show, whether the joins are exactly at the lattice points, or whether the book’s cranes are attached in some other way entirely.

The last of those is a real limitation and it cuts both ways. The arrangement modelled here — corner joins at interior lattice points — is what the ladder below’s arithmetic fits, since cranes minus joins comes out at 2n − 1 exactly. That is evidence the model is the right one and it is not a reading of the plates.

So the honest statement is conditional and it is still strong: if the joins are at the interior lattice points, the cutting is nearly forced, and the plates are showing a constraint rather than a design.

What the piece could have been instead #

The counting also says what the alternatives were, which is the other half of a claim about choice.

Fewer joins. The minimum is (n − 1)² at three-by-three — all of them — and falls below the total at four-by-four, where five of nine suffice and there is exactly one way to choose them. So a maker wanting a more open piece has one option at four-by-four and fifty at five-by-five, and none at three-by-three.

Edge joins instead. A slit grid could be joined along edges rather than at corners, which is a different arrangement with a different count: connectivity then needs a spanning tree of the grid graph, which is n² − 1 joins — twice as many, and a much less open piece.

A different lattice. Nothing here requires the grid to be square, and a hole in a sheet is an edge for the same reasons a slit is. A triangular arrangement has lattice points holding six cells and would need fewer joins still, and no tradition appears to have made one.

Which puts the book’s arrangement at a particular place: the sparsest joining that holds together, on the lattice that gives the most connection per join, at the sizes a sheet allows. That is an optimum rather than a choice, and it is reached by three separate near-forcings rather than by one.

What one join is worth, counted #

The hyperedge model has a quantity in it worth extracting, because it explains the whole shape of the counts and it is a number rather than an argument.

An edge join connects two cranes. A corner join connects four. In graph terms, adding an edge to a forest reduces the number of components by at most one; adding a hyperedge covering four vertices reduces it by at most three.

So a piece of n² cranes needs at least (n² − 1) ⁄ 3 joins, rounded up, and the minimum is exactly that whenever the geometry allows it. Nine cranes need at least three joins and the three-by-three’s answer is four, so the geometry costs one; sixteen cranes need at least five and the four-by-four’s answer is five, so the geometry costs nothing; twenty-five cranes need at least eight and the five-by-five’s answer is ten, so the geometry costs two.

The bound is tight at four-by-four and slack at the others, and the slack is the corners. A corner crane can only be reached by one join and that join is spending three of its four slots elsewhere, so corners are where the hyperedge’s efficiency is wasted.

That gives the counting a shape it did not have from the raw numbers. The piece is trying to cover n² cranes with hyperedges of size four; the theoretical best is a third of the cranes in joins; the corners force extra; and the count of arrangements achieving the minimum is a count of how many ways the covering can be arranged, which is one at four-by-four and fifty at five-by-five.

Two objects that turn out to be the same object #

The ladder’s two famous cut pieces have now both been counted and the counts have the same shape, which is worth saying because it did not have to be so.

The star is bounded by a material quantity: the symmetry order is a layer count, the layers meet a blade, and the repertoire stops where the stack does. How many wedges the paper allows computes it.

The cranes are bounded by a combinatorial one: the joins have to cover the cranes, the corners waste the covering, and almost no subset works.

Different mechanisms and the same conclusion, which is that the object is nearly forced. And there is a common form under the two: each is an operation performed once on a folded or slit whole, where the arrangement has to serve every copy at once. One cut serves 2k wedges; one lattice point serves four cranes. In both cases the economy is what makes the object possible and the economy is what removes the choice.

That is a general property of doing something once to a stack, and it is worth carrying because it is the thing that makes folded and cut objects look designed. An operation that serves many copies has few valid arrangements, so the arrangement that survives looks chosen when it was cornered.

Which theorem was checked and how #

The census is an exhaustion. Every one of the 2^((n−1)²) subsets is built, a union-find is run over the cranes, and the components are counted. Nothing is sampled and nothing is argued from a special case.

The three-by-three is asserted. Exactly one subset must work and it must be the full set, so a model that had accidentally made the joins edges rather than hyperedges would fail here rather than producing a plausible larger number.

The share must fall with the size, which is the finding, and a census in which it rose would refuse to draw.

And the largest grid must still need more than half its joins, so a size at which the arrangement had become genuinely free could not be presented as though it were forced.

Where the model stops #

Five by five is the ceiling of the exhaustion. Six by six is twenty-five join points and thirty-three million subsets, which is computable but not inside a figure, and the counts are not extrapolated.

Connectivity is the only requirement. A piece that holds together is not thereby foldable: every crane has to have enough paper at the right places, and a cut is not local — a crane whose wing tip is uncut at two corners is a different fold from one uncut at one. That constraint is not modelled and it would reduce the counts further.

The joins are points and a real join has a width. A wing tip left uncut over a few millimetres is stronger than one left uncut over a fraction of a millimetre, and nothing here prices the strength of a join or asks whether a five-join four-by-four would survive being folded.

And nothing here is a reading of the 1797 plates. The model is the arrangement the ladder below’s arithmetic fits; whether it is the arrangement the book draws is a documentary question and the essay says so rather than assuming it.

What the picture cannot show #

The table counts subsets and cannot show them. Which five of the nine joins a four-by-four needs is a definite answer and it is one arrangement out of five hundred and twelve, and a figure that drew it would be drawing a single case where the finding is about a population.

Nor can it show what the folded piece looks like. Every count here is about the flat sheet with its slits, and whether a given subset of joins produces something a folder would call a piece is a question about the folding rather than about the connectivity.

The most conspicuous absence is the strength. A piece connected through a single corner join at one point is connected in the graph and hanging by a thread in the hand, and the census treats a one-join bridge and a four-join block as equally connected. A count of what holds together is not a count of what survives being folded, and the second is what a maker is actually choosing among.

The idealisation, named #

The sheet is a grid of unit squares, the slits are along the full grid lines except at the joins, and a join is a point at which four squares remain attached.

The strongest assumption in that is the last, and it is the one the ladder below’s identity supports rather than the one this rung introduces. Cranes minus joins is 2n − 1 at every size in the book’s own arithmetic, and that identity holds exactly for the interior-lattice-point model and for no other joining scheme this collection has tried.

What is genuinely assumed here is that connectivity is the constraint, and that is a modelling choice about what the maker wanted. A maker who wanted a piece that hangs in a particular way, or that folds in a particular order, is solving a different problem with more constraints and therefore fewer solutions — so the counts here are upper bounds on the choice available.

Which is the direction that strengthens the finding. Whatever else the maker wanted, they could not have wanted anything the connectivity forbade, and at three by three the connectivity forbids everything but one arrangement.

Where the ladder goes next #

This closes the folklore ladder’s two famous objects — the star and the cranes — and both have come out the same way: an object that reads as a design turns out to be very close to forced, by a material bound in one case and a combinatorial one in the other.

What the ladder owes next is the sixth grid, and it is a real computation rather than an extension. Six by six is thirty-three million subsets and needs a smarter count than an exhaustion — the connectivity of a subset of hyperedges on a grid is the kind of thing a transfer matrix handles, and the sequence of counts would then be available for every n rather than for four values of it. Whether the share settles to something or keeps falling is not guessable from four points.

Sideways, the finding belongs beside what a cut is a licence for, which prices a cut in the design currency. That ladder asks what cutting buys; this one finds a cut arrangement with almost no freedom in it; and the two together say the famous cut piece bought very little and had almost no choice about how.

The habit worth carrying is a counting one. Before calling an arrangement a design, count the arrangements that would have worked. It is often a small computation, the answer is often much smaller than expected, and a design chosen from one is a constraint wearing a design’s clothes.