The half no notation records

The subject has had two notations and each was an advance on nothing.

The diagram — a sequence of steps, with a dashed line for a valley and a dot-dash for a mountain — records a procedure: do this, then this. It travels well, teaches well, and says nothing about a model except how to arrive at it.

The crease pattern records a state: every crease, where it runs, and which way it folds. Publishing it instead of the sequence was the change that turned design into something transmissible, because a pattern is the object a designer worked on and a sequence is a route somebody found through it.

Both of them record the paper. Neither records the stack, and the stack is the half that decides what the folded object is.

Why the pattern is not enough #

A crease pattern fixes where the paper goes. It does not fix which layer is on top of which, and which layer goes on top is not a detail of presentation — it is the difference between two objects a reader would call different models.

The site has the measurement in both directions. One marking, many objects shows a single pattern with several genuinely distinct folded states. The order does not name it either shows that even the layer order plus the outline can leave the pattern ambiguous. Between them they establish that pattern and object are different things, and that neither determines the other.

So a notation that records only the pattern has recorded half of a pair — and it is the half that is easy to draw. It is also the half that can be checked: every pattern published here has been past four conditions precisely because those conditions are about the sheet.

How big the missing half is #

The two halves can be compared once both are counted in the same unit, and the unit is bits.

A crease pattern’s size is its vertices and its letters: a Yoshimura has 45 vertices at two coordinates each and 86 creases with a letter apiece. Which ordering of its panels the folded object is takes log₂ of the number of orderings, and it has 65 panels — 302 bits.

Three hundred and two bits is thirty-eight bytes, which is nothing at all as a quantity of data and is the entire content of what a reader has to work out for themselves.

The count is an upper bound and deliberately so: most of those orderings are refused by the non-crossing conditions, and the number of legal orders is far smaller. It is not smaller in a way anybody can write down, because computing it is the intractable part of the subject — which is exactly why the field’s format has a place for the answer and no way to fill it.

Against the half that is recorded #

Three hundred and two bits is a quantity and it needs something to be a quantity of. The natural comparison is not with the coordinates, whose size depends on how many digits anybody chooses to write, but with the assignment — the one mountain-or-valley bit per crease, which is the part of a crease pattern everyone already agrees is essential and which is exactly one bit each.

Set the two columns against each other. The fold-and-cut triangle: six letters against twelve bits of order. The preliminary base: eight against fifteen. The square twist: twelve against eighteen. The Miura: thirty-eight against seventy-nine. The waterbomb tessellation: seventy-six against two hundred and twenty-six. The Yoshimura: eighty-six against three hundred and two.

The layer order outweighs the assignment on every pattern here, by a factor of one and a half at the small end and three and a half at the large one. So the missing half is not a footnote against the recorded half; it is the larger of the two on every object the shelf contains, measured in the one unit both can be put in without a convention.

And the ratio grows #

The trend down that column is not noise, and it has a reason that says the gap widens for ever.

The assignment costs one bit per crease, so it grows linearly with the pattern. The layer order costs the logarithm of the panel count’s factorial, which grows as the panels times the logarithm of the panels — faster than linearly, by exactly a logarithm.

So the ratio between the two is itself a logarithm of the pattern’s size, and it climbs slowly and without limit. One and a half at seven panels, two at twenty-four, three at fifty-two, three and a half at sixty-five, and it would be five or six on a tessellation of the size anybody actually folds.

That sharpens the essay’s closing observation rather than softening it. Three hundred bits is a trivial quantity of data and it is also more than the whole assignment, on the largest pattern here, and the share it represents is growing. A notation that carries the letters and not the order is carrying the smaller half and losing the larger one — and the more ambitious the pattern, the worse the arithmetic gets.

What the interchange format does and does not carry #

The field has a machine-readable format for crease patterns, and the files this collection publishes are written in it. It carries the vertices, the edges, the assignment of each edge, and the faces — everything a pattern is.

It also has a field for face orders: a list of which face lies above which, pairwise. It is optional, and it is the field nobody fills in.

That is not an oversight in the format. Filling it in requires solving the layer ordering, which for a general pattern is the intractable problem the whole subject organises itself around; a format that required it would be a format nobody could write. So the gap in the notation is a gap in what can be computed, wearing the clothes of a gap in what is recorded.

The files published here are in exactly that position and it is worth saying plainly rather than quietly. Every pattern on the printed shelf exports with its full geometry and its assignment; not one of them exports a layer order, because none has been computed. A reader who folds one has produced information the file does not contain.

What a diagram carries that a pattern does not #

The older notation is worth defending at this point, because it carries something the newer one loses.

A sequence of steps is a procedure, and following it produces the layers in the order the steps put them. The information is not written down as an order — it is implicit in the sequence, which is a much weaker form — but a folder who follows a diagram never has to solve the ordering problem, because the procedure solves it by construction.

That is why diagrams remain the way models are taught and crease patterns remain the way designs are transmitted, and it is a sharper distinction than the usual one about difficulty. A diagram records a solution to the layer problem in a form nobody can extract; a crease pattern records the problem.

Neither format records the solution in a form that could be checked, compared or searched, which is what a notation is for.

Three notations, three eras, one omission #

The gap is easier to see when the three records the subject has kept are put in order.

The cut-and-fold plate, which is the oldest published record in the subject: a picture of a sheet with its cuts marked and a picture of the result. It records a starting sheet and an outcome, and everything between them is left to the reader.

The step diagram, which is the nineteenth and twentieth centuries’ contribution: a sequence of pictures with a line convention that distinguishes a mountain from a valley. It records a route, and the layer order is a by-product of following it.

The crease pattern and its machine-readable form, which is the last fifty years: a complete record of the sheet in its unfolded state, with every crease and every letter, in a format a program can read.

Each is a better record than the one before it of the thing it records, and none of them records the stack. The progression is a progression in completeness about the paper, and the quantity that decides what the folded object is has been outside all three from the beginning.

Which theorem was checked, and how #

The bits are computed rather than quoted. The panel count comes from folding each pattern and counting the panels its folded state has; the bit count is the logarithm of that count’s factorial, summed term by term rather than through a factorial that would overflow at sixty-five.

The comparison is in one unit. A count of vertices and a count of orderings are not otherwise comparable, so both halves are put in bits — the pattern as coordinates and letters, the order as a choice among permutations.

The upper bound is stated as one. Not every ordering is a folded state, and the number of legal ones is not computed here because computing it is the hard problem; the figure and the table both say so rather than reporting the bound as the answer.

Every pattern on the shelf is included, so the table is not a selection.

Where the model stops #

Bits are a crude measure of a record. A notation is not judged by its information content — a diagram’s value is that a person can follow it, which no bit count captures — and the comparison above is about what is present rather than about what is useful.

The upper bound is loose. Sixty-five panels give 302 bits of orderings and the great majority of those orderings are illegal; the honest number is smaller and is not available, and how much smaller is the hard problem itself.

The count assumes a total order. Two panels that never lie over one another are not stacked in any observable sense, so counting orderings over-counts folded states — another reason the bound is loose.

Only the flat state is considered. A model that is not folded flat has no total order of its panels at all, and the layer question in that case is a different one with a different shape.

And the format is not at fault. A field left empty because filling it requires solving an intractable problem is a well-designed field; what the essay claims is that the emptiness is the notation’s largest feature and is never described as one.

What the picture cannot show #

A bar chart of bits against bits is an unusually poor picture of an argument about notation, and it is the best one available. What a reader would want to see is two folded objects that share a crease pattern and differ in their stacking, and the site has that figure elsewhere; what cannot be drawn at all is the absence — a file with a field left empty has no appearance.

The other thing no figure here can show is what a folder knows. Somebody who has folded a Yoshimura holds the layer order in their hands and could, in principle, write it down; nobody ever has, and the reason is that the form to write it in was invented long after the models were.

The idealisation, named #

A layer order here is a total order of the panels, which assumes the panels are flat, have no thickness, and lie exactly on one another. Real paper has thickness, so the “order” is a physical arrangement that a reader can see by looking at the edge of the model — which is the practical reason nobody has needed the notation.

That is the honest version of why the field has not felt the gap. A folded model is its own record of the layer order, legible to anybody holding it, and the notation was never needed until the models started being transmitted as files rather than as objects.

The generalisation #

A notation records what its inventors could compute. The diagram records a procedure because a procedure is what a folder has; the crease pattern records the creases because the creases are what a designer draws; and neither records the stacking, because nobody can compute it. So the shape of the notation is the shape of the tractable part of the subject, and the intractable part is invisible in every record the field keeps.

The consequence for a reader is worth stating as a caution. A published crease pattern is not a published model. What a record is and what a proof is is a distinction this collection has had to make about history; the same distinction applies to a file. It is a published problem, whose solution the publisher happens to have and did not include — and the fact that it usually does not matter is because the reader is a person who can fold and see, rather than a program that has to decide.

The consequence for anybody building a library of patterns is sharper. A collection of crease-pattern files is a collection whose central content is missing, and the missing content is small: thirty-eight bytes for the largest pattern here. What is expensive is not recording it — it is knowing it.

Who found it, and when #

Nothing above is a discovery about the format, whose specification names its face-order field and describes it as optional. The measurement is this collection’s own: the panel counts come from folding the printed patterns, and the comparison in bits is a way of putting a gap that is usually described in words onto an axis.

What the measurement does establish is that the gap is not large in the sense of data. Three hundred bits is nothing; the reason it is not recorded is that it is not known, and a subject whose notations all stop at the same place has said where its difficulty is.

What a record would have to contain #

If the field did fill the field in, what would it be recording? The question is not rhetorical, and answering it exposes a second gap under the first.

A layer order is a list of which panel lies above which. For a pattern with a unique folded state that list is the model, and recording it is recording everything. For a pattern with several — and several is the ordinary case — the list records one of them, which raises the question of which one, and there is no convention for saying “the one the designer meant” as opposed to “one that works”.

So a complete record needs two things the format has one of: the order, and a statement of whether the order is forced. The second is a computation nobody can perform in general, and where it can be performed the answer is often that the order is not forced at all.

That is the honest shape of the gap. It is not that the subject forgot to write something down; it is that the thing to be written down is under-determined by everything else that is recorded, and the field has never had a way to say so.

Where the ladder goes next #

The obvious continuation is to fill the field in for the patterns where it can be filled in. In one dimension the layer order is decidable and cheap; for a small two-dimensional pattern an exhaustive search over orderings would settle it, and the printed shelf’s smaller entries are within reach.

That would produce the first files in this collection that record a folded object rather than a crease pattern — and it would immediately raise the question the format leaves open, which is whether the order it records is the order or merely an order.