No format has a gluing

A file has no paper is the collection’s standing observation about crease-pattern formats: the file records geometry and the paper is an assumption. This is the same observation about a different missing thing, and this one has consequences a reader can measure.

What a format records #

The standard interchange format for crease patterns records five kinds of thing.

Vertices, as coordinates. Edges, as pairs of vertices. Assignments, one per edge — mountain, valley, boundary, flat, unassigned. Faces, as loops of vertices. And layer orders, as relations between faces.

Every one of those is a feature of the paper’s interior. Even the boundary appears only as an assignment on an edge — a way of saying this edge is not a crease — rather than as a structure of its own.

What it cannot record #

That two boundary edges are the same edge of the paper.

There is no field for it. An identification is a relation between edges, with a direction, and the format’s edge list has no place for a relation between its own entries. Adding one is a structural change rather than a new attribute.

So a file describing a rectangle of pattern describes four different objects at once, with different panel counts, different free-letter counts and — on some drawings — different answers to whether the pattern folds.

What that costs #

Three things, in increasing order of seriousness.

Two people cannot exchange a tube. Anybody sending a folded-tube pattern sends a rectangle and a sentence in an email. The sentence is not in the file and does not survive being processed by anything.

No checker can check one. A tool reading the format checks vertices, and a tube’s vertices are its rectangle’s vertices. The condition that distinguishes them is about a loop, and the tool has no loop to find because it does not know the sheet is closed.

No archive can hold one. A collection of patterns is a collection of files, and a file of a closed sheet is a file of a rectangle. The information is lost at the moment of recording.

What adding it would take #

Very little, which makes the absence the more striking.

One list. Each entry is a pair of boundary edges together with a direction — whether the first edge’s start meets the second’s start or its end. For a rectangle glued into a torus, two entries. For a cylinder, one. For a disc, none.

From that, everything downstream follows mechanically. Which creases are one crease, which panels are one panel, what the Euler characteristic is, whether the sheet is orientable, and whether the parity condition is satisfied — all computable from the drawing plus the list.

And the list is empty for every file anybody has, so adding the field breaks nothing.

The field, specified #

Since the correction is small enough to write out, it is worth writing out.

An identification is a list. Each entry names two boundary edges of the pattern and a direction, and asserts that the two are the same edge of the paper.

The direction matters and it is one bit. Either the first edge’s start corresponds to the second’s start — a plain join — or it corresponds to the second’s end, which reverses the edge and is what makes a Möbius band rather than a cylinder.

Three constraints on a well-formed list. Each edge appears at most once. Both edges of a pair are boundary edges. And the two edges have the same length, since they are the same edge of the paper.

A reader of the file can then compute everything: which creases are one crease, which panels are one panel, the Euler characteristic, whether any identification reverses orientation, and the parity of every non-shrinkable loop.

Total addition: one list, three constraints, one bit per entry.

Two ways a format could get it wrong #

Worth flagging, since the obvious implementations have problems.

Merging the edges instead. A tempting shortcut is to delete one edge of each pair and reroute the other, so that the file contains the quotient directly. That loses the drawing: the resulting object has no rectangle in it, cannot be printed, cannot be cut, and cannot be folded by anybody.

Recording it as a transformation. Another temptation is to say this pattern is periodic with these translation vectors, which is true and is not the same thing. A periodicity is a property of the drawing; an identification is a decision about the paper. The same drawing can be periodic and cut out flat, and the file has to distinguish those.

Both mistakes come from confusing the sheet with the drawing, which is the confusion the whole phase is about. The right form keeps the rectangle intact and adds a statement about its boundary, which is exactly what the object is.

What a checker could then do #

The value of the field is that a tool can act on it, and the list of what a tool could then check is worth having.

Euler’s characteristic. Vertices minus creases plus panels, computed on the quotient, compared against the value the identification implies. The cheapest possible check that the file is well formed.

Orientability. Whether any identification reverses direction, and therefore whether mountain and valley are globally defined at all.

The parity of each non-shrinkable loop. An addition per loop, and it refuses half the crease counts on some patterns.

The vertex conditions. Unchanged, since the vertices are unchanged.

Four checks, three of which are impossible today because the tool cannot know the sheet is closed. Every one of them is arithmetic and none takes measurable time.

Where the sentence lives now #

In practice, on a scrap of paper.

Somebody making a folded tube has a flat pattern in a file and a note saying which edges to join. The note is in a document, an email, a comment, or in somebody’s head, and it does not travel with the pattern.

That is exactly the situation the interchange format was created to end for the drawing: before it, patterns travelled as pictures with instructions attached, and the format’s whole value is that the picture and its meaning arrive together.

So the gluing is in the position crease patterns were in before there was a format for them — carried alongside, lost in transit, and reconstructed by whoever receives it.

Which is a reasonable argument for the field, made in the terms the format’s own creation was argued in.

Why it is not there #

The ordinary reason, and it is the same reason the hypothesis is missing from the theorems.

Every object a format was designed for is a disc of paper. The formats were written by people exchanging crease patterns for models, the models are folded from flat sheets, and the identification is empty in every case.

A field that would be empty in every file is a field nobody adds, and the decision was correct at the time and remains correct for the files that exist.

What the notation tradition has always recorded #

It is worth putting the omission in the context of what folding notation has been for, because the answer explains the shape of the gap.

Folding notation began as instructions: a sequence of diagrams, each showing the paper’s state and the next move. Its unit is the step, it records what a person does, and it says nothing about the sheet beyond its outline.

The crease pattern is the later invention: publishing the pattern instead of the sequence, which records the object rather than the process. Its unit is the crease.

Both are notations for what happens on a piece of paper. Neither has ever had a way to say what the piece of paper is, beyond drawing its outline, because in both traditions the paper is a square and saying so would be saying nothing.

So the gap is not specific to the file format. It runs through the whole notational tradition of the subject, from diagram sequences to interchange formats, and it has the same cause everywhere.

The outline, which is recorded #

A qualification, since the essay has been saying the sheet is not recorded and part of it is.

The outline is recorded, as boundary edges. A square, a hexagon, a shape with a notch, a shape with a hole: each is a set of edges assigned as boundary, and any format can hold them.

So the sheet’s shape as a flat region is fine. A file can describe a sheet with three holes in it and a scalloped edge, exactly.

What cannot be recorded is which boundary points are identified, and that is the one piece of information that distinguishes a flat region from a closed surface.

That is a narrow gap and it is the narrowest possible gap that still loses the whole distinction between the objects people fold and the objects people build.

What a reader with a file should assume #

Practical advice, since files exist and will go on lacking the field.

Assume a disc. Every file describes a flat sheet with the outline its boundary edges give, and no identification. That is the correct reading of every file anybody has.

Look for the sentence elsewhere. If the pattern is for a tube, somebody knows which edges join, and the knowledge is in the surrounding material.

Check the parity by hand. If the object is closed, count the creases a loop round it crosses and require the count to be even. That is the one condition a closed sheet adds and no tool will do it.

Three habits, all of them things somebody working with folded tubes is already doing informally, and all of them things the field would automate.

What the collection does instead #

Says it in prose, which is what everybody does.

Every figure here of a glued sheet is a rectangle with a sentence in the caption. Every printed pattern is a rectangle with an instruction to join two edges. The identification lives in the surrounding text and not in the object.

That is workable for a collection of essays and it is not workable for anything mechanical. A tool cannot read a caption, and the whole value of a format is that a tool can read it.

The general shape of a format’s silences #

The collection has now found three things a crease-pattern file does not say, and they are worth listing because they have a common cause.

It does not record which of two objects a crossing is — one vertex of degree four or two creases passing over.

It does not record a verdict — whether the pattern folds is not a field, so a file cannot say what has been checked about it.

It does not record the sheet.

Each is a case of the format recording what is drawn rather than what is meant, which is the right design for a drawing and leaves the meaning to be supplied by whoever is reading.

For the first two that is tolerable. For the third it means the file describes four objects, and the four have different answers to the question the file exists to pose.

Why a field is better than a convention #

There is a workable alternative to adding a field, and it is worth saying why it is worse.

The convention would be: put the identification in a comment, or in the file’s name, or in an accompanying document, and agree that everybody reads it.

That is what happens now and it fails in the way conventions fail. A tool processing the file discards the comment. A file passed through two tools loses it. A file in an archive has it in a document that is somewhere else. And nothing checks that the convention was followed.

A field survives processing, travels with the object, and can be validated. Those three are the whole reason a format exists rather than a convention, and the argument for the field is the same argument that produced the format.

The counter-argument — that a field nobody uses is clutter — is real and it is answered by the field being empty by default. A file with no identification is a file describing a disc, which is every file today, and the field costs nothing until somebody needs it.

The other formats #

The interchange format is the one with a specification, and it is worth checking that the gap is general rather than particular to it.

Diagram sequences. A sequence of pictures with arrows. The paper’s shape is drawn, its identification is not, and a diagram of a taped tube would show the tape as a graphical convention rather than as data.

Crease pattern images. A picture, with colour or dash carrying the assignment. No structure at all beyond what a reader infers, so certainly no identification.

Solid-model exchange. A folded tube modelled as a three-dimensional solid does record the closure — implicitly, as geometry — and loses the crease pattern entirely. That is the opposite trade and it is why such models are useless for anything about folding.

Fabrication files. A cutting path and a scoring path, sent to a machine. The join is a separate operation with its own instructions, in a different file, and nothing connects them.

Four alternatives and none of them holds both the pattern and the identification. The solid model holds the closure and not the pattern; everything else holds the pattern and not the closure.

So the gap is structural across the whole tooling, and the reason is the same everywhere: the closure happens at a different stage of the process from the folding, and the files are organised by stage.

What this collection stores #

For transparency, since the collection has been computing with glued sheets and had to store them somehow.

A glued sheet here is a patch — an ordinary crease pattern on a rectangle, with boundary edges — together with a small record saying which pairs of its edges are identified and in which direction. The record is a field on the object rather than part of the pattern, and every existing computation reads the pattern and ignores it.

That arrangement is deliberate and it has one virtue worth naming: the patch remains a patch. It can be drawn, printed, cut out and folded, and the identification is an instruction rather than a change to it.

It also means the collection’s own storage has exactly the field this essay is asking a format to add, arrived at because there was no alternative. Building the machinery forced the design, and the design is the obvious one.

Which is a mild argument that the field is right: it is what anybody implementing this ends up with, rather than a proposal made in the abstract.

Two sentences #

A crease-pattern format records what is drawn on the paper and the paper’s outline, and both are features of a flat region.

A closed sheet is a flat region plus a statement about which of its boundary points are the same point, and there is nowhere in any format for that statement to go — so every file describing a folded tube describes the rectangle it was cut from, and the difference between those two objects is the whole of what this phase has been measuring.

The correction #

One list of edge pairs with directions, empty by default.

That is the whole of it, it breaks no existing file, it makes the four sheets distinguishable, and everything downstream — the counts, the characteristic, the orientability and the parity — is computable from it without any further additions.

The reason it has not been added is that until recently nobody had an object needing it, and now every folded structure that gets manufactured does.