Two panels that are one panel

Paper cannot pass through paper is the condition the theorems do not reach. Every vertex condition can hold, every panel can be where the folding says it should be, and the folded state can still be impossible because two pieces of the sheet would have to occupy the same place.

Testing for it is straightforward in principle: take every pair of panels, see whether their folded images overlap, and if they do, check that the layer order puts one cleanly above the other.

On a glued sheet every pair of panels is not a well-defined phrase, and getting it wrong produces an error in one of two directions.

Pieces and panels #

A glued cell is a rectangle with an identification. The rectangle contains pieces of paper — the regions the creases and the rectangle’s edges cut it into — and the sheet has panels, which are the classes of those pieces under the identification.

A two-period square twist cell has twenty-five pieces and sixteen panels: nine of the pieces are second copies of panels whose other copy is elsewhere in the rectangle.

A collision test comparing pieces is comparing the wrong things.

The first error: a false collision #

Two pieces of the same panel are the same paper. They cannot collide with each other, because they are not two objects.

A test that compares pieces pairwise will nonetheless find them overlapping — a panel at the left edge and its partner at the right have the same folded image, since they are the same panel — and report a collision between a piece of paper and itself.

That is a false positive, it scales with the identification, and it would refuse every glued sheet.

The second error: a missed one #

The other direction is subtler and it is the one that matters.

On a glued sheet a panel can meet itself. Walking round the loop, the paper comes back over its own starting position — that is what the folded sheet’s own lattice vector measures — and if the vector is short enough, the panel’s image at one circuit overlaps its image at the next.

A test comparing distinct panels never asks that question, because it compares pairs and a panel is not a pair.

So the naive test refuses sheets that are fine and accepts sheets that are not, which is the least useful pair of errors available.

What the correct test asks #

Two questions rather than one, and the second is the new one.

For each pair of distinct panels, do their images overlap, and if so does the layer order separate them cleanly? That is the ordinary question, asked of panels rather than of pieces.

For each panel and each nonzero lattice step, does the panel’s image overlap its own image translated by that step, and if so does the order separate them? That is the question a sheet with a loop poses and a disc does not.

The second question has infinitely many instances in principle and finitely many in practice, since the lattice vector has a length and the panels have diameters, so only steps up to a bound can produce an overlap.

A tube meeting itself #

The second error is easier to believe with a physical object, and a tube supplies one.

Roll a sheet into a tube and collapse it flat. The paper that was at the left edge of the sheet is now lying on the paper that was at the right edge — they are adjacent through the seam — and the two are different parts of the sheet, so that is an ordinary contact between distinct panels.

Now take a longer sheet and roll it round twice before joining, so the tube has two layers everywhere. Collapse. Every point of the paper now has another point of the same sheet directly above it, one circuit along.

That second case is what a glued cell’s lattice vector describes: the folded sheet repeats, and each repetition lies over the last. Whether the paper passes through itself in doing so is a real question, and it is a question about a panel and its own translate.

The reason it does not usually arise in folding is that a physical tube of paper has thickness and simply stacks. The mathematical version — zero thickness, panels in the plane — has the panels genuinely coincident, and whether that is allowed depends on the order rather than on the geometry.

What cleanly separated means when the stack climbs #

The phrase is doing work above and it has to be redefined for a sheet with no bottom.

On a disc, two overlapping panels are cleanly separated when the layer order puts one strictly above the other and nothing between them is in the way. The order is a finite list and above is unambiguous.

On a glued sheet the order climbs: following it upward returns to the same panel one lattice step along, and continues. So there is no global above; there is a relation between panels together with a step, and the sensible statement is that a panel and its own translate are separated by however many circuits lie between them.

That is well defined and it is not a list. It is closer to a helix than to a stack, and the machinery for reasoning about helices is not the machinery for reasoning about stacks.

Which is the whole of why the collision test has not been extended: the change is not a bigger loop, it is a different notion of above.

The count of pairs #

Worth quantifying, since infinitely many images sounds worse than it is.

A panel has a diameter. The lattice vector has a length. Two images of a panel, $k$ steps apart, are $k$ times the vector apart, so they can only overlap while $k$ times the length is less than the diameter — which bounds $k$ by the ratio.

On the twist tessellations’ cells the folded lattice vector runs to about a seventh of a sheet width per period and the panels are much smaller than that, so the bound is one or two. In practice the check is a panel against its own neighbour translate and nothing further.

So the second question adds a handful of comparisons rather than an unbounded number, and the difficulty is in stating it rather than in performing it.

What a patch measurement was measuring #

There is a consequence for the collection’s existing self-intersection work, and it is worth being clear about.

Every collision test here has been run on a patch — a rectangle cut out of the plane, with a boundary. On a patch, pieces and panels are the same thing, there are no lattice steps, and the pairwise loop is exactly right.

So the existing measurements are correct measurements of patches. They are not measurements of tessellations, and a patch’s collisions and a tessellation’s are not the same set: a patch has boundary panels with fewer neighbours, and it has no repetition, so it cannot exhibit the panel-meets-itself case at all.

That is the same relationship every quantity in this phase has turned out to have. A patch is a sample with a rim, the rim changes the counts, and the property measured on it is a property of the sample.

For collisions specifically the bias has a direction: a patch has fewer collisions than the pattern, because the repetition that would produce the extra ones has been cut away.

Why the order and the collision are one problem #

A last observation about why these two pieces of machinery keep appearing together.

A collision is not a geometric fact by itself. Two panels occupying the same place in the folded image is fine if the layer order separates them; it is a contradiction only if the order does not, or if the order required to separate them contradicts something else.

So a collision is an order: the geometry says which pairs must be ordered and the ordering says whether they can be. The two are one problem, and any change to how the order works is a change to how collisions are tested.

Which is why a sheet with no bottom breaks both at once, and why repairing one without the other would not help.

Why this is the same shape as the order #

The collection has met this once already, in a different piece of machinery.

The layer order on a glued sheet has no least element, because the stack repeats and there is no bottom. The enumeration that builds an order from the bottom upward does not start.

The collision test is the same situation one level down. It assumes a finite set of panels with finitely many pairs, and a glued sheet has finitely many panels with infinitely many images.

Both are cases of a computation assuming a sheet with a boundary, and both were written without the sentence naming it — which is by now the collection’s most repeated finding.

The two errors, side by side #

Setting them out plainly, since the essay’s content is that a naive test fails in both directions at once.

False collision. Two drawn pieces belonging to one panel have the same folded image and are reported as overlapping. They are the same paper. The test refuses a sheet that is fine.

On a two-period square twist cell there are nine such pieces out of twenty-five, so the false collisions are not rare — they are most of what the test would find.

Missed collision. One panel appears at every lattice step, and the test never compares a panel with itself. The test accepts a sheet where the paper passes through itself one circuit along.

Neither error is a matter of degree. The first fires on every glued sheet with any identification at all; the second is invisible on every one.

A test with a false-positive rate that high would be noticed immediately, which is a small mercy: running the patch test on a glued sheet does not produce a plausible wrong answer, it produces an obviously wrong one. That is the better of the two ways to be broken.

Where a real folded tube sits #

Since one of these objects is manufactured, it is worth asking what happens physically.

A folded tube collapsed flat has its paper lying over itself, many layers, and no layer passes through another — the material simply stacks, because it has thickness and the layers slide.

The mathematical condition is about zero-thickness paper in the plane, where stacking means an order rather than a separation in space. A real tube’s layers are separated by their thickness; an ideal tube’s are separated by the order alone.

So the physical object never exhibits the failure, and the question the collision test asks is whether an ideal folded state exists — which is the right question for the mathematics and is not what a manufacturer is checking.

That gap is the oldest idealisation on the list and it is why the collision condition is a mathematical condition rather than an engineering one. It decides whether a folded state exists; thickness decides whether the object closes.

What has and has not been done #

The lettering search on glued sheets is correct: its consistency test reads the lattice steps, so a chain of relations returning one cell over is a climb rather than a contradiction.

The collision test has not been extended. The collection’s self-intersection machinery compares panels of a patch, and on a glued sheet it is not run at all — every glued measurement here is about letterings rather than about folded states, and a lettering is a necessary condition rather than a folded state.

So the honest position is that this collection can say which glued sheets admit consistent letterings and cannot say which admit folded states, and the gap is exactly the collision test.

What it would take #

Not much conceptually and a real amount of work.

The pairwise loop becomes a loop over panel classes and lattice steps, bounded by the panels’ diameters against the lattice vector’s length. The layer order becomes a relation that may climb, so cleanly separated has to be stated in terms that survive there.

Both changes are the same change the consistency test needed, applied in a second place, and the reason it has not been made is that nothing yet needs it: the phase’s questions have all been about letterings.

Recorded as owed.

What the identification has to supply #

For anybody extending the test, the data required is short and it is all already computed.

A map from pieces to panels. Which drawn regions are the same paper. The identification produces it as a union-find and the collection stores it.

A lattice step per identification. How far a panel’s partner is, in cells. Also stored, and it is the same step the consistency test reads.

The folded lattice vector. Where a panel’s image goes when the paper advances one cell. Computed already, since it is what distinguishes a gluing that slides from one that turns.

Given those three, the corrected test is: loop over unordered pairs of panel classes and over lattice steps within the bound; compute the two images; test overlap; consult the order.

Every ingredient exists. What does not exist is the notion of above on a climbing order, which is the piece that has to be invented rather than assembled.

One case that is already decided #

There is a family of glued sheets where the second question has an easy answer, and it is worth noting because it covers the objects people build.

If the folded lattice vector is longer than any panel’s diameter, no panel can overlap its own translate at any step, and the second question is vacuous. Every collision is between distinct panels and the ordinary test — asked of panels rather than pieces — is complete.

That is the situation on a tube whose collapsed form is long compared with its facets, which is most manufactured folded tubes: a boom collapses to a stack much longer than one facet, and the paper an inch along is not the paper at the start.

So for the practical objects, only the first correction is needed — compare panels rather than pieces — and the first correction is one line.

The second question bites on short cells, where the folded lattice vector is comparable with the panels, and those are the mathematical objects rather than the manufactured ones. Which is a fair division of labour between the two halves of this collection.

Why the phase stopped short of it #

The honest reason, since the essay records a gap rather than closing one.

This phase’s question was what a boundary costs — measured in free letters, panels, Euler’s number and search cost. Every one of those is about letterings, and a lettering is decided by the vertex conditions and the consistency test, neither of which needs a collision check.

A folded state is a stronger object: a lettering plus a layer order that separates every collision. Getting to those on glued sheets means the ordering machinery and the collision machinery both, and each needs the same conceptual change.

So the phase measured what it could measure exactly and left the rest stated. That is the right division and it means a reader should read every glued result here as being about letterings: which letterings are consistent, how many nodes finding one costs, and whether one exists at all.

Whether the corresponding folded states exist is not answered anywhere in it.

The sentence worth keeping #

A collision test compares pieces of paper, and on a glued sheet the pieces of the drawing are not the pieces of paper.

Two drawn regions can be one panel, in which case they cannot collide. One panel can appear in many places, in which case it can collide with itself. Neither situation exists on a disc, which is why the ordinary test is written as a loop over pairs and why the loop is the wrong loop.