The bottom layer is at the rim

Fold a sheet of paper flat and something is at the bottom. It is not an interesting question about most models — the bottom layer is whichever piece of paper ended up there — but it is a question with an answer, and the answer is computable from the letters alone.

Each crease says which of its two panels lies above the other. Those statements are a partial order on the panels, and a finite partial order has minimal elements: panels with nothing below them. Reading the crease list once gives them, the same way reading it once gives a proof when the order is impossible.

On patches of a twist tessellation carrying a periodic lettering, the answer is very lopsided and it is always the same.

The count #

A patch of nine periods of the square twist tessellation has a hundred and sixty-nine panels. Three of them have nothing below them.

At four periods, eighty-one panels and two of them. At one period, twenty-five panels and one. So the bottom of the stack is one panel in twenty-five, then one in forty, then one in fifty-six — shrinking as a fraction rather than growing.

The triangular tessellation gives two, three and four on sixty-nine, two hundred and thirty-three and four hundred and ninety-three panels. The honeycomb gives three, five and seven on the same panel counts. The elongated triangular tiling gives two, three and four on a hundred and five, three hundred and sixty-nine and seven hundred and ninety-three.

Where they are #

Every one of them touches the paper’s edge.

Not most. All of them, on four tilings, at three sizes, across twelve patches holding between twenty-five and seven hundred and ninety-three panels. A patch of nine periods of the honeycomb tessellation has four hundred and ninety-three panels, a hundred and nineteen of which touch the rim, and its seven minimal panels are seven of that hundred and nineteen.

The count of minimal panels grows with the rim rather than with the sheet. Panels go as the area — twenty-five, eighty-one, a hundred and sixty-nine on the square — and the minimal ones go as one, two, three. Rim panels go as sixteen, thirty-two, forty-eight, and the minimal ones are a small fixed fraction of those.

That is the shape of a quantity that lives on the boundary, and it is worth stating as such: the bottom of the stack is a rim phenomenon.

Three of a hundred and sixty-nine #

It is worth pausing on the largest square patch, because the ratio is easy to lose among the other numbers.

A hundred and sixty-nine panels of paper, folded flat. A hundred and sixty-six of them have at least one panel underneath, and many of them have a great deal underneath: the interior of a twist tessellation stacks deeply, with paper arriving from every direction. Three panels have nothing.

If the bottom of a stack were an ordinary property of a folded pattern one would expect it to scale — a bigger model, more paper at the bottom. It does the opposite. The interior grows as the square of the size and contributes nothing; only the rim contributes, and the rim grows linearly.

That is the whole shape of the finding in one sentence, and everything below is an account of why.

Why there is nothing in the middle #

The reason is short once the pattern is looked at as a pattern rather than as a patch.

The lettering these patches carry is periodic: it came from one period of the tessellation with its edges joined, and the relations it forces on the infinite sheet have a particular structure. Every loop among them travels — a chain of this above that comes back not to the panel it left but to a copy of that panel some periods away — which is the point on which the collection’s own test turns out to be a test for a disc. It is the same lettering that an exhausted search declared impossible, and its loops are the reason.

A travelling loop is an infinite ascending chain. Follow it one way and every panel has something above it; follow it the other and every panel has something below it. So on the pattern, no panel is minimal.

A panel in the interior of a patch has all its neighbours present, so it inherits exactly the relations it has on the pattern — including whatever sits below it. A panel at the rim has lost some of its neighbours to the cut, and it is only that loss that can leave it with nothing underneath.

The minimal panels are therefore precisely the ones whose below was cut away.

The measurement is a check, not an illustration #

This is a prediction with a definite failure mode, which is why it is worth measuring rather than arguing.

If a minimal panel ever turned up in the interior of a patch, one of two things would be wrong. Either the lettering is not what it is claimed to be — the transfer from the period onto the patch put a letter somewhere it does not belong — or the loops of the periodic pattern do not all travel after all, and there is a genuine bottom to the sheet.

Both would show, and neither does. Twelve patches, four tilings, three sizes, and the count of interior minimal panels is zero every time. The figure refuses to draw if it is not.

The same reading on the other tilings #

The square tessellation gives the tidiest numbers and the others give the same shape with different constants, which is worth having because a constant that is the same everywhere often means something has been miscounted.

The triangular tessellation: two minimal panels of sixty-nine at one period, three of two hundred and thirty-three at four, four of four hundred and ninety-three at nine.

The honeycomb: three, five and seven on the same panel counts, so nearly twice the triangular grid’s — because the honeycomb’s twist polygons are hexagons and its rim cuts more panels per unit of edge.

The elongated triangular tiling: two, three and four on a hundred and five, three hundred and sixty-nine and seven hundred and ninety-three, which is the largest patch here and has the smallest ratio of all.

Four tilings, four constants, one shape: the count grows like the rim and not like the sheet, and it is never zero on a patch and never positive in a patch’s interior.

What a folder actually holds #

A reader who prints one of these patches and folds it has a stack of paper with a bottom sheet, and the bottom sheet is at the edge of the model.

That is a slightly odd thing to be told and it is worth spelling out physically. In the middle of a twist tessellation the layers interleave: paper from every direction is stacked, and no piece of it is at the bottom of the pile because whatever is below it is the continuation of some panel that comes from further out. At the rim the interleaving stops, because there is nothing further out. So the outermost pieces of paper are the ones that can end up at the bottom, and they do.

Fold a bigger patch and the bottom moves further out with the rim. Fold an infinite one — which nobody can, and which the pattern is — and the bottom goes away entirely.

The count also settles a question that could otherwise be argued either way: whether the lopsidedness is about the lettering or about the pattern. If a different consistent lettering of the same patch had many minimal panels scattered through its interior, the finding would be about this particular choice of letters. Every lettering transferred from a period behaves the same way here, because every one of them has travelling loops — that is what being consistent on the joined sheet means — and travelling loops leave nothing minimal anywhere the cut has not reached.

What this does not make it #

It does not make the folded state ill-defined. Every point of the folded plane has finitely many layers over it, and those layers are ordered perfectly well by the relations among them; a reader looking at any particular spot of the folded sheet sees an ordinary stack from bottom to top.

What has no least element is the order over the whole pattern, and there is no contradiction in that. An order on infinitely many things need have no minimum; the integers do not. The folded sheet is a perfectly good folded sheet whose panels are ordered by a relation with no floor, which is a slightly unusual object and a completely consistent one.

The distinction between the layers at a point and the order on the sheet is one this collection has drawn before in a different setting: how many layers lie over a place is a local count, and which panel lies above which is a global relation, and the two need not have the same character.

Where the enumeration stops #

There is a practical consequence and it is the subject of the essay that asks what a stacking is when there is no bottom, but it belongs here in outline.

The machinery this collection uses to enumerate all the ways a pattern can be stacked works by building the order upward from the bottom: place a panel that has nothing below it, then a panel whose everything-below is already placed, and so on. That procedure needs a minimal element to start from, and on a pattern with none it has nothing to do.

So a periodic pattern is not merely expensive to enumerate stackings for; the enumeration is not defined on it. Something else has to take its place, and what takes its place is the question this thread has been circling: not which panel is at the bottom but in which direction do the layers climb.

An order can be perfectly definite and have no floor #

The idea of a stack of paper with no bottom sheet is worth sitting with, because it sounds like a paradox and is not one.

Consider a single travelling loop: panel A below the panel one period to its right, which is below the panel two periods to its right, and so on. Read the chain backwards and A has something below it, and that something has something below it, for ever. Every panel in the chain has a definite position relative to every other, and the chain has no first member.

Nothing about paper is violated. The relation lies below is transitive and never contradicts itself; the sheet is entirely stacked; there is simply no member of the chain that is under everything else. What would be impossible is a chain that came back to where it started, which is exactly the case the collection’s own test was built to catch and exactly the case that does not arise here.

The one thing that is lost is the ability to describe the folded state by listing its panels from the bottom up. That description is available for a disc and not for a periodic sheet, and every piece of machinery here that produced one was quietly relying on the sheet having an edge.

Reading a patch’s bottom layer honestly #

The general lesson is about which properties of a patch are properties of the pattern, and it is a short list.

The vertex conditions are a property of the pattern: every interior vertex of a patch is a vertex of the tessellation, asked the same questions.

The crease count per unit area is not, quite: a patch’s rim divides creases, so it reports slightly too many, and that accounting is exactly four per period of edge on the square tessellation.

The panel count is not: a patch reports the pieces its cut produced.

The bottom layer is not at all, and it is the extreme case. It exists only because of the cut, sits only on the cut, and vanishes when the cut is removed.

A number quoted for a tessellation should say which of those four kinds of quantity it is, and until this comparison could be made there was no way to tell.

A quantity nobody should have quoted #

There is a small piece of housekeeping in this, and it is the kind worth doing out loud.

Nothing in this collection ever published a bottom layer for a tessellation, so no number has to be withdrawn. But the shape of the error was available: a quantity read off a patch, reported as a property of the pattern, when it is a property of the cut. The collection has caught that shape before — the counts a patch reports move when the pitch moves it across a threshold — and it will catch it again, because a patch is the only thing most of the machinery can read and the temptation to report what it says is constant.

The test that separates the two kinds of quantity is now available and is cheap: measure it on a patch, measure it on the same rectangle joined up, and see whether it survives. Vertex counts survive. Crease and panel counts do not, by an amount that can be written down. The bottom layer does not survive at all, which makes it the clearest case in the collection of a number that belongs to the paper rather than to the pattern.

Which theorem was checked, and how #

The order is read off a folded state rather than off the pattern. Each panel is placed by composing reflections outward from a starting panel, the placement is checked for consistency — two routes to one panel must agree about where it is, and here they agree to within about one part in ten thousand million million — and each crease then reports which of its two panels the fold puts above the other.

Whether a panel is at the rim is decided geometrically, by asking whether any corner of it lies on the boundary of the patch, rather than by anything the search knows. So the two halves of the claim — which panels are minimal and which panels are at the edge — are established by machinery with nothing in common, and their agreement is evidence rather than bookkeeping.

The letters themselves come from the period and are transferred by reduction: each crease of the patch is moved by whole periods until it lands in the fundamental rectangle and takes the letter of the piece it lands on. A crease the reduction cannot place is counted, and the count is zero on every patch here.

Why the loops have to travel #

The claim that no panel of the pattern is minimal rests on every loop in its relations travelling rather than closing, and that is established rather than assumed.

The relations are taken apart one direction at a time. Choose a bearing on which no loop descends; every relation with room to spare under it can be discarded, since no chain that returns to where it started could have used it. What is left falls into smaller pieces and the next bearing is asked of those.

On the triangular tessellation’s smallest cell it takes three bearings to empty the relations entirely, and on the two-period square cell it takes two. A cell that could not be emptied would be reported as unsettled rather than as fine, and none here is.

What the picture cannot show #

A bar counting minimal panels does not show which panels they are, and the interesting part is that they are always at the edge. A drawing of a patch with its minimal panels shaded would show it directly and would be a drawing of one patch; the claim is about twelve, so the figure counts instead and refuses when the count is wrong.

Nor can a picture show the absence the essay is really about. A pattern with no bottom layer looks exactly like a pattern; the missing floor is a property of an order over infinitely many panels, and every picture here has finitely many in it.

And no measurement here says how deep the stack is over any particular point, which is a different quantity and one the collection measures elsewhere. A pattern can have no bottom layer overall and still be six sheets thick everywhere, and this one is: the layers over a point are finite and ordinary, and it is only the relation between panels far apart that has no floor. How much smaller a folded sheet gets and how many layers lie over a place are questions a patch answers honestly, and the bottom layer is not.