A bottom layer on half a rim

Fold a sheet of paper flat and there is a bottom layer: a panel with nothing underneath it, the one touching the table. Every enumeration of stackings in this collection starts there and works upward, and the move is so natural that it took a sheet with no edge to notice it was an assumption.

The bottom of a stack lives at the rim — measured on patch after patch, the panels with nothing below them are always the ones touching the paper’s boundary. And a sheet with no rim has an order with no least element, which is a perfectly good order and not one an enumeration can start from.

Those are the two ends of a scale, and the middle is a cylinder.

The measurement #

Take a rectangle of a repeating pattern and its forced order — the relations the letters compel, this panel lies below that one — and count the panels with nothing below them.

On a patch of one, four, nine and sixteen cells the counts are one, two, three and four, and every one of those panels is at the rim.

Glue one pair of the rectangle’s edges and half the rim goes. The panels that were minimal because they were at that edge are no longer at an edge, and the count falls.

Glue both and it reaches nought, which is the case with no least element at all.

Why the bottom is at the edge #

The reason is worth having, because it explains why the fact is a fact rather than a coincidence of the patterns measured.

A panel is below another when the paper between them forces it: two panels overlapping in the folded image, joined by a chain of creases whose letters put one under the other. A panel in the middle of the sheet has creases on every side, so it is under something in every direction, and the chances of it being under nothing are poor.

A panel at the rim has creases on fewer sides. In the limit — a panel with a single crease and the rest of its boundary raw edge — there is exactly one relation constraining it, and it can perfectly well be the bottom.

So at the rim is not a location so much as a description of having fewer constraints, and the rim is where constraints stop.

What half a rim leaves #

A cylinder has two circles of rim rather than four sides, so about half the panels that were at the boundary still are.

The consequences run in two directions at once, and they are worth separating.

The order still has a least element. There are panels with nothing below them, they are on the rim that is left, and an enumeration can start from one of them. Everything the collection’s stacking machinery does continues to work.

The count of candidates falls. Fewer panels are eligible to be the bottom, which means the enumeration branches less at its first step — and more at every later one, because the freedom the rim was supplying has gone.

That is the same trade the letters make: removing the rim removes choices that could not be wrong, and what remains has to be right.

What replaces the bottom #

On the sheet with no rim at all, the question which panel is at the bottom has no answer and a different question does.

The layers climb. Walking upward through the stack, the paper returns to the same panel of the pattern one cell over, and goes on. So there is no bottom and no top; there is a direction of climb, and it is a well-defined property of the folded state.

A cylinder has both. It has a bottom, because it has a rim; and in the glued direction it has a climb, because a chain of relations can return to a panel one cell round. Which is a genuinely intermediate structure and not merely a smaller version of either end.

The enumeration on a cylinder #

Whether the collection’s stacking enumeration works on a cylinder is a question with a slightly awkward answer, and it is worth being straight about it.

It starts, because there is a least element.

It does not finish correctly without a change, because the relations on a glued sheet are not a partial order on the cell’s panels. A chain returning to a panel one cell round is not a contradiction, and an enumeration checking for cycles will reject valid states — which is exactly the error the consistency test made, in a different piece of machinery.

So the enumeration needs the same repair the consistency test needed: read the lattice step, and treat a chain that returns one cell over as a climb rather than a loop. That has been done for the lettering search and not for the stacking enumeration.

Doing it with paper #

The claim that the bottom of a stack is at the edge is easy to check and slightly surprising when checked.

Fold anything flat — a Miura from a rectangle is convenient, or a simple pleated fan. Put it on a table and look at which panel is touching the table.

It is at the edge of the original sheet. Not always the very corner, but always a panel one of whose sides was a raw edge of the paper before folding.

Now try to make an interior panel the bottom one, by refolding. It can be done for a particular panel if the pattern allows enough freedom — but each time, the panel that ends up on the table is one that has an edge, and forcing an interior panel down requires the layers around it to move out of the way, which the creases usually forbid.

What the experiment does not show is the general claim, which is that every minimal panel is at the rim rather than that the one on the table happens to be. That is what the measurement is for: on patches of one, four, nine and sixteen cells, the panels with nothing below them number one, two, three and four, and every one of them touches the boundary.

A cylinder, in the hand #

The middle case is also foldable and worth doing.

Take a corrugation — a Miura or a simple zigzag — and roll it into a tube, taping the two ends of the strip together. The tube’s two open ends are its remaining rim.

Collapse it. The flattened tube has a bottom layer: a panel touching the table, with nothing under it, and that panel is at one of the two open ends.

Now look at the middle of the tube, away from both ends. There is no panel there with nothing below it, and there could not be: the paper wraps round, so every panel in the middle is under something in every direction.

So the structure is exactly as measured. A bottom exists, it is at the rim that remains, and the interior has the climbing behaviour that a torus has everywhere.

What a minimal panel is, exactly #

Since the whole essay counts them, it is worth defining the thing being counted.

A folded state induces relations between panels: where two panels overlap in the folded image, one of them is above the other, and which is determined by the letters on the creases between them. Not every pair of panels overlaps, so the relations are partial.

A minimal panel is one with nothing below it: no panel that overlaps it and lies underneath. On a folded sheet lying on a table, the panels touching the table are exactly these.

There can be more than one, and usually is. A folded state whose panels fall into two stacks — two separate piles of paper joined by a bridge — has a minimal panel in each pile, and neither is the bottom.

That is why the count is a count rather than a yes or no, and it is why the measurement is informative: one, two, three and four on patches of increasing size is a trend, and there is a bottom would not have been.

The count against the patch’s size #

The numbers deserve a second look because the scaling says something.

Patches of one, four, nine and sixteen cells have one, two, three and four minimal panels. That is the linear size of the patch — its side in cells — rather than its area.

The rim’s length is also linear in the patch’s side. So the count of minimal panels tracks the amount of rim, which is what one would expect if the rim is what makes a panel minimal, and is a check on the explanation rather than merely a consistent observation.

It also means the share of panels that are minimal falls like one over the side, which is the same behaviour every boundary quantity has here: a small patch is mostly rim and a large one is mostly interior.

Extrapolating, a patch of a hundred cells would have about ten minimal panels out of a hundred and twenty-one — a twelfth of them — and a torus of any size has none.

Where the last of the rim goes #

The scale has three rungs and it is worth being precise about what happens at each.

Four edges. Minimal panels number about the linear size of the patch, all at the boundary. The enumeration starts anywhere among them and the order is an ordinary finite partial order.

Two edges. Minimal panels number about half as many, all at the two circles of rim that remain. The order is still a partial order on the cell’s panels in the unglued direction and is a climbing structure in the glued one.

No edges. No minimal panels at all. The relations are a partial order on an infinite cover, the order has no least element, and every chain climbs.

The discontinuity is between the last two rather than the first two, which is the same shape the search costs show: having some rim is much closer to having all of it than to having none.

An order that climbs, described #

The structure that replaces a bottom deserves a description, since the layers climb is a phrase rather than a definition.

On the cover — infinitely many copies of the cell, stacked — the relations are an honest partial order. Take any panel and follow the relations upward and the chain passes through copies of the same few panels, one lattice step further along each time it comes round.

So the order is periodic: it looks the same from every copy, and it has no least element for the same reason the integers do not.

The direction of climb is the lattice step that a full circuit takes. It is a vector, it is the folded sheet’s own lattice vector, and it is computable from the folded state rather than being a description of it.

What that gives is a replacement for the enumeration’s starting point: instead of begin at the bottom, one begins at any panel and enumerates up to the climb, treating two arrangements differing by a full circuit as one arrangement. That is the change the stacking enumeration would need, and it is the same change the consistency test already made.

Two things this does not say #

It does not say a patch’s stacking is easy. Enumerating the stacked states of a folded pattern is the hard half of this subject and it stays hard. Having a place to start is not the same as being able to finish, and the layer ordering is where the theorems run out.

It does not say the minimal panels are the only interesting ones. The bottom of a stack is where an enumeration starts because it has to start somewhere; nothing about the folded object privileges it. A folded state read from the other side has a different bottom, and both readings are the same state.

That second point is sharpened by the sheets in this phase. On a Möbius band there is no consistent notion of which side is down at all, so the bottom layer is not a property even where there is a rim — the letters go the same way, and for the same reason.

So the bottom is an artefact of how a folded state is described rather than a feature of it, and the reason it matters is that every description this collection has is built on one.

What the collection would have to change #

For the record, since the essay identifies a gap rather than closing it.

The lettering search has been repaired: its consistency test reads the lattice steps, and a chain returning one cell over is a climb rather than a contradiction. That was the expensive lesson.

The stacking enumeration has not. It starts from a minimal panel, builds a linear order, and rejects on any cycle. On a cylinder it starts correctly and rejects valid states; on a torus it does not start.

Repairing it means the same change in a different place: carry a lattice step on every relation, and treat a closed chain with nonzero total step as a climb. That is not hard and it has not been done.

Until it is, the collection can say which letterings a glued sheet admits and not how many stacked states each of them has — which is the difference between counting the letterings and counting the folded states, and the second is the harder half everywhere.

The habit worth noticing #

Two pieces of machinery in this collection assumed a sheet with an edge, and both assumed it in the same way: by relying on a finite structure having a least element.

The consistency test assumed the relations have no cycles, which is true of a partial order and true because the paper has a bottom.

The stacking enumeration assumed there is a panel with nothing below it, which is the same assumption stated directly.

Both are correct for a disc, both are false for a torus, and both were written without the sentence naming the sheet — for the ordinary reason that the sentence would have said nothing.

That is now three instances of the same shape in this collection, counting the closure condition comparing against the identity. Three is enough to suggest a fourth, and the place to look is anywhere a computation walks outward from somewhere and stops.

What the measurement establishes #

Three things.

The bottom is at the rim, still. On every patch measured, all the minimal panels touch the boundary, at every size and on every tiling.

The count scales with the rim. One, two, three and four minimal panels on patches of one, four, nine and sixteen cells — which is the linear size of the patch rather than its area, and the rim’s length is linear too.

And there is a middle case. A cylinder has a bottom, and it has fewer candidates for it, and it also has a direction of climb in the glued direction. The two structures coexist on one sheet, which was not obvious before there was a sheet on which they could.

Why an order needs a bottom to be enumerated #

The general point is about enumeration rather than about paper, and it is short.

Listing the elements of a finite partial order in a linear extension is done by repeatedly taking a minimal element. That works because a finite partial order always has one.

The relations on a glued sheet are a partial order on the cover — infinitely many copies of the cell, stacked — and an infinite partial order need not have a minimal element. The integers under the usual order do not.

So the enumeration’s basic move fails not because the object is strange but because the object is infinite in a direction the enumeration assumed it was finite in. A cylinder is finite in one direction and infinite in the other, which is why it keeps the bottom and loses the tidiness.