The seam that is not a symmetry

A gluing is an instruction: these boundary points and those are the same point of the paper. It is not a fact about the drawing, and the drawing has no field in which to record it.

But the drawing has to agree with it, and the agreement is a real condition that most rectangles fail.

What the drawing has to do #

A crease running off the right-hand edge of the rectangle has to arrive at the left-hand edge, at the same height, as the continuation of itself. If it does not — if the crease leaving at one height meets nothing at that height on the other side — then identifying the two edges produces a crease that stops in the middle of the paper, which is not a crease pattern at all.

So the rectangle has to be a whole number of the pattern’s own periods in the glued direction. Two periods, or five; not two and a half.

That is the only condition, and it is why every cell in this collection is a period rather than a convenient size.

What the construction refuses #

Asked for a cell that is not a period, the construction does not produce a broken sheet. It refuses, and it refuses for a stated reason: the identification would be matching creases that are not the same crease.

There is a second condition, weaker and easier to miss, and it is about where the rectangle sits rather than how big it is. The edges have to miss every vertex, so that a crease crossing an edge does so at an ordinary interior point of itself. That is achieved by searching for a corner position, and it succeeds on every tiling here.

The case the search cannot see is a crease running exactly through a corner of the rectangle, where four edges meet and a piece ending there has no partner on any one of them. That is not a hypothetical: it turned up, produced one unmatched piece per glued pair, and was found by Euler’s number coming out at minus one.

The rectangle that will not close, in detail #

It helps to see exactly how a non-period rectangle fails, because the failure is specific rather than general untidiness.

Take the square twist tessellation and cut a rectangle one and a half periods wide. Its left and right edges are one and a half periods apart, so a crease leaving the right edge at some height is not the same crease as the one arriving at the left edge at that height — it is half a period out of step.

Identify them anyway and two things go wrong at once. Crease pieces on one edge find no partner at the matching height, so the pattern acquires creases with a loose end in the middle of the paper. And panels touching one edge are joined to panels that are not their continuations, so the folded state has two routes to one panel that disagree by half a cell.

The construction detects the second before the first. Every glued cell is folded flat and the disagreement between two routes to any panel is measured; a cell that does not close reports a distance rather than a verdict, and a distance of half a cell is not a rounding error.

So the refusal has a size, which is more useful than a refusal without one: it says the rectangle is out of step and by how much.

Two conditions on the corner, not one #

The corner position has to satisfy two things and it is worth separating them, since one of them is checked and the other was not until recently.

The edges must miss every vertex. A crease crossing an edge has to do so at an ordinary interior point of itself, so that identifying the two crossings joins one crease to itself rather than joining something at a vertex. That is what the corner search is for: it chooses each coordinate to fall in the largest gap between the drawing’s features in that direction.

The corners must miss every crease. A corner is where four edges meet, and a crease piece ending there has no partner on any one of them.

The second does not follow from the first, and the reason is that the search chooses the two coordinates independently. A crease can be far from every vertex in both coordinates separately and still pass through the point where two edges cross.

The repair is a nudge: slide the corner along the gap the vertex search already cleared, until nothing runs within a thousandth of a period of any of the four corners. That cannot move a vertex onto an edge, because the gap was chosen clear of vertices to begin with.

What is not required #

Three things a gluing does not need, each of which sounds as though it might.

The drawing need not be symmetric under the identification in any stronger sense. It needs the translation to be a symmetry, and nothing else. In particular the two glued edges need not look alike to the eye: what matters is that the drawing at one is the drawing at the other, shifted.

The two directions need not have the same period. Most of these tilings have different periods across and along, and the rectangle is a period in each direction independently.

The pattern need not be flat-foldable. The gluing is an operation on a sheet with a drawing on it, and whether the drawing folds is a separate question asked afterwards. A drawing that does not fold glues perfectly well and produces a sheet that does not fold.

That last one is worth stating because it separates two things that are easy to run together: the construction produces the object, and the object is then asked whether it folds. A refusal from the construction and a refusal from the fold are different refusals with different causes.

Why this matters for reading the counts #

Every count in this collection’s tessellation work is per rectangle, and the rectangle is a choice with two arbitrary elements in it.

The size is a whole number of periods, chosen by whoever asked. Doubling it doubles most counts.

The lattice is a rectangular sublattice of the pattern’s own, chosen because rectangles are easy to clip to. On three tilings it is index two, so those counts are twice what the smallest cell would give.

Neither affects any conclusion here, because every conclusion is a comparison between sheets built from the same rectangle. But a reader taking a number out of a table and using it as a property of the tessellation should know that it is a property of a rectangle somebody chose.

The one number that is not affected is Euler’s characteristic, which depends only on the sheet’s shape and is the same whatever rectangle is used. That is part of why it is the check.

Why a gluing is not a symmetry #

The distinction is worth insisting on because a gluing looks like one.

A symmetry of a drawing is a motion of the plane that carries the drawing onto itself. A tessellation has a lattice of translation symmetries, and the period rectangle is a cell of that lattice.

A gluing is an identification of one sheet’s boundary points with others. The sheet it produces is a different object from the plane the drawing was on, with a different shape and different conditions.

The reason the two are easy to confuse is that the gluing is only possible when the corresponding translation is a symmetry. So the symmetry is a precondition for the gluing rather than the same thing as it, and the sheets they describe are different: the plane is infinite and simply connected, and a torus is finite with two loops that cannot be shrunk.

The folded period, which is a second condition #

A gluing has to respect not only the drawing but the drawing’s folded state, and those turn out to be different requirements.

The folded state of a periodic pattern is periodic, and its period need not be the drawing’s. Folding the Yoshimura flat carries one drawn column onto the next by a turn of two hundred and forty degrees, so its folded period is three drawn columns, and a cell of one or two columns cannot be glued even though it is a perfectly good period of the drawing.

The twists do not have this problem: their folded states slide rather than turning, at every size and on every tiling, so their folded period equals their drawn period and every cell size glues.

That is a fact about the family rather than a general one, and it is the reason the whole gluing construction was built on the twists.

Which twist cylinders exist #

Putting the conditions together, the answer for this family is generous.

Every whole number of periods in either direction glues, on the square, triangular, hexagonal, elongated and rhombille tilings.

Every one of them is orientable — a path round any cell crosses an even number of creases, because a twist polygon’s pleats come in pairs — so none is refused by the parity that refuses half the grid’s cells.

Every one of them slides, so none is refused by a turn.

The only refusals are the ones about construction: a cell asked for beyond the region the drawing was generated over, and a corner with a crease through it.

The rectangle is a choice, and a coarse one #

There is a caveat that applies to every count in this collection’s tessellation work and it belongs here.

The pattern repeats under a whole lattice of translations, and the rectangle used is one cell of one convenient sublattice of it. On three of the five tilings the natural lattice is generated by steps at sixty degrees, whose smallest cell is a rhombus, and the rectangle used is twice as much paper and exactly as periodic.

That costs a factor of two in every count on those tilings and buys clipping to a rectangle rather than to a rhombus, which every piece of machinery here expects. It is a good trade and it means the counts are per rectangle rather than per smallest cell.

Nothing about the sheets depends on it — the checks hold at one period, four and nine alike — and a reader wanting the smallest description of a triangular twist tessellation will not find it here, for reasons of clipping rather than of paper.

The conditions, as a list #

For a reader wanting to know whether a particular rectangle of a particular drawing can be glued, the whole test is four questions.

Is the rectangle a whole number of periods in the direction being glued? If not, the creases do not meet and the construction refuses.

Do the rectangle’s edges miss every vertex? A corner search settles this and it has succeeded on every drawing tried.

Do the rectangle’s corners miss every crease? A separate condition, not implied by the previous one, and the one that failed.

Is the rectangle a whole number of folded periods? For most families this is the same as the first question. For the Yoshimura it is three times stricter, and for any family whose fold turns rather than slides it will be stricter by whatever the turn’s order is.

Pass all four and the gluing produces a sheet. Whether that sheet folds is then a question for the parity, the vertex conditions and the search, in that order — and those are about the pattern rather than about the identification.

Gluing something that does not repeat #

The essay has been about tessellations, and the natural question is what happens to everything else.

Nothing. A drawing that does not repeat has no period, so no rectangle of it can be glued in any direction, and the four sheets are not available.

That covers a great deal: a crumple, a design’s crease pattern, a random pattern on a grid, the four ways this collection draws patterns for its populations. For all of those the sheet is a disc and there is no alternative.

The exception that is worth noticing is a pattern that repeats in one direction only. A corrugation drawn as a finite number of courses repeats along the courses and not across them, so it glues one way and not the other — giving one cylinder and no torus.

That is a genuine object and the collection has not built one, because every family here repeats in both directions or in neither. It is the natural next case and it is the shape most manufactured folded tubes actually have: a finite number of columns round the tube, and an indefinite number of courses along it.

The conditions, in the collection’s own terms #

Each of the four checks corresponds to something already established here.

The period condition is what cutting a patch out of a plane has always required, stated as a gluing rather than as a clip.

The corner condition is what Euler’s number found.

The folded-period condition is the Yoshimura’s turn, and it is the newest of the four.

And every cell that passes them is an object the collection’s search machinery can read unchanged.

What a folder does instead #

None of the conditions above is available to somebody with a strip of paper and some tape, and it is worth saying what they do instead, because it is a different and equally valid procedure.

A folder rolls the pattern round and looks at whether the creases line up. If they do, the tape goes on. If they do not, the strip is trimmed until they do.

That is the same condition arrived at empirically: the creases line up is exactly the rectangle is a whole number of periods, and trimming to make them line up is finding a period by search. It works, it is fast, and it needs no arithmetic.

What it cannot do is the second and fourth conditions. The corner condition is invisible on a physical tube — there is no corner, since the paper is joined all the way along — and it is an artefact of describing the sheet with a rectangle. And the folded period condition is not something a folder would notice before folding, since it is about the folded state.

So the physical procedure catches the condition that matters most and misses two that only exist because the object is being described rather than made. That is a fair summary of the relationship between the two ways of working throughout this collection.

A note on the word #

Seam has been used throughout for the line where two edges are identified, and it is worth saying that the word is a convenience rather than a description.

On a physical tube there is a seam: a line of tape or glue, visible, slightly stiffer than the paper, and a real feature of the object. Everything about the parity condition treats it as an ordinary line of the paper, and on a real tube it is not quite one.

On a torus there is no seam at all. The identification joins crease pieces at ordinary interior points, leaves no mark on the drawing, and the sheet is homogeneous — every point of it looks like every other point, and the rectangle’s edges are an artefact of how it was described rather than a feature of it.

So seam names the line on the drawing rather than a feature of the sheet, and the sheet has no such feature. The collection uses the word because the drawings have to show something, and a reader should hold it as a label on a picture rather than as a part of the object.

What the seam is, once more #

Three sentences, since the whole essay is about not confusing three things.

A period is a translation carrying the drawing onto itself. It is a property of the pattern and it is what makes a gluing possible.

A gluing is a statement that two boundary edges of a rectangle are one edge of the paper. It produces a different sheet.

A seam is what the gluing looks like on the drawing: nothing at all, since the identification joins crease pieces at ordinary interior points and leaves no mark.

The last is why no drawing and no file records a gluing, and why every figure in this collection showing a glued sheet is showing a rectangle with a sentence attached.