A tessellation on a cylinder

The twist tessellations are this collection’s main tessellation work, and they have been drawn on exactly two kinds of sheet. A patch — a rectangle cut out of the plane, with a rim all the way round — and a torus, made by joining the rectangle’s opposite edges so that it has no rim at all.

Those are the two ends of a scale with one rung between them, and until now nothing could build the rung.

The cylinder #

Take one period of the square twist tessellation, cut a rectangle of it whose edges miss every vertex, and identify the left edge with the right while leaving the top and bottom as they are.

The result has two edges instead of four or none. Its interior vertices are the same interior vertices, asked the same conditions; its free letters have fallen by the number of creases the glued pair divided; its panels have merged in pairs along that pair.

That is the object, and everything else in this essay is what having it makes possible.

The numbers #

On the two-period cell of the square twist: forty free letters and twenty-five panels cut out of the plane; thirty-six letters and twenty panels with one pair glued, either pair, since this drawing is symmetric between its two directions; thirty-two letters and sixteen panels on the torus.

Sixteen interior vertices throughout.

Forty, thirty-six, thirty-six, thirty-two: two equal steps. That is the linearity the middle of the scale establishes, and it is what turns the rim costs eight letters on this cell into the rim costs four letters per glued pair.

The two cylinders #

On the square twist the two gluings give equal counts, because the drawing is symmetric under exchanging its axes.

On the elongated triangular tiling’s twist tessellation they do not. One period cut out has fifty-two free letters and thirty-three panels; glued across, forty-two letters and twenty-two panels; glued along, fifty letters and thirty panels. Ten letters saved one way and two the other, on one rectangle.

So half the rim names a topology and not a cost, and the two cylinders are two different patterned sheets on one surface.

What the search pays #

Below a certain size the four sheets cost much the same and above it they do not.

At one and two periods the square twist’s four sheets settle in five, four, four and three nodes, and thirteen, eleven, eleven and nine. Per panel that is between half and three quarters throughout, and the four columns are nearly indistinguishable.

At three periods they settle in thirty, twenty-four, eighty-five and six hundred and twenty-five.

What the rim supplies is slack — letters that cannot be wrong — and slack is useful until it runs out rather than in proportion to how much there is.

Which cylinders the construction admits #

Not every tessellation glues, and the twists mostly do, with one qualification worth stating.

The construction places the cell’s corner so that its edges fall clear of every vertex, choosing the two coordinates independently. That succeeds on the square, triangular, hexagonal and elongated tilings at every size tried. On the rhombille it succeeds too, and the rhombille’s cells are large enough that the searches on them are expensive rather than informative.

The failure the search cannot see is a crease through a corner, and it is not hypothetical — it turned up on another family and was found by Euler’s number. The repair is a nudge along a gap the vertex search had already cleared, and the check that the repair worked is the characteristic coming out at nought.

Why the family had only two sheets #

Worth a paragraph, because the gap was structural rather than an oversight.

The construction that builds a glued cell takes a rectangle and identifies opposite edges. It was written to identify both pairs, because the question it was built for was what happens when a sheet has no boundary at all, and a sheet with no boundary needs both.

Identifying one pair is one flag. The panels touching the left edge are joined to their partners at the right and the ones at the top and bottom are left alone; the creases likewise; and everything downstream — the vertex tables, the arcs, the search — works unchanged.

That it was one flag rather than a rewrite says something about how the original was built. The two directions were handled separately throughout, because they have to be: a panel identified across carries a lattice step in one coordinate and a panel identified along carries it in the other. Making the two independent was already done; making them optional was not.

So the middle rung existed as a possibility from the moment the construction did, and the reason it was not built is that nobody had asked for it until a claim about a rate needed a third point.

What the cylinder is for, beyond the rate #

Two further things the object makes possible, neither of them measured yet.

Comparison with manufactured tubes. Folded tubes are made and studied by people who are not doing mathematics, and the properties they care about — how much a tube shortens as it collapses, how stiff it is part-folded, whether it deploys reliably — are properties of the cylinder. Every measurement in this collection has been of a flat patch and is therefore about a different sheet.

Layer order with half a rim. The bottom of a stack lives at the rim, and a torus having none is why its order has no least element. A cylinder has two circles of rim and therefore a bottom, and what happens to the count of minimal panels as half the rim goes is a measurable question with an unknown answer.

Both are recorded rather than done.

The counts, tiling by tiling #

Since the linearity claim is made across the family, here are the one-period cells.

Square. Cut out: twelve free letters, nine panels, four vertices. Across: ten and six. Along: ten and six. Both: eight and four. Savings two and two.

Triangular. Cut out: thirty-four and twenty-three, twelve vertices. Across: twenty-eight and sixteen. Along: thirty and eighteen. Both: twenty-four and twelve. Savings six and four.

Elongated. Cut out: fifty-two and thirty-three, twenty vertices. Across: forty-two and twenty-two. Along: fifty and thirty. Both: forty and twenty. Savings ten and two.

Every row adds. Every row has the same vertex count across all four sheets. Every row has Euler’s number one, nought, nought, nought.

And the elongated tiling’s ten against two is the sharpest instance in the collection of why the two cylinders have to be kept apart: on that cell, one pair of edges is worth five times the other.

A note on the rhombille #

The fifth tiling is the awkward one and it is worth saying why it appears in some tables and not others.

Its repeating rectangle holds more of the tiling than the others do — twenty-four panels and forty-eight creases at one period, against the square’s four and eight — because it has two kinds of vertex and the rectangle has to contain both. So its cells are large at every size, and its searches are expensive.

At one and two periods the counts are available and behave exactly as the others do. At three the searches start hitting budgets, and a row that reports the budget ran out rather than a cost is not a measurement of the object.

So the rhombille appears wherever a count is being reported and is dropped wherever a cost is, which is a rule applied consistently and worth stating rather than leaving as an inconsistency in the tables.

The object nobody could build before #

The reason this is a rung rather than a curiosity is that the cylinder is the shape of every folded tube anybody makes.

A twist tessellation rolled into a tube is a real object. It is used in deployable structures, it is what a corrugated bellows closed on itself is, and it has been outside this collection’s reach entirely: every measurement here has been on a flat patch of the pattern, and a patch and a tube are different sheets with different counts and different verdicts.

The torus, by contrast, does not exist physically: a flat rectangle cannot be joined at both pairs of edges without stretching. It is included in every table here because the comparison needs the end of the scale, and it is a drawing with a rule attached rather than something to fold.

What a patch was standing in for #

The whole of this collection’s tessellation work has been done on patches, and the cylinder is a good occasion to ask what a patch was ever a specimen of.

A patch is a rectangle of the pattern with a rim all the way round. It is not the tessellation: the tessellation has no rim, and most of a patch is edge at small sizes. It is not a manufactured object either, since the things people make out of these patterns are tubes and closed shells rather than flat squares.

What it is, is a sample — a piece of the pattern small enough to draw and search — and the standing worry about samples is that the rim makes them unrepresentative.

The measurements say the worry is well founded and points the other way from the intuition. The rim does not make a patch harder or stranger; it makes it easy, because a crease the rim divides is a letter that cannot be wrong. A search on a patch spends most of its time in the part that is not the pattern.

So a patch was standing in for the tessellation and reporting a systematically low difficulty, and the cylinder and the torus are what say by how much.

Printing one #

The paper rule in this collection is that a pattern which folds gets printed at a stated size, and a cylinder raises a small question about what to print.

The answer is: the rectangle. A glued cell is drawn as a flat rectangle with a crease pattern on it, exactly as a patch is, and the gluing is a sentence rather than a mark. So the printed sheet for a cylinder is the printed sheet for the corresponding patch, with an instruction to join two edges.

That is not a dodge. It is what the object is: a rectangle of paper together with a statement about which of its boundary points are the same point, and the statement is not something a drawing can carry. No file format has a field for it either.

The practical version, for a reader with a printer: print the patch, cut it out, crease it, and tape the left edge to the right. What comes out is the object every count in this essay is about, and whether it presses flat is decided by arithmetic that can be done before the tape goes on.

The three sheets, ranked by how much they say #

For a reader deciding which object a claim should be made about.

The patch is what can be drawn, printed and folded. Every figure in this collection is one, and it will go on being the thing a reader sees.

The cylinder is what gets manufactured. If a claim is meant to be about a folded tube, this is the sheet, and no measurement on a patch settles it.

The torus is the pattern itself, as nearly as a finite object can be. It has no rim, so nothing about it is an artefact of where somebody cut, and a claim about the tessellation should be made here or not at all.

That ranking is new. Before the middle rung existed the choice was between a patch and a torus, which is a choice between what can be drawn and what the pattern is — and the object anybody actually builds was in neither category.

What is unchanged #

Everything about the drawing, which is the point of a controlled comparison.

The pattern is one pattern. Its vertices are in the same places at the same angles, its creases are the same creases, and every condition the subject checks at a point holds or fails identically on all four sheets.

The twist tessellations, checked at one and two periods on four tilings, are orientable under every gluing: their pleats come in pairs, so a path round any cell crosses an even number of creases, and the parity that refuses half the grid’s cells never fires here.

They also slide rather than turning: the folded state of one cell is carried onto the next by a translation, so their folded period is one drawn period and every cell size glues. That is not true of every family — the Yoshimura’s folded period is three of its drawn ones — and it is why the twists were the family the whole gluing construction was built on.

The scale, and its three rungs #

To have the whole thing in one place: a rectangle of repeating pattern can be given four edges, two, or none, and that is the complete list.

Four edges — cut out of the plane. The most letters, the most panels, the cheapest search, and the least like the pattern.

Two edges — one pair glued. Two ways of doing it, giving two different sheets on anisotropic drawings. Half the letters gone, half the panels merged, a bottom layer retained, and a physical object.

No edges — both pairs glued. Every letter that the rim divided gone, no bottom layer, and the closest a finite object comes to being the tessellation.

There is no rung between four and two, and none between two and none, because an edge is glued or it is not. A cell’s edges cannot be partly identified: the pattern has to match up along the whole of a glued edge or the identification joins creases that are not the same crease.

So the scale has exactly three points, and having all three is the difference between an interval and a rate.

What the twists have that the other families do not #

Among the drawings this collection can glue, the twists are the well-behaved ones, and it is worth saying what their good behaviour consists of.

They never flip. A path round any cell crosses an even number of creases, at every size and on every tiling, because a twist polygon’s pleats come in pairs — one entering and one leaving. So the parity that refuses half the grid’s cells and half the Miura’s never fires.

They never turn. The folded state of one cell is carried onto the next by a translation, at every size and on every tiling, so their folded period equals their drawn period and every cell size glues. The Yoshimura is the counter-example and it is a genuine one.

Their period is known exactly. The construction that generates a twist tessellation knows its own repeat rectangle, because it is built from a tiling with a stated lattice. Families drawn as finite patches have to have their period worked out, and getting it wrong produces a cell that will not close.

Those three are why the whole gluing apparatus was built on the twists and why the other families arrived later. A family that flips, turns, or has an unstated period is a family the construction has to be careful about, and the twists are careful about none of it.

Which is also a reason to be suspicious of any generalisation drawn from them alone. Everything the collection knew about glued sheets before the other three families arrived was knowledge about the best-behaved case.

Where this leaves the family #

Three sheets, of which two are physical, and a rate rather than an interval.

The counts are settled: what a glued pair of edges costs is the creases it divides, the two pairs add, and Euler’s number checks the identification. Those hold at every size and on every tiling tried.

The costs are not settled, and the honest position is that the four sheets are indistinguishable below a threshold and a factor of twenty-six apart above it, on one family at one size under one variable order. That is a direction rather than a law, and the measurement that would settle it has not been run.