The period nobody measured

Everything this collection knows about a repeating pattern is quoted per cell. Panels per cell, creases per cell, vertices per cell, free letters per cell of rim, nodes of search per panel of a cell. The cell is the unit, and it is the rectangle the drawing repeats in.

There is a second period, it belongs to the folded object rather than to the drawing, and nothing anywhere records it.

Two periods #

A pattern’s drawn period is the smallest rectangle whose translate lands the drawing back on itself. It has to be known before anything can be generated, so it is recorded for every family, and it is the number every measurement is divided by.

A pattern’s folded period is the smallest rectangle whose translate lands the folded state back on itself — not merely into the same shape, but into the same place, related by a slide rather than by a turn.

Every folded period is a drawn period, since a folded state that repeats has a drawing that repeats. The converse is what fails.

Measuring it #

The measurement is a small search and it terminates for a reason.

Build a cell of one drawn period, identify its opposite edges, and ask whether the two panels the identification joins have folded motions that differ by a translation. If they do, the folded period is one. If they differ by a rotation, try two periods, then three.

The turn per period is a fixed angle, so the turn per $n$ periods is $n$ times it, and the search ends at the first $n$ making that a multiple of a full turn. If the angle is not a rational part of a full turn there is no such $n$, and the pattern has no finite folded period in that direction — in which case no cell of it can ever be glued.

What the three families come out at #

The grid: folded period one, in both directions. Its cells slide onto each other, and what alternates instead is whether the paper comes back the same way up — a parity rather than a turn.

The Miura: folded period one, in both directions. The four creases a vertical loop crosses compose to a rotation by twice their alternating sum, and the alternating sum is nought by the zigzag’s own symmetry.

The Yoshimura: folded period three across its columns and one along its courses.

The twist tessellations, measured on four tilings at one and two periods, also come out at one — which resolves an apparent contradiction with the thirty-six-degree turn their shrink is known for, since that turn is between two lattices rather than between two panels.

What a period is for #

It is worth pausing on why a pattern’s period is such an important number in the first place, because that is what makes a second one worth having.

A tessellation is infinite and every computation about it is finite. The bridge is the period: whatever is true of one cell is true of every cell, so a measurement on one cell is a measurement on the whole plane. That is why panels, creases and vertices are quoted per cell rather than in total, and why a figure of one cell is a figure of the pattern.

Every one of those inferences depends on the cell being a genuine period of the thing being measured. Count panels per drawn cell and the inference is sound, because panels are a feature of the drawing. Count something about the folded object per drawn cell and the inference is sound only if the folded object also repeats there.

On the Yoshimura it does not, so the bridge is broken for exactly the quantities that are about folding — and it is broken quietly, because a third of a period is still a rectangle and still produces numbers.

The search that finds it, in detail #

The procedure is short and worth writing out, since it is the whole of the new measurement.

For each $n$ from one upward: build a cell $n$ drawn periods across, place its corners so that its edges miss every vertex, and identify the left edge with the right. Fold the resulting rectangle flat by composing reflections, and for each pair of panels the identification joins, compare their two motions.

If every pair’s motions differ by a translation, $n$ is the folded period and the search stops. If any pair’s motions differ in the sign of the determinant, the sheet has been turned over and the failure is a parity rather than a turn; that case is reported separately and the search continues, because a parity at $n$ says nothing about $n+1$ except that it flips.

If the determinants agree and the matrices differ, the difference is a rotation; its angle is recorded and the search goes to $n+1$.

The angle recorded at $n = 1$ is the turn per drawn period. Every later angle is a multiple of it, which is both a check on the computation and the reason the search terminates.

What the numbers looked like #

For the equilateral Yoshimura across its columns: two hundred and forty degrees at one period, one hundred and twenty at two, nought at three, two hundred and forty at four, one hundred and twenty at five, nought at six, two hundred and forty at seven.

That is a rotation of order three, walked round twice. It is also a check: had the sequence read two hundred and forty, then something other than one hundred and twenty, the computation would have been wrong, because the turn at $n$ periods has to be $n$ times the turn at one.

For the same pattern along its courses, and for the grid and the Miura in both directions: nought at every size.

For the twist tessellations on four tilings at one and two periods: nought.

So of ten pattern-and-direction pairs measured, one has a folded period other than one. That is a rare enough phenomenon to have been missed and common enough to matter.

Why the Yoshimura and not the others #

The structural reason is visible in the drawings and worth stating, since it predicts where else to look.

A turn between consecutive cells is a composition of reflections in the creases crossed on the way from one to the next. Reflections in parallel lines compose to translations, so any pattern whose crossings are all parallel slides — the grid’s are, being all vertical or all horizontal within one loop.

Reflections in lines at an angle compose to a rotation by twice that angle. So a turn requires crossings at different angles, and the turn is nought only if the alternating sum of those angles happens to vanish. The Miura’s does, by the symmetry of its zigzag: each course is crossed going in and coming out at equal and opposite slants.

The Yoshimura’s does not. Its diagonals slant the same way within a column and the horizontal courses contribute their own reflection, and the alternating sum comes out at a hundred and twenty degrees of rotation per crossing pair.

So the rule of thumb is: a pattern whose creases cross a loop in mirror-symmetric pairs slides, and one whose crossings are chiral turns. That is a property of the drawing that can be read off it without folding anything.

A caution about the folded period being small #

There is a tempting reading of the folded period is three as the folded object is three times bigger, and it is wrong.

The folded object repeats every three drawn columns. It does not have three times as much in it as a one-column cell has; it has exactly three columns’ worth, which is the same paper. What is three is the unit in which its repetition should be described, not the amount of anything.

The distinction matters when a per-cell quantity is being converted. A folded footprint quoted per drawn column has to be multiplied by three to become a footprint per folded cell — and a footprint per folded cell is the number that means something, because it is the area of one tile of a pattern that actually tiles.

Getting that backwards divides where it should multiply, which is a factor of nine.

Which statements are in the wrong unit #

The correction is narrower than it might sound, and saying which statements it touches is the useful part.

Anything about the drawing is unaffected. Panels per cell, creases per cell, vertices per cell: those count features of a rectangle and the rectangle is the drawn period, correctly.

Anything about the folded object quoted per cell is quoted per third of a unit on the Yoshimura. The folded footprint, the layer count, the shrink factor, the folded lattice vector: each is a property of an object whose own period is three columns, and dividing by one column gives a number that is a third of the honest one.

In practice almost nothing in this collection is affected, because Yoshimura measurements here are taken over whole patches rather than per cell. That is luck rather than care, and the fix is to record the number rather than to rely on the luck continuing.

Why the distinction was never forced #

A quantity that has been equal to another quantity in every case anybody looked at does not get a name, and this collection has met that shape before.

The families it could glue were twist tessellations, and they all slide. The families it drew but could not glue — the grid, the Miura, the Yoshimura — were never asked the question, because asking it requires an identification and there was no way to make one.

So the drawn period was the only period there was, in the practical sense that no computation could have distinguished it from any other, and it inherited the unqualified name.

The shape of the omission #

This collection has a small catalogue of quantities that went unnamed because they were always equal to something else, and it is worth setting the folded period beside them, since the shape recurs.

A test was imported from a literature where sheets are discs, without the sentence saying so, and applied for a long time to sheets whose whole interest is that they repeat.

The closure condition compares a composition against the identity, because on a disc the identification is trivial and the identity is what one writes when there is nothing to write.

The two-colouring’s rule is stated as an even count, because on a sheet with two sides the seam’s contribution is one and a factor of one is a factor nobody records.

Each of those is the same story: a general statement collapsing to a special case on every object available, the special case acquiring the general name, and the omission becoming visible only when an object arrives that separates them.

The folded period is the fourth. It differs from the drawn period on one family out of ten measured, and until a family could be glued at all there was nothing that could have told them apart.

What is not claimed #

Two limits worth naming.

Ten pairs is not a survey. Three families and the twists, in two directions each, is what has been measured. A general claim about how often the two periods differ would need many more, and the honest statement is that one case in this collection has it and the rest do not.

The measurement is per direction and per rectangle. A pattern’s translation lattice may be generated by vectors that are not the sides of the rectangle being used — three of the tilings here are laid out on a rhombic lattice and glued on a rectangular sublattice of it at index two, for the ordinary reason that clipping to a rectangle is easier than clipping to a rhombus. So the folded periods above are folded periods of the rectangles being used, and a finer cell might behave differently.

That is a real caveat and it is the same one the drawn periods already carry, so at least the two numbers are qualified identically.

What a pattern’s record should carry #

The proposal is small: two numbers per direction rather than one.

The drawn period, as now, because the drawing cannot be generated without it.

The folded period, as a multiple of the drawn one, with the turn per drawn period beside it. Three and two hundred and forty degrees for the Yoshimura across; one and nought for everything else measured.

The turn is worth recording alongside the multiple because it is the thing that explains it. A folded period of three with no turn recorded is a fact to be remembered; a turn of two hundred and forty degrees is a fact that computes the three, and computes it again if the pattern is drawn at a different proportion.

A folder’s version of the same fact #

There is a way to meet this without any of the apparatus, and it is worth having because it makes the abstraction concrete.

Fold a strip of Yoshimura — two courses, six or eight columns — and collapse it. Before collapsing, draw a pencil line along one panel, in the direction of the strip. Collapse, and look at where that line points.

Now find the panel one column over. Its line points a hundred and twenty degrees away from the first. Two columns over: another hundred and twenty. Three columns over: back to where it started.

That is the folded period, read directly off the paper, and it is the sort of thing that is hard to believe until it is on the table. The pattern looks as though it repeats every column, and it does — as a drawing. As a folded object it repeats every third.

Try the same experiment on a Miura and the lines all point the same way, which is why nobody has ever had reason to think about this.

What it would take to record #

The change is small enough to describe and it has not been made, so it is recorded here as owed rather than done.

Every pattern in this collection’s shelf carries a name, a construction, a size and a set of measurements taken from its drawing. Adding a folded period means running the search above once per pattern per direction, storing two numbers, and quoting them wherever a folded quantity appears.

The cost is one gluing per candidate size per pattern, which is seconds. The obstacle is not cost; it is that most of the shelf’s patterns cannot be glued at all, because a gluing needs a plane drawing and a stated period and most of the shelf is drawn as a finite patch with a boundary.

So the honest position is that the folded period can be measured for the families that have a plane construction — three of them, plus the twists — and cannot yet be measured for the rest. Recording it for four families and leaving a gap for the others would be a worse record than none, since a missing entry reads as no folded period rather than as not measured.

That is the state: the quantity is defined, it is measurable where a plane drawing exists, one family has a value other than the obvious one, and the shelf does not carry a column for it.

Where periods are already used #

The drawn period is load-bearing in several places here and it is worth listing them, since each is a place a folded period might be wanted instead.

A cell of a tessellation is what every count is quoted per. Cutting a patch out of a plane takes a whole number of them. The rim’s price is per cell of rim, which is per drawn period.

Every one of those is about the drawing and is correctly quoted per drawn period. What is not is anything about the folded object, and the shrink is the clearest case.

The general form #

Written once, because the Yoshimura will not be the last family with two periods.

A repeating pattern has a lattice of translations preserving its drawing. Folded flat, it has a lattice of translations preserving its folded state, and the second is a sublattice of the first. The index is finite exactly when the turn per drawn period is a rational part of a full turn, and the index is the order of that rotation.

Everything about the drawing is quoted per drawn cell. Everything about the folded object is quoted per folded cell. The two coincide on most families, and most is not all, which is the whole content of this.