Folding it flat is one similarity

How much smaller does a folded sheet get? It is one of the first questions anybody asks about a tessellation and one this collection has answered twice, both times by measuring a patch: the directional shrink factors read off a folded state, and the area of paper the pleats take up.

Both are measurements of a piece of paper. A patch has a rim, the rim’s polygons are half-drawn, and the folded footprint of a patch includes a fringe of partly-folded material that is an artefact of the cut. So the answers have always been about a specimen, and the question is about a pattern.

Joining a period’s opposite edges answers it exactly, and the answer is one sentence long.

Where a cell of paper goes #

A panel touching one side of the cell and the panel touching the opposite side are the same panel of the sheet, a period apart on the paper. Folded, they must also be the same panel, some fixed distance apart — so the rigid motion the fold gives one differs from the motion it gives the other by a translation, and that translation is where a period of paper goes.

Two sides, two translations, two vectors. They are the folded sheet’s own lattice, and nothing in this collection could compute them before, because computing one needs a panel and its own image a period away, and on a patch there is no such thing: a panel at the rim has no partner.

The answer #

At the turn, pitch and fill this collection draws its patches at, the folded lattice is the flat lattice scaled by 0.410373441 and turned by 36.62 degrees.

Along the cell and across it, the same scale. On the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the rhombille, the same scale and the same turn — agreeing to eight decimal places, on cells of one, four and nine periods.

Fifteen cells, five tilings, three sizes: one number.

The vectors come out of the identification rather than out of a formula. Each panel is placed by composing reflections outward from a starting panel, so each carries a rigid motion; a panel and its partner across a side carry two, and the difference between them — after allowing for the period that separates them on the flat sheet — is the vector. Nothing is fitted and nothing is measured off a drawing.

Why that is surprising #

The five tilings have very little in common. Their tiles are squares, triangles, hexagons, mixtures of squares and triangles, and rhombi. Their repeating rectangles are $1 \times 1$, $1 \times \sqrt3$ and $1 \times (2 + \sqrt3)$. Their twist polygons have four, six, three, five and mixed numbers of sides. The rhombille’s vertices are not even all alike, so its polygons come in two sizes with two different side distances.

And the collection’s other answer to how much smaller they get gives them five different numbers. Measured as the fraction of the sheet that is not taken up in pleats, the square keeps 0.42, the elongated 0.50, the honeycomb 0.56, the triangular 0.58 and the rhombille 0.61 — a spread of nearly half.

So one quantity separates the tilings completely and the other does not distinguish them at all, and both are correct.

The two questions #

They are different questions and it took the second object to see that.

How much paper is not in a pleat is a question about the sheet before it is folded, and it depends on how many pleats a tiling has per unit area. A tiling with many short edges has many pleats and loses more paper to them than one with few long edges. That is a fact about the tiling and the five answers are five facts.

How much plane the folded sheet covers is a question about the folded state, and it is set by the pleat’s own geometry. Every pleat here has the same width relative to the edge it lies along, because the construction fixes that with one number — the fill — and folding a pleat of a given proportional width contracts the edge it lies along by a fixed factor whatever the tiling.

Contract every edge of a lattice by one factor and turn it by one angle and the lattice has been mapped by a similarity. The tiling does not enter, because every edge is treated the same way — which is the same reason the sector a twist corner is judged on is the same number on every tiling.

Reading the two answers together #

It is worth putting the pair of numbers for one tiling side by side, because the temptation is to think one of them must be wrong.

The square twist tessellation keeps 0.42 of its paper out of the pleats, and its folded state covers 0.17 of the plane the flat sheet covered.

Both are true and they are not in tension. The first says: of every hundred square centimetres of paper, forty-two end up as visible surface and fifty-eight are folded away inside pleats. The second says: those hundred square centimetres now occupy seventeen square centimetres of table.

The second is smaller because the forty-two that remain are stacked on top of each other rather than laid out flat. Dividing gives about two and a half layers of visible surface over the footprint, and counting the pleated material as well gives about six layers of paper — which is the number a folder feels in the thickness of the finished piece.

Three quantities, then, and the collection now has all of them: paper not in a pleat, plane covered, and layers. Any two determine the third.

What a similarity means here #

The word is worth taking literally. A similarity of the plane is a rigid motion composed with a uniform scale: it preserves every angle and multiplies every length by one number.

So the folded twist tessellation is the flat one, shrunk and rotated. Any two points a period apart end up $0.41$ of a period apart, in a direction turned $36.62°$ from the original. The shape of the lattice — square, rhombic, tall and thin — is unchanged, and the folded pattern of twists has exactly the same arrangement as the flat one it came from.

That is a stronger statement than “it draws in equally both ways”, which is what a patch measurement gives and which is what the collection knew before. Equal contraction in two directions is a necessary consequence of a similarity and does not imply one, because a contraction can be equal both ways and still shear.

The rotation is the part a patch measurement cannot see. It is the same turn, present at every scale, and it is why a twist tessellation folded flat looks like a smaller copy of itself rotated rather than like a different pattern.

What it depends on #

Not the tiling. The turn and the fill.

At a turn of $0.25$ radians and a fill of $0.62$ the similarity is $0.33953365$ at $32.45°$. At $0.5$ and $0.45$ it is $0.48739138$ at $71.73°$. At $0.7$ and $0.8$ it is $0.79095544$ at $14.43°$. Every one of those is the same on all five tilings, to eight figures.

So the construction has two parameters and the folded lattice is a function of both of them and of nothing else. A designer choosing how tightly a tessellation packs is choosing exactly two numbers, and the tiling is a choice about what the pattern looks like rather than about how far it shrinks.

Which is a genuinely useful thing to be able to say, and it is the opposite of what the pleat-area measurement suggests.

Area, and the number that is not the linear one #

The area a folded cell covers is the square of the linear factor: $0.410373441^2 = 0.168406$.

That is much smaller than the pleat-area answers, and the difference is not an error in either. The folded sheet overlaps itself. A cell of paper covers $0.168$ of a cell of plane, so on average it is about six layers deep — the paper has not gone anywhere, it has been stacked.

The pleat-area number does not count that, because it is a statement about the flat sheet: this much of the paper ends up inside a pleat and this much does not. Both fractions are of the flat sheet, and the folded footprint is a different denominator entirely.

Setting the two side by side gives a third quantity for free. Divide the paper that is not in a pleat by the plane the folded sheet covers and the answer is the average layer count over the footprint, which is a number a folder can feel and which neither measurement gives on its own.

How the two vectors check each other #

The lattice is not read once. Every pair of identified panels gives a vector, and there are many pairs — one for every panel the cell’s sides divide — so the measurement is over-determined and its consistency is the check.

Two things could fail. The two motions could differ in their linear part, which would mean the fold turns the paper over between one period and the next and the folded state is not periodic at all. And the translations from different pairs could disagree, which would mean there is no single vector.

Neither happens. No pair is turned over, on any cell here, and the vectors from different pairs agree to about one part in a hundred million million. And the vector from a cell of $n$ periods is exactly $n$ times the vector from a cell of one, which is a third check across sizes that nothing forces.

Why nothing could have measured this before #

The obstacle was not arithmetic. It was that the measurement needs two copies of one panel and a patch has none.

A folded state is built by walking outward from a starting panel and reflecting, so every panel gets a motion. To learn where a period of paper goes, two panels are needed that are the same panel a period apart — and on a patch every panel is itself, once. The panels at the left edge and the panels at the right edge are different pieces of paper with no relationship the machinery can see.

Joining the edges creates exactly the relationship. A panel at the left edge and its partner at the right are now one panel with two motions, and the difference between them is the answer. The quantity was not hard to compute; it was undefined on the objects available.

That is the general shape of what a sheet with no edge makes possible, and it is worth noticing that the shrinkage measurement is the least contentious of the three things it gives. It does not correct anything and it does not depend on any argument about layer orders. It is a new number that the old objects had no place for.

Which theorem was checked, and how #

The scale along the cell and the scale across it are computed separately, from two different families of identified pairs, and they come out equal — which is what makes the map a similarity rather than a squash, and is asserted rather than assumed.

The independence from the tiling is checked by computing the whole thing five times and comparing, with the figure refusing to draw if the spread exceeds a part in ten million. The independence from the cell size is checked by computing it at one, four and nine periods and dividing by the period count.

And the pleat-area answers are recomputed alongside, so that the two quantities are printed together rather than one being remembered from elsewhere. Two measurements that disagree by a factor of three are worth having in one place.

The turn, and where it comes from #

The scale is easy to believe and the rotation deserves a paragraph, because it is the half of the answer a patch measurement never suggested.

A twist tessellation is built by putting a polygon at every vertex of a tiling and rotating each polygon against the tiling’s own directions by a fixed angle — the turn — with pleats taking up the mismatch. Folding the pleats brings the polygons together, and the polygons arrive in their rotated orientation rather than the tiling’s.

So the folded pattern is the tiling’s arrangement, contracted, in the polygons’ orientation. The angle between the two is not the turn itself: at a turn of $0.35$ radians, which is $20.05°$, the folded lattice is rotated by $36.62°$. It is a function of the turn and the fill together, and the four measurements here — $32.45°$, $36.62°$, $71.73°$ and $14.43°$ at four combinations — do not obviously suggest a simple formula.

That is worth recording as unfinished. The scale and the turn are both smooth functions of two parameters, both measured at four points, and neither is derived. A closed form would predict the folded lattice of a tessellation before it is drawn, which is exactly the sort of thing a designer would use.

What a folder takes from it #

Two things, and the second is the more useful.

A twist tessellation folded flat is 0.41 of its size, at the proportions this collection draws, whatever tiling it is built on. So the sheet a folder needs is about two and a half times the finished piece in each direction, and six times its area — and that arithmetic does not change if the pattern is moved from a square grid to a honeycomb.

And the finished piece is rotated relative to the sheet, by nearly thirty-seven degrees. That is not obvious from the crease pattern and it matters for registration: a pattern laid out square to the paper’s edge does not fold to something square to the paper’s edge, and a design that needs the finished piece aligned has to be drawn at an angle.

What it means for a patch’s own numbers #

Everything measured on a patch about shrinkage has a correction attached, and this gives its size.

A patch’s folded footprint is the pattern’s footprint plus a fringe. The fringe is the material at the rim whose pleats are cut and which therefore does not fold up — most of a patch is edge, so on the sizes this collection draws the fringe is not a small correction. A shrink factor read off a patch is therefore an underestimate of how much the pattern contracts, by an amount that falls as the patch grows and is never zero.

The similarity is the limit those measurements were approaching, and it is available exactly rather than as an extrapolation. Which is the general use of the object: not to correct the patch numbers one at a time, but to have the answer the patches were approximating and to know how far off they are.

What the picture cannot show #

Two pairs of arrows show a scale and a rotation and they do not show that the map is the same map everywhere on the sheet. A similarity is a statement about every pair of points, and what is measured here is a statement about pairs a whole number of periods apart — which is all the identification can reach, and is enough to fix the lattice and not enough to say anything about what happens inside one period.

Nor can a picture show the six layers. The folded footprint is drawn as a region and the paper stacked over it is drawn as nothing at all, which is the standing difficulty with every figure of a folded state here and the reason layer counts are sampled rather than drawn.

And nothing here shows what happens at the turn and fill where the construction stops working. As the fill approaches one the pleats close to nothing and the scale approaches one — the sheet stops shrinking because there is nothing left to take up — and past it the polygons want the same paper. The similarity is a smooth function of two parameters on the region where the construction draws a pattern, and the figures show four points of it.