A count is not a length

How much crease is in a tessellation? There are two obvious answers — how many creases there are, and how much total length of crease there is — and this collection has always taken both off a patch, because a patch was the only thing there was to take them off.

One of them is exact. The other has never been right and gets closer to right as the patch grows — which is the worst way for a measurement to be wrong, because it looks like convergence and is a bias.

The two measurements #

Take a rectangle exactly one period of a square twist tessellation, and read it two ways: as a piece of paper with an edge, which is what a patch is, and with its opposite sides joined so that a crease running off one side is the crease running on at the other, which is what the pattern is.

Crease count. Twelve on the patch, eight on the pattern. Four creases are divided by the rectangle’s sides and each half is counted as a crease of its own.

Crease length. The same on both, exactly — because the two halves of a divided crease are the two halves of a crease and their lengths add up.

Divide each by the area and the difference is a density that is wrong and a density that is right.

The numbers #

On the square tessellation the patch reports 103.8 creases per unit area at one period, 86.5 at four and 80.7 at nine. The pattern’s figure is 69.2.

Fifty per cent too many, then twenty-five, then seventeen. The error falls as the reciprocal of the patch’s size, because the divided creases go as the perimeter and the honest ones go as the area — but it never reaches zero, and at the sizes this collection draws its patches it is between ten and fifty per cent.

The length per unit area is 12.6751 at one period, 12.6751 at four and 12.6751 at nine. And 12.6751 is the pattern’s.

The same on four more tilings #

The square gives the tidiest numbers and the others give the same shape.

The triangular tessellation’s rectangle reports forty-two per cent too many creases at one period, twenty-one at four and fourteen at nine, and its length per unit area is 16.9380 at all three. The honeycomb’s errors are the same and its length is 17.6911. The elongated triangular tiling reports thirty, fifteen and ten per cent too many at a length of 14.6536. The rhombille reports twenty-five, twelve and eight at 23.5143.

Five tilings, fifteen rectangles, five different lengths and one behaviour: the length is a constant of the tiling and the count is a function of how big a piece of it was cut.

The five lengths are worth a glance on their own, since they are the honest answer to how densely creased each tessellation is. The rhombille carries nearly twice the crease per unit area of the square, which is what having two kinds of vertex and correspondingly more polygons per unit area buys — and it is the only number in this paragraph that could have been read off a patch all along.

What that says about the older measurements #

Two of the collection’s crease measurements were taken on patches and this sorts them.

How much line is on the paper is a total length, and a total length divided by the area of the sheet is a density. That measurement is exact and always was, on any patch, at any size. Nothing about it needs revising.

Anything counting creases is not. A crease count over a patch is a count of pieces, and the difference from a count over the pattern is exactly the number of creases the rim divides. That number is four per period of edge on the square tessellation, ten on the triangular and honeycomb, twelve on the elongated and the rhombille.

The mean crease length is the ratio of the two and is therefore wrong in the other direction. A one-period square patch reports a mean of 0.122 where the pattern’s is 0.183 — a third short. At four periods it is 0.147 and at nine 0.157, still fourteen per cent short.

What a patch would have to be to be honest #

The obvious response to a bias that falls as the reciprocal of the size is to draw a bigger patch, and it is worth pricing.

To get the square tessellation’s crease count within one per cent of the truth the rectangle would have to be a hundred periods on a side: ten thousand twists, a hundred and twenty thousand creases, on a sheet a hundred times the pitch. At the pitch this collection draws — about a third of a sheet — that is a piece of paper thirty-three times the size of the one the pattern is printed on, and nothing here has ever drawn one.

The patches actually drawn are three to five periods across. Their crease counts are between ten and twenty per cent high, and their panel counts similarly. So the error is not a residual to be neglected; it is a fifth of the answer.

Reading the count off the joined rectangle instead costs nothing and is exact at one period. Which is the general argument for the object: not that it corrects a number, but that it corrects it at the smallest size rather than the largest.

Why length survives and count does not #

The rule that separates them is short and applies past this example.

A measurement that is a sum over the paper survives a cut, because cutting the paper divides the sum without changing it. Total crease length, total area, total paper in pleats: each is an integral over the sheet, and an integral does not care where the sheet is divided.

A measurement that is a count of objects does not, because cutting turns one object into two. Creases, panels, and anything derived from them.

That is the whole of it, and it applies to every quantity this collection reads off a patch. Length densities are honest; count densities are inflated by the perimeter-to-area ratio; and ratios of the two inherit the error of whichever is a count.

The angle that also survives #

There is a third crease measurement worth setting beside those two, and it survives for a different reason.

At every corner of every twist polygon four creases meet, and the sectors between them are the polygon’s own angle, the tiling’s angle across the corner, and two sectors belonging to the pleat. On the square tessellation those come out at $46.15°$, $90°$, $133.85°$ and $90°$.

The pleat’s $46.15°$ is the same number on the triangular tiling, the honeycomb and the elongated triangular tiling, whose other sectors are $60°$, $120°$ and mixtures. It is the same at a pitch of 0.5, 0.42, 0.34, 0.25 and 0.2 of a sheet.

It is a function of the construction’s turn and fill and of nothing else. Not the tiling, not the pitch, not the size of the patch, not where the patch was cut.

Why the angle is a pleat’s and not a tiling’s #

The construction puts a polygon at each vertex of a tiling and turns it, and the pleats take up the mismatch. A pleat is therefore a strip of a fixed proportional width, and the two creases bounding it leave the polygon’s corner at an angle set by that width and by the turn.

The tiling decides how long the strip is and how many of them meet at a corner. It does not decide the angle at which one leaves, because that is a local matter between one polygon’s side and its neighbour’s.

So the sector angle is what a pleat looks like, magnified — a property of the construction’s two parameters, appearing identically wherever a pleat appears.

And it matters more than an angle usually does, because it is the smallest of the four sectors at every twist corner, which makes it the sector the big-little-big lemma reads and therefore the one that decides how many labellings a vertex of a twist tessellation admits.

Where the pleat’s angle is not the smallest #

There is one place the third measurement’s usefulness runs out, and it is worth flagging because it is where the interesting behaviour of these patterns lives.

The pleat’s sector is the smallest of the four when it is smaller than the tiling’s own angle at that corner. On the square tessellation the tiling contributes ninety degrees and the pleat’s sector never reaches it over the range drawn, so the pleat’s is always the smallest. On the triangular tiling and the honeycomb the tiling contributes sixty degrees, and the pleat’s sector passes sixty as the turn closes.

Past that crossing the smallest sector is a different sector, and the condition it imposes falls on a different pair of creases. That is the transition where a patch goes from having no consistent lettering to having one across a hundredth of a radian, and it is now visible as an intersection of two lines rather than as an unexplained threshold.

So the angle is exactly measurable on a patch and what it means depends on the tiling after all — not because the tiling changes the angle, but because the tiling supplies the number it has to be compared against.

The fill is the lever #

The angle moves a long way with the fill and less with the turn, which is worth having because the fill is a designer’s parameter.

At a fill of 0.3 the pleat sector runs from $73.85°$ at a turn of 0.2 radians down to $29.05°$ at 0.9. At a fill of 0.62 the same range is $61.92°$ to $16.78°$. At 0.95 it is $13.86°$ to $2.27°$.

A fill near one means the pleats have closed almost to nothing, and the sector goes with them: at a fill of 0.95 and a turn of 0.9 radians the smallest sector at a corner is two and a quarter degrees, which is a crease pattern with an almost invisible wedge in it at every vertex of the sheet.

That is where a folder’s difficulties come from rather than a search’s. A two-degree sector is a fold nobody can make accurately, and the pattern remains perfectly flat-foldable on paper of zero thickness — which is the standing gap between what a theorem says and what a hand can do.

Which of these is a material fact #

The essay sits in a thread about how much crease a sheet carries, and the practical version is worth stating plainly, because that thread is about paper rather than about mathematics.

Length per unit area is what a folder feels, and where that length sits on the sheet is a separate question a patch also answers honestly. It is how much creasing has to be done per square centimetre, it is what makes a pattern tiring to fold, and it is exact on any patch at any size.

Creases per unit area is what a drawing shows. It is inflated on a patch, by ten to fifty per cent at the sizes drawn here, and the inflation is entirely at the rim.

The smallest sector is what decides whether a fold can be made at all. It is exact on any patch — a sector angle is a local measurement and a cut cannot change one it does not pass through — and it is the same on every tiling at a given turn and fill.

So of the three, two are safe on a specimen and one is not, and the unsafe one is the one most often quoted — including in this collection, where a crease count is the first number any figure of a patch prints.

There is a fourth quantity in the same family worth a mention because it is the shortest crease rather than the mean one. Twelve creases a micrometre long turned up on one published patch, and every one of them was produced by the cut: a pleat clipped a hair’s breadth from a corner leaves a fragment that is a crease by the list’s reckoning and a smudge by anybody else’s. Those are the extreme case of a count error — objects that exist only because of where the paper stopped.

The rule, as a checklist #

The essay’s content is a sorting rule, and it is short enough to give as one.

A quantity read off a patch is a quantity about the tessellation when it is:

• a total over the paper — crease length, area, paper in pleats;
• a local angle — a sector, a polygon’s corner, a pleat’s width;
• a count of interior vertices, since the rectangle is placed to miss them.

It is a quantity about the patch when it is:

• a count of creases or of panels, both inflated by the rim;
• a mean whose denominator is one of those, deflated correspondingly;
• anything about the layer order’s extremes — the bottom of the stack is entirely a fact about the cut;
• the cost of finding a lettering, which the rim reduces by orders of magnitude.

The first list is safe on a specimen of any size. The second needs the size stated with it, and the error on each falls as the reciprocal of the size and never to zero.

That is the practical value of having built a pattern with no edge: not a correction to any one number, but a way of telling which numbers needed one.

Which theorem was checked, and how #

The length is computed by summing the lengths of the segments in the crease list, and the area by multiplying the rectangle’s two sides. Neither reads anything from the identification, so the agreement between the three sizes is an independent fact rather than a consequence of the joining.

The figure refuses to draw if the length per unit area moves with the cell size on any tiling, since that would mean the drawing does not repeat — and a drawing that does not repeat is not a period of anything, which would invalidate every other measurement here.

The counts come from two separate procedures: the patch’s from its own crease list, the pattern’s from a union-find over crease pieces matched by their endpoints on the rectangle’s sides. A mismatch would show as a count error that does not fall as the reciprocal of the size, and it falls exactly so on all five tilings.

The sector angles are measured off the drawn pattern by sorting the creases at each vertex by the angle at which they leave and differencing, so the claim that the pleat’s sector is the same on four tilings is a comparison of four measurements rather than of four evaluations of a formula.

What the picture cannot show #

A density is a ratio and a bar chart of ratios does not show which of its two parts moved. Here the numerator moves and the denominator does not, and the reason the numerator moves is a handful of creases at the edge of a large drawing — which is exactly the sort of detail a chart of a summary statistic exists to hide.

Nor can any picture show that the length is exact rather than nearly exact. Three numbers agreeing to four decimal places is what the figure prints, and the reason they agree is an argument about sums over a divided region, which is prose.

And nothing here measures the quantity a folder would most like, which is how much creasing a finished piece costs rather than how much a unit of sheet does. That depends on how far the pattern shrinks as well as on how densely it is creased, and the two are separate numbers with separate behaviour under a cut — the first being a similarity that the tiling does not enter and the second being the five different lengths above.