A fragment counts creases, not length

No placement cleans every drawing established what a fragment is and what it is not. A fragment is a piece of crease in a list too short to matter, made where a corner of the pattern sits close to the rim, and it belongs to a placement of the sheet rather than to the construction that drew the pattern. Moving the sheet a tenth of a millimetre clears any one drawing, no single movement clears all of them, and at random about one drawing in nine carries a fragment.

All of that rested on a threshold, a thousandth of the sheet’s side, and the essay admitted what kind of threshold it was. It was a choice, made for being smaller than a pencil line, and it decides the fault rate. The essay left a better question standing: what is the number if the threshold is set by the paper?

The answer turns out to be less interesting than what comes with it. A fragment rate can be derived, from the drawing alone, with no placement drawn. It comes out as a width times a count, and the count is of creases.

The width a paper sets #

A crease is not a line. It is a rounded band a few sheet thicknesses across, and the density a paper allows put the width at about six thicknesses: 0.16 mm for foil-backed tissue, 0.24 mm for washi, 0.42 mm for kami and 0.60 mm for copier paper. That six is stated rather than derived, and it carries through everything here. A piece of crease shorter than that band cannot be formed in that paper as a separate crease. It will not fold along its own length; the fold arriving at one end of it and leaving at the other simply rounds the corner.

So the fragment gets a physical threshold in place of the thousandth. At a width of 0.15 mm the old test and the new one nearly coincide. At kami’s 0.42 mm the new test is nearly three times coarser, and copier paper’s four times.

Before any of it can be measured there is a suspicion to clear. A survey that moves a drawing a long way under its sheet might leave the part of the tessellation that had been computed, so that placements out at two periods would be drawing a thinner pattern and counting fewer pieces for a dull reason. That was checked directly. Mean total crease length on a moved drawing stays within half a per cent of the centred one’s, at every shift from half a period to three. The drawings are the tessellation’s own at every placement used here. And four ways to draw a pattern defined the clipped construction these are drawn by, which a stub is never alone and how deep is a crossing showed makes neither of the other two faults, so a short piece at the rim is the only thing left to count.

What a rim does to one crease #

The map at the top shows the structure before any number does. Each point in it is one placement of the same drawing, moved up to a period either way; dark means that placement leaves a piece of crease shorter than the width. The dark points are not scattered. They are straight bands, running exactly along the two directions of the sheet’s edges, and turning the dial thickens every band in proportion. The few isolated points look like the pieces described below that are cut across a corner of the sheet, though that reading of them was not checked point by point.

That is a direct picture of the mechanism the earlier essay found. Moving the sheet sideways takes a corner of the pattern across the left or right rim at one particular offset, wherever that corner is vertically, so a corner makes an upright band on the map. The width of the band is how far the rim can sit from the corner while cutting a piece short enough. Moving the sheet the other way does nothing to it.

The drawing says the same in plain sight. Six pieces too short to form, all six on the rim, and they are not randomly placed either. Four sit along the bottom edge at the same spacing, because the tessellation repeats and the rim runs parallel to one of its rows, and when it catches one corner of the row it catches them all. Every ring is the same event repeated.

A single crease is then easy to account for. Take a crease of length $L$ pointing in the direction $(u_x, u_y)$, and a rim of unit length running upright. If the rim falls across the crease, the piece left on one side is shorter than the width $w$ exactly when the crease’s end lies within a band of width $w \cdot |u_x|$ beside the rim, or $L \cdot |u_x|$ if the crease itself is shorter than $w$. The chance of that, over placements, is the crease’s density times that band’s area. Add up the four rims and every crease:

$$E(w) \approx \sum_{\text{creases}} \rho \, \cdot 2\,(|u_x| + |u_y|)\, \min(w, L)$$

The length $L$ appears only inside the minimum, and on these drawings almost every crease is far longer than any paper’s crease width, so it drops out. A long crease and a short one contribute the same. Each crease has two ends, and only ends can be left stranded by a rim; nothing that happens along the crease’s length can leave a short piece. Two more terms complete the count, and both are small. A crease shorter than $w$ counts whole wherever it sits, and on these drawings that is rare. A crease running past a corner of the sheet is cut by two rims at once, and the piece left is short across a band of width $w \cdot |u_x u_y|$; this is the one term that reads length, since a longer crease is more likely to pass a given corner, and it is about three per cent of the total.

The rate against the survey #

The densities and directions were read off the same tessellation on a sheet eight sides across, from the middle four, with no placement drawn and no piece measured. Against that, the survey draws 1,024 placements uniformly over a box two periods wide, for every one of the 120 clipped twist drawings, and counts what each leaves.

At the width of a crease in kami, measured 1.00 pieces a placement, derived 0.99. Single drawings scatter around the diagonal and none misses by more than a factor of 1.25. Most of the scatter is on the drawings with fewest creases, where a thousand placements see only a few pieces and the sampling noise is large; a drawing’s own departure is not something this measurement can separate from that noise.

The natural first estimate would have been something else. A rim crossing more crease makes more cuts, which suggests multiplying the rim’s length by the density of crease length, and that is right about cuts. Not every cut makes a short piece, though. A piece is short only when the cut lands near an end, and that depends on how many ends there are. Across a whole population the two can look alike, because drawings with more creases also tend to carry more crease length. They are nevertheless different quantities, and it is the ends that the survey follows.

The count, and only the count #

Pulling the rate apart gives the plainest statement available.

A crease pointing in a random direction has an average $|u_x| + |u_y|$ of $4/\pi$, so a sheet whose creases point evenly every way has a rate of $8/\pi$, about 2.55, for every crease on it. On all 120 drawings the rate per crease lies between 2.42 and 2.74. Five different tilings, three periods and eight turns, a sevenfold range in crease count from under fifty to over three hundred and fifty, and one straight line through all of it. The honeycomb’s creases and the rhombille’s look nothing alike, and what they have to say about fragments is how many of them there are.

This gives the number a designer would want, and it comes out as arithmetic. Expect about $2.5\,N w / S$ too-short pieces on a sheet of side $S$ carrying $N$ creases at crease width $w$. A hundred-crease twist drawing on a 150 mm sheet of kami has $2.5 \times 100 \times 0.42 / 150$, about 0.7 of a piece per placement; the same drawing in foil-backed tissue has a quarter of one. A finer tessellation of the same sheet raises the rate directly, through $N$, and a larger sheet of the same paper carrying the same creases lowers it, through $S$.

It also corrects a sentence in the earlier essay. There, the order of the tilings by fragment rate was said to be “not simply” how many polygon corners each tiling packs into the plane, since the honeycomb has fewer than the triangular grid and carries fragments slightly more often. Counting corners was close but not quite the right count. The rate reads crease ends and their directions, and the honeycomb and the triangular grid at the same settings have almost the same number of creases; the line through the figure above has them lying on top of each other. The small difference in their rates at a thousandth of the sheet is inside the scatter.

The rate does not give the share #

A designer may care less about how many fragments a placement has than whether it has any, and the rate cannot say that.

If pieces fell independently of one another, a mean of $m$ a placement would leave $e^{-m}$ of placements clean. At the width of copier paper the mean is a little over 1.4, which predicts 30%. The measured share is 63%. Pieces do not arrive independently: they come in rows, as the ringed drawing showed, because a single placement that puts one corner near the rim puts every corner of its row there. So a placement’s pieces arrive in clusters of about three, the same mean is spread over fewer unlucky placements, and more placements escape altogether. The straight bands on the map are this fact drawn directly. Independent pieces would make a speckle, and the map has lines.

The tilings separate here where the rate did not need them to. At copier paper’s width about 80% of placements of a square-grid drawing are clean, about 65% of triangular-grid and honeycomb drawings, and 44% of the rhombille’s. Those averages follow the crease count, though not all the way: nothing in the formula above gives the clean share, because that depends on how the corners of the tessellation line up along a row, which is a property of the tiling’s lattice and not of its crease count. A theory of the share would need the lattice, and none is offered here.

A clean placement exists for every drawing #

The original question was how many printed twist patches carry a crease the paper cannot form. Taken at a fixed placement the answer is a share. Taken over placements the answer is better: a designer can move the sheet.

Every one of the 120 drawings has a placement clean at the crease width of every paper in the table. The tightest, a rhombille at period 0.28 and turn 0.45, reaches 1.57 mm and the next above it nearly 2.5 mm, more than twice copier paper’s width. Most drawings sit between about 4 and 10 mm. So no clipped twist drawing here needs a thinner paper for its rim’s sake: it needs a placement, which can be chosen by reading a map like the first figure and picking a white square.

That completes a small arc with two faults, not four, which reduced a checker’s four tests to two independent faults and called the fragment one of them. The fragment was a fault of the placement, the earlier essay showed; now it is a fault with a known rate and, for each drawing, a known way out at any paper’s width.

The surprise worth carrying sits in the shape of the formula. How much line is on the paper and every other density measurement on this subject counts length: metres of crease per square metre. The ceiling a paper sets on a pattern, from the same crease width, also reads length. And the failure that same width produces at the rim does not read length at all; it reads a count of ends, a dimensionally different quantity. Two limits come from one crease width, and they read two different properties of the pattern. A drawing with long creases far apart and one with short creases close together can have the same density and very different fragment rates.

What the law rests on #

A rim with no width. The rim is a line, and a crease is a line with a nominal width applied only as a threshold on length. A real cut edge wanders, and a real crease’s rounded band reaches the rim before its centre does. The law says when a piece is shorter than a stated width, which is a statement about the drawing, and it does not say a folder would see any particular thing happen there. The crease has a radius is what the width comes from, and the six thicknesses are a stated figure.

Only the rim is treated. Everywhere away from the rim, two creases closer than a crease is wide are an entirely separate problem, and this survey does not ask about it. A too-short piece at the rim is the only failure counted, and a drawing with every placement clean here is not thereby foldable in that paper. The junction where three creases meet inside a sheet also needs room, and the density a paper allows is the nearest treatment of that.

The placement, the drawing and the width are the whole model. Nothing here looks at how a pattern folds, whether a lettering exists after the move, or the fact that one nudged drawing in ten came back from the search with letters forcing a loop. The picture can show which placements leave a short piece and cannot show whether a sheet with one would fold any worse. That a too-short piece does harm has been argued from the crease’s width and has not been measured on paper.

How the survey was checked #

Two calculations that share nothing. The rate is computed from a sheet eight sides across with no placement drawn, and the survey from 1,024 placements of the 150 mm sheet with no rate in sight. The figure requires them to agree within five per cent over all 120 drawings and within a factor of 1.6 on every one.

Moved drawings are still the tessellation. Moved by one whole period, the square and triangular drawings must be the centred drawing segment for segment, as the earlier essay requires. And the mean crease length at a moved placement was compared directly with the centred one, at five shifts in eight directions on all 120, and stayed within half a per cent.

Placements are a sample. The widest clean crease is a lower bound, since a better placement than any of the 1,024 may exist, and the tightest figure, 1.57 mm, could rise if more were drawn. The map draws a grid of 127 placements a side on one drawing, a prime so that it cannot fall into step with the tessellation; regular grids of 64 and 96 read 42% and 22% clean at copier paper’s width where random placements read about 30%, and the share printed on the map comes from the random ones. All sampled figures are seeded and reproduce exactly.

Still open: whether the white squares can be chosen by rule #

The map suggests its own next question. A clean placement is a point outside a set of bands, and bands are computable from the drawing’s corners without drawing anything, so a best placement for a given width should come from arithmetic alone rather than from a survey. The largest empty rectangle among the bands, for each drawing, would give both a chosen placement and a tolerance on it: how far the sheet could wander before a fragment appeared. A tolerance is a direction found that a tolerance on a mesh has a shape, and this would be another one, with the two axes of the sheet as its directions.

The other direction is the corner-of-the-eye result about clusters. The clean share sits far above the independent value and depends on how a tiling’s corners line up along a row, which suggests the share is really a statement about the lattice, and the count of bands per period along each axis is a number each tiling carries. Predicting the share from the lattice would finish what the rate started, and it would say why the square grid escapes so much more often than its crease count alone explains.

The third is the second lead the earlier essay left, not touched here: whether the fraction of a millimetre of paper at the edge changes the lettering. What the rim was doing found that a sheet’s edge decides how hard a lettering is to find, and a map of placements coloured by whether a lettering exists would sit naturally beside the one at the top of this essay.

The habit worth carrying is about estimates that match on average. Before believing a rate over a population, check it one case at a time, because a correlated quantity can match the mean and be the wrong thing. Across a whole population crease length and crease count travel together closely enough to hide which of them a fault is following. One drawing at a time is what says it is the count.