Twelve creases a micrometre long

The instrument these measurements are built on has one requirement that looked like housekeeping when it was written. Every fold in the folded sheet must be matched back to exactly one crease in the pattern, and a fold that matches none, or matches two, stops the whole construction.

The requirement is there because of a failure that had already happened. Matching folds to creases by the vertices at their ends is correct only when no crease has been cut into pieces by another crossing it, and on a pattern where eight half-creases meet at a point, seven of the eight matches failed and were quietly dropped. A dropped arc cannot close a circle, so the effect was a search that cheerfully returned letterings with contradictions in them and reported them as clean.

With the requirement in place, the construction is refused rather than approximate — and running it over every pattern in the collection turned up something else.

The twelve #

The hexagonal tessellation patch — the one built by putting a twist polygon at every vertex of a honeycomb and joining them with pleats — has a hundred and forty-two creases and produces a hundred and thirty arcs.

Twelve creases carry no arc. And the reason is not the interesting-sounding one.

A crease can legitimately carry no arc if it has paper on one side only: it joins a panel to the outside of the sheet, so it makes no statement about which panel lies above which. That is a real category and one would expect a clipped patch to be full of it. It is not what these twelve are.

They are between eight millionths and six hundred-thousandths of a sheet long. On the hundred-and-sixty-millimetre sheet this patch is printed at, that is between one and nine micrometres. A six-hundred-dot-an-inch laser printer puts down a dot forty-two micrometres across. Four of these creases would fit inside one dot with room to spare.

They come in pairs, and the pairs sit at six vertices on the top and bottom edges of the sheet. That is exactly where the construction clips: the pattern is generated across an unbounded tiling and then cut to a square, and at those six points the cut lands almost exactly on a corner of a pleat. Almost, and not exactly — so instead of removing the corner it leaves a fragment of it, twice, once on each side.

The number they were being counted in #

Nothing had noticed, and it is worth saying precisely what “nothing” covers, because a great deal was being measured on this patch.

Every count of the patch’s creases has been a count of a hundred and forty-two, of which twelve are not creases. The lettering space has been two to the power of a hundred and forty-two rather than two to the hundred and thirty, which is a factor of four thousand of letterings that differ only in what letter is written on something a micrometre long. The sampler drew them, the propagation gave them values, and the shares reported for this patch were computed over a set four thousand times larger than the real one.

The shares themselves are unaffected — every real lettering appears with all four thousand of its meaningless variants, so proportions come out the same — which is why nothing looked wrong. This is the failure mode that has no symptom.

What a crease is, for the purpose of a count #

The fragments make an awkward question unavoidable, and this collection has answered it implicitly a hundred times without ever writing the answer down. What counts as a crease?

For the folder, the answer is operational: a crease is a line the paper is folded along, and a line a micrometre long is not one. For the file format, a crease is an edge in a graph with an assignment on it, and an edge is an edge however short. For the four conditions at a vertex, a crease is a spoke contributing an angle, and a fragment collinear with its parent contributes nothing and might as well not be there.

Three definitions, agreeing everywhere except on twelve objects, which is why nothing caught them. The definition that disagrees is the folder’s, and it is the one this site’s whole paper rule is written to serve: a pattern here is meant to be foldable by a reader with a printer, so an object a printer cannot render is not part of the pattern in the sense that matters.

That gives the rule for counting. A crease is a line a reader could fold. The threshold is not a matter of taste — it is whatever the printed sheet can resolve, which for a hundred-and-sixty-millimetre sheet at six hundred dots an inch is about four hundredths of a millimetre. Everything in this collection except these twelve is orders of magnitude above it; these twelve are two orders below.

The other patches, checked #

The obvious worry is that the hexagonal patch is not special and the others simply have their fragments below whatever threshold was being looked at. It is worth ruling out, and it rules out cleanly.

Across five tilings at four turn angles each, the shortest crease anywhere else is nineteen ten-thousandths of a sheet — three tenths of a millimetre on paper, small but perfectly foldable — on the rhombille at its shallowest turn. The next shortest is on the triangular patch at a half-radian turn, at two and a half thousandths. Everything else sits between a hundredth and a twentieth.

So there is no continuum running down into the fragments. There is a population of real creases spanning about one order of magnitude, and then a gap of two orders, and then twelve objects at one configuration.

The claim they explain #

There is one published statement in this collection that the twelve fragments account for, and it was written as an observation with a plausible explanation attached.

Measuring how many independent closed chains of panels each pattern has — the quantity that predicts how often a lettering agrees with itself — the count came out equal to the number of interior vertices on twelve of thirteen patterns. The exception was this patch: sixty interior vertices, fifty-four chains. The explanation offered was that six of its vertices sit where the clip has taken the paper away on one side, so their panels no longer close a ring.

That is the right count of vertices and the wrong mechanism. The six vertices are the six where the fragments are, and each of them is a vertex whose two fragment creases join no panels — so the vertex is not a place where the paper ran out, it is a place where the paper is present and the creases at it are numerical debris.

The correction does not change the number. It changes what the number is evidence for. Six vertices lost their ring to the clip would be a fact about clipping a tessellation, and it would be expected to recur on the other patches, which are clipped the same way. Six vertices are carrying fragments is a fact about one construction at one angle, and it recurs nowhere: the same patch at three other turn angles has no fragment at all, and neither does any other tiling at any angle tried.

Why the pattern still folds #

A reasonable worry, given all this, is whether the patch has been folding on paper at all — whether the verified pattern is verified in some sense that a real sheet would not recognise.

It has and it is. A fragment of a crease is a crease with two vertices at its ends, and both of those vertices satisfy every condition the subject has: developability, the alternating angle sum, the count, the smallest-sector lemma. They satisfy them because the fragment is collinear with the pleat it is a fragment of, so the angles at its ends are a straight line and a straight line contributes nothing to any of the sums.

That is also why nothing refused it. A degenerate crease of this kind is invisible to every vertex condition, not merely tolerated by them.

And on paper the fragments do not exist at all. Nothing draws them at a micrometre; the printed sheet shows the pleat corner as a corner, which is what it geometrically almost is. A folder folding this patch has been folding the pattern the collection meant to draw, and would not be able to fold anything else.

What it cost, and what it did not #

It is worth separating the two questions a defect like this raises, because they have different answers and running them together produces either complacency or panic.

Was anything published wrong? One thing: the explanation of why this patch has fewer chains than interior vertices, corrected above. Every share, every ratio and every proportion quoted about this patch is unaffected, for the reason given — the redundancy multiplies numerator and denominator alike. Every statement about its panels, its twists, its shrinkage and its pleats is about the geometry, which the fragments do not touch. The arc count of a hundred and thirty that appears in one essay is, as it happens, exactly right, because arcs were always counted from the folded sheet rather than from the crease list.

Was anything being computed on nonsense? Yes, cheaply. Every draw on this patch spent effort assigning letters to twelve objects, every enumeration of its lettering space was four thousand times too large, and every one of those twelve letters was then propagated into the vertex conditions, which had nothing to say about them.

The gap between those two answers is the interesting part. A defect can be entirely real, present for a very long time, and cost almost nothing — and the reason it costs almost nothing is the same reason it survived, which is that it is invisible to every quantity anybody was measuring.

What this says about the boundary of a patch #

The general lesson is about clipping, and it generalises past this pattern.

A tessellation is an infinite object. Everything this collection prints of one is a patch: the pattern generated over a region and then cut to a square sheet, with whatever the cut produces at the edges. That cut is where a construction’s arithmetic meets an arbitrary boundary, and it is the one place where the geometry is not determined by the pattern’s own rules.

Most of a patch is edge already established the scale of the problem — on the smaller patches the majority of twist polygons touch the rim — and treated it as a question about which measurements the rim distorts. This adds a second kind of distortion, and it is a different kind: not a panel that is smaller than it should be, but an object that should not be there.

The general rule that follows is short. A construction that clips should refuse a fragment rather than emit one. A crease shorter than some fraction of the smallest real feature is not a crease, and the honest response is to merge its endpoints and drop it, exactly as a planariser merges two vertices a rounding error apart.

The measurements that stand #

It is worth walking one measurement through, to show what “unaffected” means in practice rather than asserting it.

The patch’s tangle — the panels lying on some circle under a contradictory lettering — was measured at forty-eight of seventy-seven panels and seventy-two of a hundred and thirty arrows. Both numbers are computed from the folded sheet, which is built by placing panels, so neither of them ever saw a fragment: a fragment joins no panels, contributes no arc, and cannot lie on a circle.

The same is true of everything measured through the folded sheet: the loop a walk finds, the panel count, the number of independent chains, and the search costs reported here. What went through the crease list is the shorter list, and it is the one to be careful with: the crease count, the buried count, the size of the lettering space, and anything that divides by any of them.

That division line is worth carrying forward. Two objects describe every pattern here — the drawing and the folded sheet — and they agree about almost everything. Where they disagree, the folded sheet is the one that has been through a construction that can refuse, and the crease list is the one that will hold whatever it was given.

What has not been done about it #

The repair has not been made, and the reason is worth stating rather than leaving implicit.

Dropping the twelve fragments changes the patch’s crease count from a hundred and forty-two to a hundred and thirty. It probably also changes its interior vertex count from sixty to fifty-four, since the fragments’ shared endpoints stop being vertices — and if it does, the anomaly the fragments explain disappears with them: the patch would then have fifty-four chains and fifty-four interior vertices, and would stop being the exception in a table that several essays here discuss at length.

That is a good outcome and a large edit. It touches every number quoted about this patch across four rounds of work on it, and re-measuring them is work there was no room for here. So the fragments are recorded here, the mechanism is named, the mistaken explanation is corrected, and the patch is left as it is until the repair can be made properly rather than in a hurry.

What is not left ambiguous is the status of the twelve. They are not creases, no reader should fold along them, and any future count of this patch’s creases should say a hundred and thirty.

What found it, and what could not have #

Every gate this site runs has looked at this patch and passed it, every time, and none of them was wrong to.

A gate that asks whether a pattern folds asks the vertex conditions, and the fragments satisfy them. A gate that asks whether a figure’s labels fit measures ink on a canvas, and the fragments draw nothing. A gate that asks whether a claim in a caption matches the data recomputes the claim, and the claim was about proportions, which the redundancy does not move.

What found it was a requirement that had nothing to do with any of that: every arc must name exactly one crease. The twelve turned up as creases that no arc named, which is a question nothing here had ever asked, and it was asked only because a different failure — seven dropped arcs on the preliminary base — had made the matching worth being strict about.

That is the second time recently that a defect here has been caught by a check written for an unrelated reason, and the pattern in both cases is the same. The gates ask whether what is drawn is right. Neither of these was a wrong drawing; both were something present that should not have been, and the only checks that see that kind of fault are the ones that insist two independent accounts of an object agree exactly.

The collection has now met this shape often enough to state it as a habit rather than a coincidence. A tick function that returned an empty array left two figures with no gridlines at all and passed every gate, because every gate asked whether a label fitted and none asked whether it existed. A cap computed with a shift reported a pattern of thirty-eight creases as having sixty-four letterings, and nothing refused, because the answer was a plausible number. Absence and silent success are the two failure modes a gate does not see, and the instrument that sees them is always a second, independent account of the same object, required to agree exactly rather than approximately.

Twelve creases with no arc is exactly that: the crease list says a hundred and forty-two, the folded sheet says a hundred and thirty, and the disagreement is the finding.