No placement cleans every drawing

Two faults, not four counted every way a drawn crease pattern can differ from the crease list it was drawn from, on a hundred and twenty twist patches drawn two ways, and reduced a checker’s four tests to two that carry independent information. One of the two was the fragment: a crease in the list too short for any pencil to draw. The clipped construction, which draws the tessellation large and cuts it to the sheet, makes no crossings and no stubs, and on five of its hundred and twenty drawings it makes fragments and nothing else.

That essay ended with a proposal. The fragments occur where the sheet’s edge passes within a fraction of a millimetre of the pattern, so moving the sheet by that fraction should remove them, “without changing anything a reader could see”. Whether a small stated shift removes every fragment on every setting was called a short computation.

It is a short computation, and the answer is no. Each drawing can be cleaned by a shift of about a tenth of a millimetre; no one shift cleans all of them; and the reason has to do with what a fragment is, which the earlier account had slightly wrong.

What a fragment is #

The earlier essay said a fragment is made “where a corner of the sheet falls within a fraction of a millimetre of a corner of a pleat”, and that fragments come in pairs because a pleat is two parallel creases of equal length, so a clip that catches one catches the other.

Neither is how the fragments arise. Magnified, the rhombille patch’s fragment is nothing to do with the sheet’s corner, and nothing to do with a pleat’s two creases. A corner of a twist polygon — a vertex where a polygon side and two pleat creases meet — happens to sit 0.125 mm inside the left edge of the sheet. One of the creases leaving it runs toward the edge at a shallow angle, and the rim cuts it 0.131 mm from the corner. The crease list holds that piece as a crease, with a letter and two vertices. The drawing holds a mark shorter than a pencil line is wide.

A corner near the rim is all it takes, and the angle of the creases leaving it decides how many fragments one corner makes. On the rhombille patch the polygon side runs off at a shallow angle and is cut short, while the pleat crease beside it runs off more steeply and crosses the rim half a millimetre away — long enough to be a crease. On the honeycomb patch at period 0.34 the corner sits a thousandth of a millimetre from the edge, and both creases leaving it toward the rim are cut to fragments, of 0.0012 and 0.0088 mm. One corner, one or two fragments, and never the paired creases of a pleat.

The dial makes the mechanism visible. Move the pattern a little further in and the corner draws away from the rim: the cut piece lengthens past the threshold and is a crease. Move it out and the corner crosses the rim: the crease that made the fragment now starts outside the paper, and what is left inside is long. A fragment exists only while a corner sits in a band a seventh of a millimetre wide along the edge.

Every fragment has a partner #

The pairing the earlier essay noticed is real. Its cause is the sheet, not the pleat.

Every fragment lies on an edge of the sheet, never at a corner of it, and every one has a partner of exactly the same length on the opposite edge. On four of the five drawings the partner is the fragment’s image under a half-turn about the sheet’s centre. These drawings are made by laying a vertex of the tiling on the sheet’s centre, and a vertex of the square, triangular, honeycomb and rhombille tilings carries a half-turn of the twist pattern with it; so whatever the rim does at one edge, it does at the opposite edge too. A corner a hair inside the left rim has a twin a hair inside the right one.

The elongated triangular patch is the instructive exception. Its tiling has no half-turn about a vertex, its drawing is not symmetric about the sheet’s centre, and its two fragments still have equal lengths on opposite edges. The half-turn that exchanges them is about a point a seventh of a period above the centre — the middle of one of the tiling’s squares — and any half-turn whose centre lies on the sheet’s vertical midline carries the left edge onto the right. The pairs are symmetry, not mechanism, and the earlier essay’s even counts were the even counts of a symmetric drawing.

A tenth of a millimetre clears each drawing #

With the mechanism right, the repair the earlier essay proposed can be tested where it was aimed.

Each drawing, moved across the edge its fragments sit on, loses every fragment within 0.127 mm of where it was made, in either direction. That is a shift no reader could see on a 150 mm sheet, which is what the earlier essay predicted. The fragment counts on the way are worth reading: the honeycomb’s twelve fall to six, then three, then none, and the elongated patch’s two pass through one. Part way, the counts are odd. A nudge breaks the symmetry that made the fragments come in twins, so one twin leaves the band before the other.

The moved drawings are still the same patterns in every respect that the subject checks. Moved 0.13 mm either way, all ten pass every vertex condition at every vertex and close round every vertex, exactly as the centred drawings do. One thing does change, and it is a reminder that moving a sheet is not nothing: on one of the ten, the honeycomb at period 0.28 moved 0.13 mm to the left, the lettering the search finds forces a loop of panels where the centred drawing’s did not. The sheet holds slightly different paper after a move — a sliver of polygon more at one edge, a sliver less at the other — and the letters that pass every vertex are a property of the paper held.

No placement cleans them all #

Each drawing has its own clean placement. The question the earlier essay asked was whether one placement is clean for every drawing — a single shift that could be written into the construction once.

It is not. Laid at twenty-four random placements across a period of the pattern, the sheet leaves between three and twenty-four of the hundred and twenty drawings with fragments; half the placements leave between ten and eighteen. None leaves none. The centred placement, at five, is bettered by one of the twenty-four and beats the other twenty-three.

The reason is arithmetic about many drawings at once. A shift that moves a corner out of the band on one drawing moves some other drawing’s corner into it, because the hundred and twenty patches have their corners at a hundred and twenty unrelated sets of positions along the rim. A shift that cleared every drawing would have to put every corner of every patch outside a band a seventh of a millimetre wide on all four edges, and nothing about the patterns arranges their corners to make that possible.

Why the centred placement does so much better than a typical one is not something the survey explains, and it is worth saying so rather than reaching for a reason. What it does show is the shape of the trade. The centred placement leaves fragments on fewer drawings than a random one, and on the drawings it does leave them on, it leaves them in pairs — fewer faulty drawings, each with twice the faults. A checker that counts faulty drawings and one that counts faults would rank the same placements in opposite orders.

One in nine at random #

The survey also measures how often a placement leaves a fragment at all, drawing by drawing, and how that depends on what counts as a fragment.

At random, 11.2% of drawings carry a crease under a thousandth of the sheet — 0.15 mm on a 150 mm sheet. The tilings differ by a factor of four. The square grid’s patches carry one 4.7% of the time and the rhombille’s 18.8%, with the triangular grid, the honeycomb and the elongated triangular tiling between, at 10.4%, 11.6% and 10.4%. The order is not simply how many polygon corners each tiling packs into the plane — the honeycomb has fewer than the triangular grid and carries fragments slightly more often — and nothing here settles what it is.

Cut the threshold tenfold, to a ten-thousandth of the sheet, and the share falls to 1.8% — six-fold, not tenfold. A drawing is counted once however many of its corners lie in the band, and at the wider band more drawings have several, so the count at a thousandth is already short of what the band’s width alone would give. The fragment rate is set by the threshold, and the threshold is a choice. Four ways to draw a pattern set it at a thousandth because the centred drawings had a gap of more than a factor of ten above it with no crease in it; at random placements that gap is not guaranteed, and a thousandth is simply where the line was drawn.

A fault of the placement, not of the construction #

Put together, the measurements move the fragment from one category to another.

Two faults, not four treated it as a fault of the clipped construction, the one it still made after giving up crossings and stubs. The construction does not make it. A fragment is made by a placement, a threshold and the corners of the pattern, and every construction that cuts a pattern to a sheet inherits it in the same proportion: a drawing whose corners happen to sit near the rim, at a threshold fine enough, has one, and moving the sheet a tenth of a millimetre changes which drawings do. The earlier essay found the extension construction carrying the same fragments at two of the same settings, and that is the same fact seen from the other side — both constructions lay the sheet in the same place.

So the clean clipped construction exists, and it is not a construction with a better rule at the rim. It is the same construction with one more step: move each drawing until no corner lies within the threshold of the rim, which on every patch here takes at most 0.127 mm. The step is cheap and deterministic. It costs the drawing its symmetry about the centre, since a moved drawing is no longer centred on a vertex; and it changes the paper held by a sliver, which on one drawing in ten was enough to change the letters the search returns.

Placement is a tolerance like any other #

The earlier essays on tolerance were about errors in the paper: a crease placed a little wrong, which error is folded too found turning everything beyond it by twice as much, and a mesh cut a little wrong, which solved is not built found losing its closure. A placement is an error of a different kind — nothing about the pattern is wrong, only where it sits on the sheet — and it behaves like the others in one respect and unlike them in another.

Like them, it has a direction. Moving a drawing along the edge its fragments sit on does nothing; moving it across that edge clears them within a tenth of a millimetre. The honeycomb at period 0.34 has its fragments on the top and bottom edges and is untouched by any sideways shift, while the rhombille, with its fragments on the left and right edges, is untouched by a small vertical one. A placement tolerance is a pair of numbers, one per axis, and a drawing is sensitive to the axis its corners crowd.

Unlike them, it has no right answer common to a family. A solved mesh has a surface of good configurations that every member shares; the hundred and twenty drawings have a hundred and twenty sets of good placements that do not overlap in any placement the survey found. The tolerance is a property of each drawing, and a construction that wants every drawing clean has to read the drawing to place it.

Where the measurements stop #

The fragment is a fault of a list, not of paper. Nothing here says a fragment would cause a folder trouble. A crease 0.13 mm long cannot be folded on its own, and the earlier census found that deleting one leaves a pattern whose panels do not close; but a printed sheet with such a crease folds as the paper allows, and whether a folder would ever notice is a question about hands, not lists.

Placements were sampled, not exhausted. Twenty-four random placements found none clean; that makes a clean common placement unlikely, not impossible, and the argument for why there is none is an argument about how corners are scattered, not a proof.

And the threshold is fixed at two values. At a thousandth and a ten-thousandth the shares are measured; the shape of the curve between and beyond them is not, though the six-fold fall for a tenfold threshold says it is not a straight line.

A rim with no width, and a threshold that is a choice #

The rim is a line of no width and the creases are lines, so a corner is either inside the paper or not and a crease’s cut length is exact. Real printing gives a rim and a crease a width of a few tenths of a millimetre, which is larger than every fragment measured here; on a printed sheet the fragment and the corner beside it are one smudge. The fragment threshold, a thousandth of the sheet’s side, is the one the census of drawing faults used; it is a statement about the list and about what a drawing can show, not about what a folder can do.

How the placements were checked #

Moving a drawing does not re-solve it. The twist geometry of a setting is solved once and the polygons are translated, which is what makes thousands of placements affordable; the translated drawing is required to be identical, segment for segment, to the centred drawing when it is moved by a whole period of the square and triangular tilings, where the tessellation repeats exactly. The magnified window requires the pieces it draws as fragments to be exactly the ones the reading counts. The five fragment drawings are required to have an equal-length partner on the opposite edge for every fragment, exchanged by the sheet’s half-turn on exactly four of them. The moved drawings are lettered and checked as the centred ones are, and are required to pass every vertex condition, close round every vertex and carry no fragment. And the survey is required to find no placement clean on all hundred and twenty drawings.

Still open: a threshold set by the paper #

The fragment threshold has been a fraction of the sheet since it was introduced, and the survey makes plain that its value decides the fault rate. A threshold set by the paper — the width of a fold’s radius, say, below which two creases on a printed sheet cannot be told apart — would make the fragment a statement about folding rather than about lists, and the density a paper allows has measured that width. Re-running the survey at it would say how many printed twist patches carry a crease the paper itself cannot form, which is the number a designer would actually want.

The second lead is the lettering that moved. One nudged drawing in ten came back from the search with letters forcing a loop. What the rim was doing found that a sheet’s edge decides how hard its lettering is to find, and cutting a patch out of a plane that where the paper stops decides what is drawn at all; a sweep of placements, lettering each, would measure how often a tenth of a millimetre of paper at the edge changes the answer, which is a tolerance on placement in the sense a tolerance is a direction measured tolerances on a mesh.

Sideways, the same structure — each case fixable, no single fix for all — is how a stub is never alone and how deep is a crossing measured the other drawing faults, one drawing at a time. Whether a placement exists that clears the extension construction’s crossings and stubs together with the fragments is a question the same survey could answer for the other construction.

The habit worth carrying is about repairs proposed in the singular. When one adjustment repairs each case, ask whether one adjustment repairs every case before writing it into the method. A tenth of a millimetre cleaned every drawing it was aimed at; aimed at all of them together, the same tenth of a millimetre cleaned none.