Two faults, not four

Four ways to draw a pattern separated the ways a drawing can differ from the crease list it was drawn from. A crossing is two creases passing through each other at a point neither lists; a stub is a crease with a free end in the middle of the paper; a junction is a crease ending on the interior of another; a fragment is a crease too short to see. The first three are the drawing having more than the list; the fourth is the list having more than the drawing shows.

How deep is a crossing gave the first a distance and a stub is never alone gave the second one, and found something about the second that changed what it was worth: every drawing with a stub also had a crossing, so a test for crossings already refused everything a test for stubs would. It ended by naming the distance the third fault ought to have — how far a junction lands from the nearest vertex of the crease it ends on — and by proposing the census that would settle all four at once.

That census is below. It answers the question about the third distance in the least expected way, and it finds the fourth fault in the one place it was assumed not to be.

Two constructions, two hundred and forty drawings #

The population is the same one the stubs were counted on, drawn both ways. Five tilings — the square grid, the triangular grid, the honeycomb, the rhombille and the elongated triangular tiling — at three periods and eight turns of the twist, each built as a patch of paper with a twist at every vertex of the tiling.

The two constructions differ at the edge of the sheet. The extension construction draws the twists whose polygons lie on the sheet and runs every pleat that heads off the paper along its own line until it reaches the rim. The clipped construction draws the tessellation over a region larger than the sheet and cuts the drawing to the square, so every crease ends where the paper does. The earlier essay recorded that the clipped construction has neither crossings nor stubs, and that is confirmed: none of its 120 drawings carries either.

Each drawing is read twice, as its list of segments and as the ink a reader would fold, and the four faults are the four ways the readings can disagree. The counts are of drawings carrying at least one of each.

The third distance has nothing to measure #

The junction count comes back at 240 of 240, which looks like the commonest fault of all, and is not a fault at all.

Every junction the reading finds is a crease meeting the rim. The crease list records the sheet’s boundary as four long edges, one per side, and does not split an edge where a crease arrives at it. So a crease ending on the rim ends on the interior of a listed edge, which is exactly what the reading’s definition of a junction says. It is also exactly where a crease is supposed to end: a fold line runs from edge to edge of whatever it folds, and the rim is the edge.

The fault the name was meant for — a crease ending on the interior of another crease, a T where no vertex was listed — occurs on none of the 240 drawings, and on none of the eight printed patterns or the seven fold-and-cut outlines either. The distance the earlier essay asked for, from such an endpoint to the nearest vertex of its host, has no instance to be measured on.

That is a result about the constructions, and it has a reason. A twist tessellation’s creases are built polygon by polygon: each pleat runs from a corner of one twist polygon to a corner of the next, and each polygon’s corners are listed vertices, so any two creases that meet in the interior of the sheet meet at a listed vertex by construction. The only place a crease ends somewhere the construction did not put a vertex is where it meets the rim — and there the construction deliberately did not split the rim. The fold-and-cut outlines drop perpendiculars onto outline edges and split the edges at the feet, so they have no interior junction either; the reading finds their rim junctions and nothing else.

So the category is empty, and emptiness is the finding. A junction distance would have been a measurement of a failure these constructions cannot commit. The earlier essay was right that such a distance would be hard to interpret — a junction near a vertex and one far from any mean opposite things — and the census makes the difficulty moot rather than resolving it.

The stub count is unchanged by the census and it is worth seeing again beside the others, because it is the fault whose redundancy was found first. Fifteen drawings, sixty-six stubs in thirty-three pairs, and not one drawing where a stub is the only thing wrong. The census adds the other three faults to that picture and asks the same question of each: does it ever occur where nothing else does?

Which faults travel together #

With junctions set aside, three faults remain, and the second figure groups every drawing by the combination it carries.

Only five combinations of three faults occur: nothing wrong; a crossing alone; a crossing and a stub; a crossing, a stub and a fragment; and a fragment alone. A stub never occurs without a crossing — the earlier essay’s finding, now seen alongside the rest. A fragment occurs either with both of the others or on its own, never with a crossing and no stub.

The two drawings with all three are both extension drawings at the settings where the clipped construction also makes fragments: the honeycomb at period 0.34 and turn 0.35, and the rhombille at period 0.28 and turn 0.3. The five clipped drawings with a fragment have nothing else wrong. On them the fragment is the only difference between the drawing and the list, and a checker without a fragment test would pass them.

So the four tests a checker carries reduce to two that carry independent information. The crossing test refuses every drawing the stub test refuses, and sixty-one more. The junction test, read as a fault, refuses nothing. The fragment test is the only one that refuses a drawing none of the others would, and it is also the only one the clipped construction needs.

Fragments come in pairs, as stubs did #

The fragments have a structure of their own, and it is the stubs’ structure.

Every drawing with fragments has an even number of them — two, four, eight or twelve — and every length occurs an even number of times. The honeycomb’s twelve are six pairs at two lengths, 0.0012 mm and 0.0088 mm on a 150 mm sheet; the clipped rhombille’s eight are two lengths four times each. The mechanism is the one a stub is never alone gave for its pairs. A pleat is two parallel creases of equal length, and a clip that catches one of them almost exactly at the corner of the sheet catches the other at the same place, leaving two slivers of the same length.

And the lengths are far below anything a pencil or a printer draws. The longest fragment is 0.13 mm, a third of a pencil line’s width; the shortest is 0.0012 mm, about a hundredth of a sheet of paper’s thickness. The reason a fragment cannot simply be deleted is that each separates two panels a fold places differently — remove the honeycomb’s twelve and two routes to one panel disagree by 2.3 sheet widths — so these are not blemishes to tidy away. They are creases the list needs and the drawing cannot show, which is the one fault that cannot be seen by looking at the paper.

The same slivers from both constructions #

The fragments’ settings say where they come from, and it is not the construction’s rule at the edge.

The extension construction’s two drawings with fragments are the honeycomb at period 0.34 and turn 0.35, and the rhombille at period 0.28 and turn 0.3. The clipped construction makes fragments at both of those settings too, and on the honeycomb they are the same twelve slivers at the same two lengths, to the last digit. The two constructions disagree about almost everything at the edge of the sheet — one runs pleats across one another to reach the rim, the other cuts them where the rim falls — and they agree exactly about these.

What they share is the sheet. Both place the square in the same position on the same tessellation, so both put the sheet’s corners at the same points of the pattern, and a fragment is made where a corner of the sheet falls within a fraction of a millimetre of a corner of a pleat. The extension rule adds nothing there and removes nothing: the sliver is a property of where the sheet is laid on the tessellation, which neither construction chooses and both inherit. The clipped construction’s three other fragment settings have no counterpart in the extension drawings, which do not draw the pleats that head off the paper at all and so have nothing there for the edge to slice.

That turns the fragment into a tolerance question of exactly the kind this line of argument has been asking. A tolerance is a direction found that a solved mesh forgives errors in some directions and not others; the fragment is a drawing that fails because its sheet sits a thousandth of a millimetre from a coincidence. The fault is real in the list and unmeasurable on the paper, and the account above predicts that moving the sheet by a little more than the fragment’s length, in the right direction, removes it — which is the experiment the last section proposes.

What the clipped construction’s cleanliness was #

The earlier essays reported the clipped construction as the clean one — no crossings, no stubs — and it is, by those two tests. It is not clean by the fourth.

The reason is where the clip falls. The clipped construction cuts the tessellation at the sheet’s edge wherever the edge happens to be, and at five of its 120 settings the edge passes within a fraction of a millimetre of a pleat’s corner. The cut then leaves a sliver of crease inside the sheet, too short to see and too long to be nothing, and its pair beside it. The extension construction trades that fault for two others: it never cuts a pleat, so it never leaves a sliver, but it runs pleats across one another and stops them short.

That makes the choice between the constructions a choice between fault profiles rather than between a faulty one and a clean one. The extension construction fails on 76 drawings of 120, nearly all of them visibly; the clipped construction on 5, every one invisibly. A visible fault is a refusal a reader can check; an invisible one is a refusal a reader would argue with, which is the same point how deep is a crossing reached about the shallowest crossings, now made about a whole construction.

A checker smaller than it looks #

A checker that has accumulated tests over years has more tests than independent questions, and nobody finds out which until the tests are run side by side. This census is that comparison for the drawing faults, and it is short: of four tests, one refuses a subset of another’s refusals, one refuses nothing, and two are independent.

That is not an argument for deleting the redundant tests. A stub test is cheap, it names its own repair — extend the crease by the distance measured — and a crossing test does not; a junction test would catch a construction that did end creases on creases, and the next construction written might. What the census establishes is how much independent evidence a clean result carries. A drawing that passes all four tests has passed two questions, not four, and a report that it “passed four checks” overstates what is known by exactly the redundancy.

The same comparison applies to every other check a pattern here is put through. A tolerance is a direction and the allowance is spent at the end measure how far a built sheet may depart from its drawing; solved is not built separated two reasons a mesh folds. Each is a test with its own reason for existing, and none has been run beside the others to see which of them ever refuses alone.

What the census cannot show #

It is one family of patterns. Every drawing is a twist tessellation built one of two ways, plus the printed shelf and the fold-and-cut outlines. A construction that places creases by a different rule — a tree-method base, a hand-drawn design — could end creases on creases freely, and the empty junction row says nothing about it.

The fragment threshold is a thousandth of a sheet. A crease shorter than that is counted as a fragment, and four ways to draw a pattern found a gap of more than a factor of ten above it with no crease in it, so the threshold is not doing much work here. On a different population it might be.

And a count is not a severity. A drawing with one crossing and one with a hundred and eighty-four are both counted once; a fragment of a thousandth of a millimetre and one of a tenth are both fragments. The census says which faults occur together, not how bad any of them is.

The drawing the census reads #

A drawing is its segments as ink: every listed crease drawn edge to edge between its endpoints, with no line weight and no gap. A crossing is a proper crossing of two creases’ interiors; a stub is an interior endpoint with one crease and nothing else touching; a junction is a listed vertex lying on the interior of another listed edge; a fragment is a crease shorter than a thousandth of the sheet’s side.

Distances are on a 150 mm sheet, the size these patterns print at; every length scales with the sheet.

And the rim is listed as it is listed. Four long boundary edges is how every pattern here records its sheet, and a format that split the rim at every crease would report no junctions at all; the census reports the convention rather than hiding it, because the convention is what made the category look full.

How the census was checked #

Every stub is required to lie on a drawing with a crossing, every junction on a crease to be absent, and every drawing to have creases ending on the rim; the figure is refused if any of the three fails. Every clipped drawing with a fragment is required to carry no crossing and no stub, every fragment length to occur an even number of times, and every drawing to fall in one of the five combinations drawn, so a drawing with a sixth combination would stop the figure rather than be left out of it.

Still open: a construction that makes junctions #

The empty row is a property of how these patterns are built, and the natural test of the account is a construction that builds differently. A pattern whose creases are placed by a rule that does not route every crease through listed corners — a tree-method base drawn from its molecules, or a pattern read off a photograph — could end one crease partway along another, and the junction distance the earlier essay defined would then have instances. Whether they cluster near vertices, as a drawing that nearly listed a vertex would, or spread along the host crease, is the first measurement such a population would allow.

The other direction is the fragments on the clipped construction. They occur where the sheet’s edge passes within a fraction of a millimetre of a pleat’s corner, and nudging the clip by that fraction would remove them without changing anything a reader could see. Whether a small, stated shift of the clip removes every fragment on every setting — and what it does to the patterns’ symmetry — is a short computation that would make the clipped construction clean by all four tests.

Sideways from here, the rim convention is worth stating wherever crease-pattern files are compared. A file has no paper found the format silent on where the paper is; the census adds that how the paper’s edge is listed decides whether every drawing has a hundred junctions or none, which is the kind of difference two programs reading the same file would disagree about without either being wrong.

The habit worth carrying is about categories that come back full or empty. When a test fires on everything, or on nothing, look at the definition before the data. The junction test fired on every drawing because of how the rim was listed, and on the drawings its name was meant for it fired on none; both answers were facts about the test, and only the second was a fact about the drawings.