One choice with eleven answers

The field is empty where it would say nothing filled in the crease-pattern format’s layer-order field for every pattern whose folded states could be listed, and found most of them with exactly one. On those the field would restate what the crease pattern already implies. It ended by proposing a better record than a sign for every overlapping pair: a folded state is a small set of free choices — which side a flap lies on, where a rim panel sits — and every sign follows from the crease pattern plus the choices. It asked whether the four-by-three Miura’s five states were two independent choices of two and a half options, or something less tidy.

The question has an answer, and finding it turned up something the earlier table could not have shown. The choices are never independent. And the five states were not five.

A blind spot a third of a panel wide #

The folded states of a pattern are listed by ordering its panels. Two panels need an order only if they share paper once folded, so the first step is to decide, for every pair of folded panels, whether they overlap — and every constraint and every state downstream is built on that list of pairs.

The test that made the list was a reasonable one. Two polygons overlap if an edge of one properly crosses an edge of the other; and if none does, they overlap only if one lies inside the other, which is checked by testing whether the other contains any of five probe points — the polygon’s centroid and four points pulled three quarters of the way toward it from each corner. That catches crossings, and it catches the commonest case in folding, two panels folded exactly onto one another.

It does not catch the Miura’s.

Folding a Miura collapses each row into a zigzag, and panels two columns apart land on the same slanted line, offset along it by a fixed amount. Their slanted edges are collinear, so no edge crosses another properly — each ends exactly on the other’s line. Their horizontal edges cross the other’s slanted edges exactly at corners, which is not a proper crossing either. And the strip they share, 35.6 per cent of a panel’s area at the shear these essays draw the Miura at, runs between the two panels’ probe points: the nearest probe misses it by two hundredths of the panel’s height.

So every pair of panels two columns apart was reported as not overlapping, and every constraint that pair would have imposed was never imposed. On the three-by-three patch that was nine of its thirty-six overlapping pairs; on the four-by-three, eighteen of sixty-six. The printed six-by-four Miura lost sixty-four.

The repair is to ask the question the test was standing in for. Every panel of every pattern folded here is convex, and two convex polygons overlap exactly when the region they share has area, which can be computed by clipping one against each edge of the other. That is now how the overlap is decided, with the probe test kept only for a panel that is not convex; no printed pattern has one.

What the corrected counts are #

With every overlapping pair present the Miura’s folded states come out as the first figure draws them. A two-by-two patch has one state; the three-by-two has three; the three-by-three has six; the four-by-three has eleven. A patch two columns wide has one state at every length measured, from two rows to six. A strip two rows high has exactly $2c - 3$ states for $c$ columns, from three columns to seven — three, five, seven, nine, eleven.

The earlier table had the two-by-two right, since its four panels all lie on one another and the probe test sees that. Everything larger it undercounted. It reported the three-by-two and three-by-three with one state each and every sign forced; they have three and six, with four and nine signs varying. It reported the four-by-three with five states and nine varying signs; it has eleven, with twenty-seven varying.

That reverses the earlier essay’s conclusion for the Miura. It found the field redundant on the smaller Miura patches and informative only from twelve panels, one size before the search gave out — a window “one size wide”. With the pairs present, the field carries information on every Miura patch with three or more columns, from six panels up: 1.58 bits on the three-by-two, 2.58 on the three-by-three, 3.46 on the four-by-three. The preliminary base, the square twist and the hexagon twist keep their single states — none of them has panels slid along a shared edge — so the earlier finding stands for them, and the claim it was generalised to does not.

The smallest strip is two letter folds #

The three-by-two strip is small enough to read its states by eye, and what they turn out to be connects the Miura to the plainest fold there is.

Each row of the strip is three panels in a zigzag, and a zigzag of three is a letter fold: the two end panels fold onto the same side of the middle one and lie against each other. One sheet down found that a letter fold has two folded states, because nothing decides which end panel is on top — the two ends are not joined to each other by a crease, so no rule orders them. The Miura strip is two such letter folds, one per row, joined along the zigzag crease that runs between the rows, so each column’s two panels move together.

Listed top to bottom, the three states are these. The right-hand column lies over the left-hand column; or the left-hand column lies over the right-hand one; or the left-hand column is tucked between the right-hand column’s two panels, so that in the top row the right end is uppermost and in the bottom row the left end is. The middle column is underneath in every state.

Two letter folds would give four combinations, and the strip allows three. The fourth — the other interleaving, left end uppermost in the top row and right end in the bottom — would need the two columns to pass through each other along the crease that joins the rows, and the rule against two folds interleaving in the same place forbids it. That one forbidden combination is the whole reason the choices are not independent: the rows are two choices that share the crease between them, and sharing it is what ties them into one.

It is also why the old test could not see any of it. The two end panels of a letter fold on a square sheet lie exactly on top of one another, and the probe test finds that. On the Miura each row is sheared, so the ends land offset along the shear — slid along a shared edge, the one arrangement the test was blind to. The Miura’s layer order was invisible for the same reason it is interesting: the shear that makes the pattern fold in one motion is what lays its end panels side by side along a line.

Are the choices independent? #

With the states listed correctly, the question the earlier essay asked can be answered properly. A state is a sign for every overlapping pair; the signs that never vary are forced; the ones that vary are the choices. Two varying signs that always flip together, or always opposite, are one fact about the stack. So the varying signs collapse into a smaller set of facts, and a state is a row of bits, one per fact.

The three-by-three patch’s nine varying signs collapse into four facts. Four independent binary facts would give sixteen states; the patch has six. Its facts are not independent, and the pattern of which combinations occur is not a product of anything: the rows that fold are 0000, 1100, 0100, 1111, 1101 and 0101 — every state with the third fact set also has the second and fourth; the fourth may be set only with the second.

The decisive test is whether the facts split into groups that do not touch. Two facts involving disjoint sets of panels could, in principle, be chosen separately — a flap at one end and a flap at the other. Facts sharing a panel cannot, since moving that panel moves both. Merging every pair of facts that share a panel gives the independent choices, and on the three-by-three there is exactly one: all four facts share panels, and six panels of the nine take part.

The four-by-three is the same shape of answer at a larger size: twenty-seven varying signs, twelve facts, one chain of shared panels, eleven states out of the 4,096 the facts could make. On every Miura patch listed, the choices are one choice. There are no independent flaps anywhere in the family. What varies between the states is a single region of the stack — on the three-by-two strip the panels of its two end columns, on the larger patches most of the sheet — and each state is one of its positions.

What that means for a record #

The proposal was to replace the layer-order field’s many signs with a few free choices. The answer is that there is nothing to factor. A Miura patch’s folded state is best recorded as a single index into its list of states, and the list is short: $2c - 3$ for a two-row strip, so the index costs $\log_2(2c - 3)$ bits, which grows as the logarithm of the patch’s length.

That is a better economy than the proposal hoped for, not a worse one. A list of independent choices would need a bit for each, and the four-by-three’s twelve facts are twelve bits; its eleven states are 3.46. The field itself stores sixty-six signs. A reader who knows the crease pattern needs three and a half bits to know the folded object, and neither of the two records the format has — the signs, or a list of flap positions — says it that cheaply. It is also a record a person could use: the first thing about layers found no layer-order rule simple enough to teach, and on the smallest strip “the end columns go over, under, or between” is one sentence.

It also changes what a file has no paper and the file records no verdict implied about the Miura in particular, and it is the answer no height to swap gave for the printed patterns, extended: where a pattern has a choice at all, the choice is a single rearrangement of one region rather than several independent swaps. A crease pattern file for a small Miura was taken to describe one folded object; it describes several, and a designer who publishes one has published a family. On a strip two rows high the family grows by two for every column added, and nothing in the file says which member the folded model is.

Why a blind spot survives #

A test that is wrong on a whole family of patterns should have been noticed, and it is worth asking why it was not.

The test was checked against an independent result, and passed. Every sheet-ordering count is required to agree with the independent count for strips on patterns whose creases are parallel, where the answer is known by other means, and it does. But a strip’s folded panels lie exactly on top of one another, which is the case the probe test was written for. The Miura’s panels are the case it was not, and the agreement check never visits it.

The wrong answer was the plausible one. One folded state for a small Miura is what the preliminary base and both twists also gave, and it fitted the account the earlier essay built — that freedom comes from flaps and rims and a small pattern has none. A result that confirms an explanation is read less suspiciously than one that breaks it, and this one confirmed two.

And the failure is silent in the direction that matters. A missed pair removes constraints, and removing constraints can only let more orderings through; but a state is counted by its signs over the pairs present, so the same missing pairs that loosened the search also blinded the count to the orderings it let through. Five states on the four-by-three was an undercount produced by a test that was, in its own terms, too permissive.

What the listing still cannot settle #

The search still stops. It lists every Miura strip two rows high to seven columns, and the three-row patches to four columns, inside its usual budget. Given two hundred times that budget the four-by-four gives nineteen states in one chain and the eight-by-two still does not finish. That a two-row strip has $2c - 3$ states is measured on five lengths and is not proved; that every patch’s choices are one chain is measured on every patch listed.

The states are the rules’ states. A folded state here is an order of the panels that satisfies the crease rule and the two non-crossing rules. Whether every one of the three-by-three’s six can be reached by folding paper, rather than merely drawn without contradiction, is not something the listing can say — and the new states are exactly the ones in which panels two columns apart are ordered against their neighbours in ways a folder would have to arrange deliberately. The fold a machine can make asks that question of strips folded one crease at a time, and the Miura’s states are the natural next object for it.

And the other patterns are not re-examined here in full. The printed patterns other than the Miura were checked against the exact test and none of them loses a pair; patches built from other tilings, and sheets glued into tubes, rest on the same test and are the next place a slid pair could be hiding.

The record the counts assume #

Two panels overlap when the region they share has area. A pair that touches along an edge or at a corner shares no paper, needs no order, and is not a pair; this is the same convention the earlier listing used, now applied exactly.

A state is its signs over the overlapping pairs, so two orderings that differ only in panels that share no paper are one state, because nobody could tell them apart by looking.

And a fact is a set of signs that always flip together. Two signs that flip together in every listed state are one fact about the stack; two facts sharing a panel are one choice. The grouping depends only on the states listed and would change if a longer listing found a state that separated them.

How the counts were checked #

The exact overlap test was compared with the old probe test pair by pair on every printed pattern and on Miura patches to six by four. They agree everywhere except on the Miura, and every pair the old test missed there — nine on the three-by-three, eighteen on the four-by-three, sixty-four on the six-by-four — is a panel and a copy of it slid along one of its own edges. The blind-spot figure is refused unless every probe point of each panel really does lie outside the other. The strips’ counts are required to be $2c - 3$, and every patch’s varying signs are required to form one chain — so a patch whose choices split would stop the figure rather than be drawn as one.

Still open: a count that does not need the listing #

The $2c - 3$ law is too regular to be an accident, and an argument that produced it from the zigzag’s structure would replace the listing on the strips the search cannot reach. Each added column adds two states, which suggests each column contributes one new position for the single chain on either side of the ones already there; the three-row patches grow faster, six to eleven for one more column, and whether they are linear in the columns too is the first thing a longer listing would show.

The other direction is reachability. Now that the Miura has several states, it is a question which of them a folder produces. The half no notation records found the crease pattern silent on the folded object; a diagram sequence, which what a dashed line can say describes, builds the stack one fold at a time and so picks a state. Which of the eleven a standard collapse sequence picks, and whether any of them needs a sequence nobody has written, is a small and definite question.

Sideways from here, the blind spot is a pattern worth looking for in every test of this shape. A test that probes a region by a few sample points will miss a thin region that runs between them, and the thin regions are exactly where two shapes slide along one another — which is what a folded tessellation does everywhere. Publishing the pattern instead of the sequence moved the burden of finding the folded object onto the reader; a reader with a sampled test would have found one state where there are eleven.

The habit worth carrying is about checks that agree. A check that passes against an independent answer has been tested on the cases that answer covers, and no others. The strip agreement said the ordering was right on strips, and it was; the Miura is not a strip, and a result that confirmed a tidy account was left unexamined because nothing asked the test the one question — do these two panels share paper — in the one arrangement it could not see.