The field is empty where it would say nothing

The half no notation records found a field left empty. The subject’s interchange format has a place to record which panel lies above which in the folded object — a sign for each pair of panels that overlap — and no file anyone publishes fills it in, including the files for the patterns printed here. It measured the size of what was missing as the number of bits needed to pick one ordering of the panels out of all of them, which for a pattern of $P$ panels is $\log_2 P!$, and it closed by proposing the obvious repair: fill the field in for the patterns where that can be done, and see whether the order recorded is the order or merely an order.

That repair is now possible on half the printed shelf, and it answers a different question from the one it was asked for.

What the field holds #

A crease pattern file lists vertices, the edges joining them and a letter for each edge. The layer-order field, where a file has it, lists triples: two panels that overlap in the folded state, and a sign saying which is above. A complete field has one triple for every overlapping pair.

So the field’s size is fixed by the pattern’s geometry. The preliminary base folds its eight panels into a stack where every pair overlaps — twenty-eight signs. The square twist has thirty-six, the hexagon twist sixty-six, the fold-and-cut triangle twenty-one. The Yoshimura’s sixty-five panels overlap in 2,055 pairs.

What the field says is a different number. A folded object is one of the pattern’s legal folded states, and naming which one takes $\log_2 S$ bits for $S$ states. The signs that carry that information are the ones that differ between states; a sign that is the same in every state is one the crease pattern already decides.

Four patterns listed, three with one state #

The stackings of a pattern can be listed exhaustively by placing panels from the bottom up and refusing any placement that breaks a rule — a crease that would open the wrong way, a panel that would sit between the two panels of a crease running through it, two folds in the same place crossing each other. The search refuses above eighteen panels, which leaves four printed patterns in reach.

The preliminary base has one folded state. Its twenty-eight signs are all forced. The square twist has one, and so does the hexagon twist, with thirty-six and sixty-six signs forced. On all three, a file that filled in the layer-order field would record a list of signs every one of which a reader could compute from the letters already in the file.

The fold-and-cut triangle has two folded states. Of its twenty-one signs, six differ between the two, and they differ together: choosing the state is one bit, and the six signs are that one bit written six times.

The answer to the question as it was put — the order or an order — is therefore the order on three of the four patterns that can be checked, and an order on the fourth. On none of them is it much of a record.

The measure that was out by the whole field #

The earlier measure of the missing half set $\log_2 P!$ beside each pattern: fifteen bits of order for the preliminary base’s eight panels, eighteen for the square twist’s nine, thirty-three for the hexagon twist’s thirteen. That is the number of bits needed to pick an ordering of the panels if every ordering were possible, and almost none of them are.

On the preliminary base, the rules leave exactly one ordering standing, so the information the field would carry is zero bits, not fifteen. On the hexagon twist it is zero, not thirty-three. The measure counted the ignorance of a reader who knows nothing about folding, and a reader holding the crease pattern knows almost everything the field would tell them.

That changes what the empty field means on the patterns that can be checked. It is not half the object going unrecorded; it is a redundant restatement left out. The earlier essay was right that no published file says which state its object is, and wrong about how much there is to say, and the error was the size of the whole field. The correction does not reach the large patterns, where nothing is known either way, but it removes the only measurement that had been made, and it replaces it with one that can be checked: the number of folded states, which is either listed or refused and never estimated from the panel count alone.

Where the field would earn its place #

The pattern with more than one state is the instructive one. The fold-and-cut triangle folds a sheet so that a single straight cut releases a triangle, and its seven panels stack with one flap free to lie on either side of its neighbour. Six signs flip together when the flap moves, because the flap overlaps six other panels and its position relative to all of them changes at once.

A file that recorded the triangle’s folded object would need to say which side the flap is on. The layer-order field says it by storing all twenty-one signs, fifteen of which are forced; the information is one bit. The format’s representation is correct and its economy is about twenty to one, which matters little for a seven-panel pattern and a great deal for a sixty-five-panel one.

The Miura, as it grows #

The Miura is the pattern where the question matters most, because it is the one that gets built, and its patches can be listed up to twelve panels.

A two-by-two Miura has one folded state: six signs, all forced. From three columns up the stack has room to rearrange. The three-by-two has three states among fifteen signs, four of which vary; the three-by-three six states, nine varying signs of thirty-six; and the four-by-three eleven legal orderings and eleven distinct states, 3.46 bits carried by the twenty-seven signs that vary among sixty-six. At four by four — sixteen panels, a hundred and twenty pairs — the listing exhausts its budget before finishing.

These counts replace ones this essay first gave, which were wrong. The overlap test that decides which pairs of panels share ground had a blind spot: two panels of a folded Miura two columns apart are the same parallelogram slid along its own slanted edge, no edge of one properly crosses an edge of the other, and the strip they share missed every point the test probed. So those pairs were never ordered, and the counts came out as one state on the three-by-two and three-by-three and five on the four-by-three. One choice with eleven answers draws the blind spot and repairs the test; the printed patterns other than the Miura lose no pair to it, and their counts above are unchanged.

So the Miura has a real layer-order question from six panels, and becomes unlistable at sixteen. On every Miura patch that can be listed with three or more columns the field carries information, from 1.58 bits on the three-by-two to 3.46 on the four-by-three; the pattern that gets built is the one on which the field is never redundant.

Why small patterns have one state #

The three one-state patterns are not accidents. A pattern’s folded state has freedom only where a panel, or a group of panels, can move past its neighbours without crossing a crease, and a small pattern mostly has none.

In the preliminary base every panel is joined to its two neighbours by creases that alternate mountain and valley round the centre, and a crease fixes the order of the two panels it joins. Eight panels round one vertex with eight such creases leave no pair unconstrained. The twists are similar: every panel is either joined to its neighbours or pinned between the images of creases running through the stack.

Freedom appears when a panel is joined to the rest of the stack along only one crease and overlaps panels it is not joined to — a flap. The triangle has one such flap and gets two states. The Miura patches have end columns that fold like the two flaps of a letter fold, lying against each other with nothing to order them, and get three, six and eleven. A file has no paper observed that a file’s three arrays cannot say where paper is; the flaps are where the paper’s position is not said by the creases either.

What that implies for the patterns nobody can list #

The four unlisted printed patterns — the Miura, the tapered corrugation, the waterbomb and the Yoshimura — have twenty-four to sixty-five panels and hundreds to thousands of overlapping pairs. Nothing here says how many folded states they have.

What the small cases suggest is that the answer is governed by flaps and rims rather than by the number of panels, and that it grows much more slowly than $\log_2 P!$. A sixty-five-panel Yoshimura has 302 bits of orderings by the earlier measure; its real layer-order information is whatever freedom its rim panels have, which might be tens of bits, and is certainly not three hundred. A file that recorded it would be recording a small number of real choices in a large number of signs, and the right format for that is a list of the choices, not a list of every pair.

That is also the reason the field is empty in practice, and it is a better reason than neglect. The file records no verdict set the layer verdicts beside a crossing sweep and found the sweep the only check cheap enough to require of every file, because a layer verdict is a search. The listing adds the other half of why. Where the field is cheap to fill, it is redundant, and where it would carry information, it is expensive to compute. Nobody fills a field that is always one or the other.

What a partial field would look like #

The format allows a layer-order field to be partial — to list some overlapping pairs and not others — and the listing says which subset would be enough.

A partial field listing only the varying pairs is complete in information. On the fold-and-cut triangle it is six signs instead of twenty-one; on the four-by-three Miura twenty-seven instead of sixty-six. A reader holding the crease pattern and those signs can recover every other sign by the same three rules the listing used, because every other sign is the same in every state and the rules force it. That makes the partial field both smaller and more honest: its entries are exactly the decisions the letters did not make.

It is also a teachable object, which the complete field is not. The first thing about layers found that no rule about layer order was ever simple enough to teach, because the question is global and every answer was a search. A list of the few pairs a pattern leaves free is a list of the places a folder actually has to choose, and on the small patterns that list is empty or a single flap. Where the list is empty, there is nothing to teach; where it has one entry, the teaching is one sentence.

The awkwardness is that identifying the varying pairs requires listing the states, which is the expensive part. A format can permit a partial field; it cannot tell its writer which pairs belong in it.

A notation’s history, read against this #

The Yoshizawa–Randlett symbols, which what a dashed line can say found to express exactly one simple fold at a time, never needed a layer order: a diagram sequence builds the stack one fold at a time, and the order is whatever the sequence produced. Publishing the pattern instead of the sequence is where the order was lost — a crease pattern replaces the sequence with the letters, and the letters decide the order only where the pattern has one state.

So the shift from diagrams to crease patterns lost the layer order exactly on the patterns with flaps — the patterns where a sequence of folds makes a choice the finished letters do not record. For a traditional base, one state; for a complex design with many flaps, many. The information the format could not hold grew with the designs the format was adopted for, which is a sharper version of the earlier essay’s observation that publishing patterns moved the difficulty onto the reader.

What the listing assumes #

A folded state is a flat stack that obeys three local rules. A crease fixes its two panels’ order; a panel cannot sit between the two panels of a crease running through it; two folds in the same place cannot interleave. The listing finds every ordering satisfying all three. That the rules are the whole of flat foldability is not claimed, and a state the rules allow might still not be reachable by folding.

States are counted by the signs of overlapping pairs. Two orderings that differ only in panels that do not overlap are one state, because nobody could tell them apart by looking.

The field is taken to be complete. A file could record a partial layer order — only the pairs that vary — and the format permits that; the comparison is with the complete field because that is the one whose absence was measured.

What the listing cannot settle #

It stops at eighteen panels. The limit is the search’s and not the subject’s: a pattern of twenty panels has more orderings than the search can refuse its way through, and the refusal is reported rather than replaced by a sample. The four printed patterns anybody would want a layer order for are past it, and the claim that their information is small is an extrapolation from patterns of seven to thirteen panels and Miura patches of four to twelve.

It does not address where lines meet. The reader decides the junction found a separate ambiguity in the same files — whether a crease ending on another crease meets it or passes through — and a layer order is computed after that question has been answered; a file that left it open would not have one folded state or several but a different pattern.

It says nothing about which state a folder produces. The triangle’s two states are both legal; a folder following a sequence makes one of them, and the file cannot say which without a record of the sequence or of the object.

And it does not measure the cost of computing the field. A listing that took under a second on twelve panels exhausted two million nodes at sixteen, and whether a cleverer search reaches the printed Miura’s twenty-four is a question about search rather than about notation.

Still open: the choices, not the signs #

The small patterns point at a better record. A folded state is a small set of free choices — which side a flap lies on, where a rim panel sits — and every sign in the layer-order field follows from the crease pattern plus those choices.

What would settle how small that set is on real patterns is a listing that works flap by flap: find the panels whose order is not forced by any crease or through-crease rule, and enumerate only their placements. On the four-by-three Miura the twenty-seven varying signs come from eleven states, and whether those eleven are the product of independent choices or something less tidy is visible from the states already listed — one choice with eleven answers lists them and finds a single choice.

Sideways from here, the one-state result says something about attribution. No format has a gluing found the format unable to say which sheet a pattern is on. A pattern with a single folded state is one whose object is fully determined by its file, and a pattern with many is one whose file describes a family of objects. Whether two designers who publish the same crease pattern have published the same model is a question with a definite answer on the first kind and none on the second.

The habit worth carrying is about measuring what a record omits. Count the information against a reader who knows the rest of the record, not against one who knows nothing. $\log_2 P!$ is what a layer order would cost to someone without the creases; with the creases, it cost nothing on three patterns in four, and measuring the gap against the wrong reader made a redundancy look like half the object.