A slit tip keeps one condition

A cut is a licence turned what a cut buys into a count. Cut one crease of a crease pattern and the vertices at its ends stop being interior vertices; none of the four conditions the subject states at an interior vertex — developability, Kawasaki, Maekawa and the big-little-big lemma — applies at them any more; and the share of letterings the pattern admits rises by a factor of two for each vertex released. It measured the factor on the square twist, cutting each of its twelve creases in turn, and found it exact on every cut. Its ledger said so plainly: a cut changes the share “by exactly the factor of two per vertex it releases — no more, and not less either”.

It also named what it had not checked. A released vertex is “subject to the strip condition rather than to nothing”, and a pattern whose released vertices are constrained by it “is a pattern whose cut bought less than the ledger says”. Measuring the gap meant asking, of every fan a cut produces, whether it admits every lettering — a question with an exact answer that had not been asked at scale.

This asks it at the scale of the printed shelf, and the gap is not small, not everywhere, and not where the ledger was measured. It is the latest turn of an argument that began with what one cut buys: a fold cannot change how much paper surrounds a point and a cut can — and a cut that removes no paper at all still changes which conditions the point answers to.

What a slit tip still is #

A slit is a cut of no width, so it removes no paper. At the end of one — a slit tip — there is a full turn of paper, exactly as at an interior vertex, and a ring and a line put the difference precisely: the paper at a slit tip does not close round, because a walk round the point starts on one side of the cut and finishes on the other. The sectors come in a line rather than in a ring. The four conditions are all statements about a ring, so none of them is defined there.

But a line of sectors is a one-dimensional crease pattern — a strip, a full turn long, creased at the angles where the remaining creases leave the vertex — and whether a strip folds flat under a given lettering is a question with an exact answer, the one the vertices nobody checks built for the vertices on the edge of the paper. A strip has conditions of its own. The best known is the strip’s version of the big-little-big lemma: a segment strictly shorter than both of its neighbours has to be folded with its two creases opposite, mountain and valley, or the neighbours collide over it. A released vertex is subject to that lemma, read on its sectors, and the ledger counted it as subject to nothing.

Every slit on the shelf #

The test is direct. Take each printed pattern, cut each crease with an interior vertex at both ends — a buried crease, the kind the ledger recommended cutting — and ask, at each of the two vertices the cut releases, how many of its letterings the strip admits.

Of the 298 vertices released, 81 still refuse some of their letterings, on four of the six patterns that have a buried crease at all. On the Miura, 12 of 44. On the hexagon twist, 6 of 12. On the tapered corrugation, 15 of 54. On the waterbomb tessellation, 48 of 84 — its degree-six vertices, released, become fans of five creases that admit 24 of their 32 letterings. The square twist releases 8 vertices and none refuses anything; the Yoshimura pattern releases 96 and none does either.

So the ledger’s factor of two is exact on two patterns of six and an overstatement on four. And the pattern it was measured on is one of the two on which it is exact, which is the ordinary way a rule measured on one example acquires a word like “exactly”.

One rule for every refusal #

The refusals look various — six of eight at the Miura’s tips, twenty-four of thirty-two at the waterbomb’s — and they are one rule. At every one of the 298 released vertices, the letterings the strip refuses are exactly the ones that fold three neighbouring creases alike round two equal sectors, each shorter than the sector beyond it. At a degree-four tip there is at most one such pair, between the two long end sectors, and the three creases round it are all three creases: the refused letterings are the two in which everything folds alike. At a waterbomb tip of five creases the pair is two 45-degree sectors between two of 90, the three creases round them are three of the five, and the refused letterings are the eight in which those three agree, whatever the other two do.

The rule says at once why two patterns are clean. The square twist’s released vertices are fans of four 90-degree sectors, and the Yoshimura’s are fans of six 60-degree sectors: every sector equal to every other, so no pair is shorter than anything beside it, and the strip has nothing to refuse. A released vertex can keep a condition only if its sectors are unequal, and the pattern the ledger was measured on is one whose released vertices are as equal as sectors get.

The Miura’s slit tip #

The refusing vertices on the Miura, the hexagon twist and the tapered corrugation all refuse the same way, and the Miura shows it most cleanly.

A Miura vertex has sectors of about 110 and 70 degrees: its two large sectors lie on either side of one half of its straight crease, and its two small ones on either side of the other half. Slit along a straight crease, one of the two vertices it releases is left with the slit between its two large sectors, and its strip runs 110, 70, 70, 110: long at both ends, short in the middle. It admits six of its eight letterings, and the two it refuses are the two in which all three creases fold the same way — a roll, every fold turning the strip the same way, for which the strip has no stacking at all.

That is the strip’s big-little-big lemma meeting a tie. Neither middle sector is smaller than the other, so neither is strictly smaller than both its neighbours, and the lemma’s usual conclusion — those two creases must differ — does not fire. What survives of it is weaker: the pair of short sectors together, against the long ones on either side, forbid only the roll. The other vertex the same slit releases has the slit between its two small sectors, its strip runs 70, 110, 110, 70, short at the ends, and it refuses nothing.

Where in a fan the lemma bites #

The shelf’s patterns are drawn with symmetric angles, which is what puts two equal sectors in the middle of a released Miura vertex. A fan drawn without that symmetry behaves differently, and a map of every fan of the same kind shows how.

Over 1,296 such fans the rule is exact, and the tie on the diagonal refuses only the rolls at every size of difference tried, from end sectors of 95 degrees against middles of 85 to ends of 130 against middles of 50. A fan admits four of its eight letterings when one of its middle sectors is strictly smaller than both of its neighbours — the strip’s big-little-big lemma, forcing the two creases round that sector to differ and refusing every lettering in which they agree. It admits six when the two middle sectors are equal to each other and smaller than the ends: the Miura’s case, and the hexagon twist’s, and the tapered corrugation’s. And it admits all eight otherwise, when the smallest sector is at an end, against the cut, where a short segment has nothing to collide with on one side.

So the shelf’s quarter is a consequence of its symmetry. A released vertex of a generic pattern — a crease pattern drawn from a tree, say, or a crumpled sheet — refuses half of its letterings whenever a middle sector is strictly smaller than both its neighbours, and there the release buys half of what the ledger counts at that vertex.

The figure above is that case: a slit tip whose second sector, 50 degrees, sits between 120 and 90. Every lettering with the two creases round it alike is refused, and the vertex the ledger counted as admitting all eight of its letterings admits four.

Three where the ledger says four #

The census counts vertices; the ledger’s claim is about whole patterns. On a pattern small enough to letter every way, the two can be compared exactly.

Uncut, the four-by-three Miura admits 2,048 of its $2^{17}$ letterings, a share of one in sixty-four for its six interior vertices — two to the minus six, as the ledger predicts. Every slit through a buried crease releases two vertices, and by the four conditions alone every slit multiplies the share by four, as the ledger also predicts. With the strip’s condition at the two slit tips, the four slits along straight creases multiply it by three, and the three slits along zigzag creases by four.

The difference is the one the fan figures explained. A straight crease of the Miura runs between its vertices’ two large sectors at one end and its two small sectors at the other, so slitting it releases one vertex that refuses the roll and one that refuses nothing: a factor of two from the free one and one and a half from the other. A slit along a zigzag crease has a large sector on one side of it and a small one on the other, at both of its ends, so neither released vertex has two short sectors between long ones, and both are free.

Where the cut goes along a crease matters as well as how many vertices it releases, and the ledger’s instruction to count vertices is right about zigzags and three quarters right about straight creases.

What the ledger got right #

Nothing the earlier essay measured was wrong. The four conditions do stop applying at a released vertex, and by those conditions a slit multiplies the share by exactly two per vertex released — on every pattern, as the four-by-three Miura’s ledger column shows. A ring and a line proved the companion fact: cutting a ring open never costs a lettering, because restricting a ring’s folding to the line is defined everywhere and never collides. Both are about what the release removes.

What the ledger left out is what the release leaves. A slit tip keeps exactly one condition — the strip’s — and that condition is not always vacuous. Whether it bites depends on where the cut sits among the vertex’s sectors, and on the printed shelf that is a quarter of the released vertices, all of them where a slit runs between two large sectors.

What a designer takes from it #

The earlier essay’s three-line rule was: count interior vertices, cut through vertices rather than across panels, prefer buried creases. A fourth line belongs with them. Before cutting along a crease, look at the sectors at each of its ends: a slit that leaves a short sector between two long ones has released a vertex that still decides something. On a symmetric pattern that costs a quarter of the letterings at that vertex; on a generic one, half. Cutting the crease on the other side of that sector, or along a crease whose ends leave the short sectors against the cut, buys the full factor.

On a Miura the instruction is concrete: cut along the zigzags. A zigzag slit leaves a long and a short sector beside the cut at both of its ends, every released vertex is free, and a row of such slits releases everything the ledger promised. The same cut turned up in a different argument for a different reason. A cut along the zigzag needs a tether found that the only way to divide a Miura into independently folding modules is to cut through the line of degree-four vertices a zigzag makes, because a join along a crease couples the modules through those vertices. The cut that divides a Miura’s mechanism and the cut that frees its letterings are the same cut, and for related reasons: both are about the zigzag being where the Miura’s vertices hold their two sides together.

A cut is not local found that a cut’s effect on which letterings can reach which spreads past the vertices it touches, and one cut removes one arc found which single cuts repair a pattern whose letters contradict themselves. Both counted with the ledger’s free vertices. On a pattern where a slit leaves a short sector between two long ones, each would count fewer letterings at that cut, and nothing here says whether that changes either conclusion.

What the counts cannot show #

Whether the pattern folds. Every count here is of letterings that pass local conditions — the four at interior vertices and the strip’s at slit tips — and passing every local condition is necessary and not sufficient for a crease pattern to fold, as how little the conditions decide measured. A released vertex that admits every lettering locally can still be part of a pattern that folds under none of them.

How many slit tips a design has. A slit creates two boundary vertices where there were interior ones, and most of a patch is edge found that boundary vertices are already the majority on the printed patterns. Every one of them answers to the strip’s condition, cut or not, and nothing here counts the ones that were on the edge to begin with.

Fans of other degrees, beyond the shelf. The map is drawn for released degree-four vertices with equal end sectors. The waterbomb’s released degree-six vertices refuse eight of thirty-two letterings, in sets that depend on where the slit fell, and no map of five-crease fans is drawn.

A slit of no width #

A slit of no width, cut exactly along a crease to an exact vertex, in paper of no thickness. A real slit has a width and a rounded end, and a real slit tip is not a point; the strip model is the limit of a very fine cut.

Every strip decided exactly #

Every strip is decided exactly by the same solver that decides one-dimensional crease patterns, at every vertex every slit releases. On the shelf, every three-crease tip that refuses two letterings is required to refuse exactly the two rolls. The map’s rule is required to hold on every one of its 1,296 fans, with the ties on the diagonal required to admit six. The small Miura is counted by trying every lettering: the ledger’s factor is required to be exactly two to the number of vertices released on every slit, and the strip’s factor to be three on some slits and four on others.

Still open: the slit array #

The ledger’s own next step was the slit array — a lattice of cuts releasing a countable number of conditions, with the count available before the sheet is made. The count now has a second term. Every slit in an array leaves its tips with sectors, and each tip that has a short sector between two long ones keeps a condition, so the conditions an array releases are its interior vertices minus the tips that still decide something. For an array cut along a Miura’s straight creases that is a quarter of every pair; for one cut along its zigzags, nothing. Which arrays of cuts release the most for the fewest slits is a question this count can now answer before anything is cut.

Sideways from here, the cut that changes nothing separated slits, which leave a sheet’s topology alone, from closed cuts, which add a condition round a hole. A slit tip keeping a condition of its own is a third case: a cut that removes four conditions at a point and leaves one, which is neither nothing nor a new loop.

The habit worth carrying is about the word “free”. When a change removes the conditions a theory states, ask what conditions the new object has that the theory never needed to state. A released vertex is free of the ring’s four, and it is a strip, and strips have a lemma; the ledger priced the release by what it removed and never asked what was left.