An array is priced by its vertices

A cut is a licence gave a cut a price. Cut one crease of a crease pattern and the vertices at its two ends stop being interior; the four conditions the subject states at an interior vertex no longer apply there; and the share of letterings the pattern admits rises by a factor of two for every vertex released. A slit tip keeps one condition found the price too generous in one place. A vertex at the end of a slit is a strip of paper a full turn long, and a strip has a big-little-big rule of its own, so a slit tip with a short sector between two long ones still refuses some letterings. On a Miura that made a straight slit worth a factor of three rather than four; a zigzag slit kept its four.

Both essays priced one slit. The question both ended on was an array: a lattice of cuts, released conditions counted before the sheet is made, and the best arrangement for the fewest slits. Answering it needs one thing neither essay had to face, which is two slits arriving at the same vertex. Once they can, the price stops being a product over slits, and it becomes something more useful.

Two slits that meet #

The four-by-three Miura is the pattern the single slit was priced on, and it has seven buried creases — creases with an interior vertex at each end — four straight and three zigzag. Every pair of them can be slit together and the letterings counted exactly, and there are twenty-one pairs.

Eleven of them behave as the single-slit prices say. Two straight slits that do not touch multiply the share by nine, three times three; a straight and a zigzag slit apart, by twelve; two zigzag slits apart, by sixteen. The other ten pairs share a vertex, and none of them does. Two straight slits along one row, end to end, multiply the share by six where the prices predict nine. A straight and a zigzag slit meeting at a vertex multiply it by eight or by six where the prices predict twelve.

The shortfall is not mysterious once it is looked at from the vertices rather than the slits. Each single slit released two vertices, and its factor was what releasing those two bought. Two slits that meet release three, not four, because the vertex they share is released once. The product of the slits’ factors counts that shared vertex twice — and a vertex released twice is still a vertex released once. The price of a cut was always a price per vertex, and multiplying per slit was a shortcut that worked only while no two slits touched.

What a vertex two slits meet is #

That leaves the question of what a vertex at which two slits meet actually admits, and here the strip picture from the single slit generalises at once.

At an uncut vertex the paper goes round in a ring. At a slit tip it goes round once and stops, and the sectors make a strip a full turn long — the distinction a ring and a line drew, and the one that lets a strip’s rule be applied where the ring’s four no longer can. At a vertex where two slits meet the paper is cut in two directions, and what is left is two separate wedges, one on each side of the pair of cuts. Each wedge is a strip of its own, spanning the angle between the two cuts, and the vertex admits a lettering exactly when both of its wedges fold.

On the Miura, every such vertex is free. Cut along a straight crease and a zigzag one, it leaves one empty wedge and one wedge carrying two creases whose sectors run 70, 70 and 110 degrees — the two short sectors tie, so neither is strictly smaller than both of its neighbours, and nothing is forced. Cut along both halves of a straight line, it leaves two wedges of one crease each, and one crease always folds. Cut three ways, the wedges hold one crease between them. Every vertex that two or more slits meet admits every lettering of what is left at it, on every array of the four-by-three Miura.

The strip rule itself is the one the vertices nobody checks built for vertices on the edge of the paper, and a wedge between two cuts is exactly such a vertex: a fan of creases bounded by paper’s edge on both sides, which here happens to be the edge of a cut rather than of the sheet.

So meeting slits cost nothing at the shared vertex and gain nothing there either. The arithmetic of the pairs follows directly. A straight and a zigzag slit that meet release three vertices: the shared one, which is free; the zigzag slit’s far end, which is free; and the straight slit’s far end, which is free if the slit arrived there between a long sector and a short one and refuses a quarter of its letterings if it arrived between the two long ones. Each free vertex doubles the share and the refusing one multiplies it by one and a half, which gives eight or six — exactly the two values counted.

Every array, priced by its vertices #

The rule that came out of the pairs can be tested on every array the small Miura has. Seven buried creases give $2^7 = 128$ ways to slit some of them, from none to all seven.

Priced by the vertex, all 128 are exact. Take each vertex of the slit pattern, ask what share of its own letterings its own rule admits — half, at a Miura vertex nothing has cut; three quarters, at a slit tip between two long sectors; all of them, at any other released vertex — and multiply. The product is the exact share of letterings that pass every local test, on every array from one slit to seven.

Priced by the slit, only 22 are exact, and they are exactly the arrays in which no two slits meet: the empty array, the seven single slits, the eleven pairs apart, and three triples. Beyond three slits on a sheet this small, every array has slits that touch, and the slits’ factors overprice every one.

That sounds like a criticism of the earlier pricing and is not one. The earlier essays priced slits one at a time, where the two rules coincide, and stated the price per vertex in the first place. What the census adds is the reason the price must be stated that way: a slit is not a unit of release. A vertex is.

The cheapest way to release everything #

With the price per vertex, the question the earlier essay left — which arrays release the most for the fewest slits — becomes a question about how slits can cover vertices.

Every slit has two ends, so a slit releases at most two vertices, and it releases two only if neither end is shared with another slit. A sheet with $V$ interior vertices therefore needs at least $V/2$ slits to release all of them, and it achieves that bound only with slits that never meet. Slits that pair up vertices without sharing any are what graph theory calls a matching, and releasing every vertex with the fewest slits is the problem of finding a perfect one.

A Miura six columns by five rows has twenty interior vertices, in five columns of four. Each column’s zigzag creases pair its vertices one and two, three and four, and ten such slits release all twenty. Every one of the twenty is a slit tip between a long sector and a short one, so every one is free. The dial adds the slits one at a time, and each multiplies the share by four: after ten, the share has risen by $2^{20}$, from one lettering in a million to every lettering. Every assignment of mountains and valleys to the remaining creases passes every local test on the sheet.

The slits never reach the rim. Each zigzag slit runs between two interior vertices, so the sheet is still one piece, held together by the paper between the end of each slit and the rim — the material one cut short of falling apart found carrying everything a cut sheet can do.

Three ways to lay the same slits #

Not every way of laying slits does this well, and the difference is visible slit by slit.

Zigzag slits in pairs climb two powers of two per slit and reach the top in ten. Zigzag slits laid end to end, up each column, need fifteen: the first slit in a column releases two vertices, and every later one shares a vertex with the one before and releases one, so two slits in three buy a factor of two instead of four. They end in the same place — every vertex free, every lettering passing — having spent half as many slits again.

Straight slits in pairs never get there. A straight slit is worth only three, because one of its tips lies between the vertex’s two long sectors and still refuses the roll, so each buys about 1.58 powers of two instead of two. And the rows of this Miura have five interior vertices each, an odd number, so pairs along a row release four and strand the fifth; eight straight slits release sixteen vertices and stop, at a factor of $3^8 = 6{,}561$, about $2^{12.7}$.

The zigzag’s advantage is not new. A cut along the zigzag needs a tether found that the zigzag lines are where a Miura’s vertices hold its two sides together, which is why cutting along them divides the mechanism; here the same lines are where a cut leaves a long and a short sector beside it at both ends, which is why cutting along them frees the letterings. The difference between the two results is length. A cut that divides the mechanism runs the whole zigzag line to the rim; the slits that free the letterings are short, stop at interior vertices, and leave the sheet whole.

So the design rule the single-slit essay gave — cut along the zigzags — survives the move to arrays, and gains a second clause: cut along the zigzags, and never let two slits meet. The first clause keeps every tip free; the second makes every slit release two vertices.

When the columns are odd #

A perfect matching does not always exist, and when it does not, the rule’s second clause has to give.

The printed six-by-four Miura has fifteen interior vertices in five columns of three. A column of three cannot be paired up by its own zigzag creases, so pairs release two of each column’s three and strand the third: five slits, ten vertices. To release the stranded ones, the slits must share — two end to end up each column, ten slits for fifteen vertices — and every vertex then ends free, the share multiplied by $2^{15}$.

Fifteen is odd, so no array at all can release this sheet with fewer than eight slits, and with only zigzag slits it takes ten. Whether a mixed array — some straight slits joining the stranded vertices across a row — can do better depends on whether the price of a straight tip is worth the vertex it saves, and that is a trade the per-vertex price makes directly calculable rather than a matter of judgement.

Where the count can be made in advance #

Everything above rests on one fact about the Miura that has been used without being stated: that a product over vertices is the count at all. A vertex’s own share says how many of its letterings its own rule admits, and multiplying those shares gives the pattern’s share only if what one vertex admits is independent of what its neighbours admit — if no vertex’s verdict changes the odds at the next.

On four of the five printed patterns tested it is. The Miura’s fifteen interior vertices admit exactly $2^{-15}$ of its letterings, the square twist’s four exactly $2^{-4}$, the hexagon twist’s six $2^{-6}$, the tapered corrugation’s eighteen $2^{-18}$ — the product of the vertices’ shares every time. On the Yoshimura pattern it is not. Its counted share is $2^{-24.059}$ against a product of $2^{-24.048}$, three quarters of a per cent less, and the difference is the cost of its vertices not deciding independently: a lettering that satisfies one of its degree-six vertices changes the odds at the vertices it shares creases with.

That is a small number with a large consequence. On the Miura an array can be priced before anything is cut, by reading each vertex’s sectors and the cuts that will reach it; on the Yoshimura the same reading gives an estimate, close but not exact, and the exact price needs the count. The ledger’s promise — the count available before the sheet is made — holds on patterns whose vertices are independent, and the shelf does not say in advance which those are.

What the counts leave out #

Folding. Every count here is of letterings that pass local tests, the four ring conditions at uncut vertices and the strip rule wedge by wedge at cut ones. How little the conditions decide measured how far passing every local test is from folding. A fully released Miura, every lettering passing, is not a Miura on which every lettering folds; it is a Miura on which the local tests have nothing left to say, which is where the global question — the one that is hard in general — begins.

Why the Yoshimura’s vertices are not independent. The shelf shows that they are not, and by how much; it does not say which structure in the pattern causes it. The Yoshimura is the only pattern tested whose interior vertices have six creases, but one pattern is not enough to make that the reason.

Which letterings can reach which. A cut is not local found that cutting one crease of a square twist changes how its letterings fall into pieces that single moves cannot join, and the change spreads past the vertices the cut touches. Every count here is of how many letterings pass, not of how they connect, and a fully released Miura could still have its letterings split into pieces.

Arrays on other patterns. The census of every array was made on the Miura, where every vertex two slits meet turned out to be free. On a pattern with smaller sectors, a wedge between two cuts could carry a short sector between two long ones and refuse letterings, and the price of meeting slits would then be different, though still a product over vertices wherever the vertices are independent.

Slits of no width, letterings counted exactly #

The slits are cuts of no width, run exactly along creases from vertex to vertex, in paper of no thickness; a vertex where two slits meet is a point at which the paper is divided into wedges that touch only there. A real slit has a width and a rounded end, and two real slits meeting at a vertex leave a small region of paper whose behaviour is not a strip’s.

Every share is counted, not estimated: the letterings are swept across the sheet column by column, carrying only the letters of creases whose vertices are still waiting, and every vertex’s rule is checked the moment its last crease is lettered. The sweep is checked against plain enumeration of every lettering on the small Miura before it is trusted on the larger ones.

The checks behind the counts #

On the four-by-three Miura every array is counted three ways, and the per-vertex product is required to equal the exact count on all 128, and the per-slit product on exactly the arrays in which no two slits meet. The pairs figure requires the same split on the twenty-one pairs. On the larger sheets, every zigzag slit laid apart is required to multiply the share by exactly four and every straight one by exactly three, the end-to-end array to release every interior vertex, and the drawing of each array to carry a factor equal to the product of what its cut vertices admit. The shelf requires at least one pattern on which the product is exact and at least one on which it is not.

Still open: what makes vertices independent #

The shelf splits four patterns from one, and the next measurement is the reason. The obvious candidate is the structure of the creases the vertices share. A Miura’s interior vertices meet their neighbours along single creases arranged in a grid; a Yoshimura’s meet along a triangulated lattice, three vertices pairwise joined round every small triangle. Whether a loop of three vertices is what lets one vertex’s verdict reach another’s odds — and whether a pattern with no short loops always decides independently — is a question the per-vertex count can answer pattern by pattern, by slitting one crease of each triangle and watching the ratio return to one.

The second lead is the mixed array on the six-by-four Miura. The stranded vertices in its odd columns can be reached by straight slits across a row, which are worth three rather than four, or by zigzag slits end to end, which share a vertex and are worth two. Which is cheaper, and whether any array releases all fifteen vertices with fewer than ten slits, is a small search over matchings with a price on each edge.

Sideways, one cut removes one arc asked which single cuts repair a pattern whose letters contradict themselves, and counted with the ledger’s price per slit; an array of repairs that meet at a vertex would be priced per vertex, and whether the cheapest repair of a contradictory pattern is ever a pair of slits sharing an end is the same matching question asked of a different cost. The cut that changes nothing separated slits, which leave the sheet’s topology alone, from closed cuts, which add a condition round a hole. The arrays here are all slits, so they release without adding; a full release is a sheet on which the local tests have nothing left to refuse and the topology has nothing new to impose, which is as close as a cut sheet gets to being free.

The habit worth carrying is about units. When a price is measured one item at a time, check that the item is the thing being bought. A slit looked like the unit, because a slit is what gets cut; the thing bought was always a vertex, and the price per slit was right for exactly as long as no two slits shared one.