A cut is not local

A cut is a licence counted what a cut buys in letterings: cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released — exactly, on every cut tried.

This is the other question about the same operation, and the answers are different in kind. Not how many letterings a cut adds, but which of them it makes reachable.

The letterings of a pattern fall into pieces no local change can cross, one for each crease with an interior vertex at each end. Cutting such a crease removes it from that count. So a cut does not merely add letterings to a set — it joins pieces that were unreachable from one another, and a folder can now get from one to the other without unfolding the sheet.

That much is expected. What is not expected is how far the effect reaches.

What the cut is #

A cut along an existing crease removes no paper. The two panels that met along it are still there and still the same size; they are simply no longer joined. As a change to the drawing it is one edge whose assignment goes from a letter to a raw edge, and everything else follows by itself.

The vertex at either end of it now has paper on one side only. So it stops being an interior vertex, and every condition that was being evaluated there — developability, Kawasaki, Maekawa, the big-little-big lemma — stops applying. That is the release, and it is the operation the whole anchor is about.

Nothing here is a slit through solid paper, which is a different object with different arithmetic. A cut along a crease is the minimal thing that can be called a cut, and it is the one whose consequences can be traced exactly.

Sixteen pieces to two #

The square twist has twelve creases, four interior vertices and four buried creases — the four sides of its central square. Two to the fourth is sixteen, and its 256 letterings fall into sixteen pieces of sixteen.

Cut one of those four. Two vertices are released, which is the earlier rung’s number. And the number of buried creases falls from four to one.

Sixteen pieces to two.

The arithmetic is worth doing slowly because the surprise is in it. The cut crease itself stops being buried — that is one. The two vertices it released were each the far end of another crease that had been buried, and those two creases now have a boundary vertex at one end — that is two more. Three creases leave the settled set for a cut made along one of them.

And a cut that was never settled still quarters it #

The stronger half of the result is about the creases a cut is not expected to matter at.

Eight of the square twist’s twelve creases are not buried: each has at least one end on the edge of the paper, and each therefore had a letter a hand could already argue with. Cutting one of those changes nothing about that crease’s own status.

It still takes the count from sixteen pieces to four.

One vertex is released, and that vertex was the far end of two creases that were buried. Both stop being buried, and the count falls by a factor of four.

So there is no cut anywhere on this pattern that leaves the piece count alone. A cut anywhere rearranges the whole space, and cutting the crease that looks least consequential still removes three-quarters of the barriers.

The same numbers on every pattern that has any #

Two of the printed patterns are small enough to cut every crease of in turn and have anything to cut, and the arithmetic is identical on both.

Ten buried creases and thirty-four reaching ones were cut, one at a time, across four printed patterns. Every buried cut released two vertices and took three creases out of the settled set. Every reaching cut released one vertex and took two out. No exceptions.

The regularity is a consequence of both patterns being made of degree-four vertices with every crease running between two of them or out to the paper’s edge. It is not a theorem about all patterns, and the essay does not claim it is: a vertex of higher degree has more creases meeting at it, so releasing it un-buries more of them.

And nothing at all where there was nothing #

The control is the two patterns with no buried crease.

The preliminary base has eight creases, one interior vertex and nothing buried, so its letterings are one piece. The fold-and-cut triangle is the same. Cutting any crease of either releases a vertex, admits more letterings, and leaves the piece count at one — because there was never a barrier for the cut to remove.

That row matters more than it looks. Without it the result would be indistinguishable from “cutting reduces a number”, which is true of many numbers for uninteresting reasons. What the zero row shows is that the reduction is specifically about barriers, and a pattern with no barriers has none to lose.

What the two rungs measure, and why the numbers differ #

The earlier rung reported a factor of two per vertex released and this one reports factors of eight and four. Those are not in conflict and it is worth saying exactly why, because a reader who does not check will conclude one of them is wrong.

The earlier number is about how many letterings the pattern admits. Releasing a vertex removes the conditions at it, so more letterings pass, and the factor is two per vertex — exactly, on every cut tried.

This number is about how those letterings are divided. It counts barriers rather than members, and barriers are creases rather than vertices, and a released vertex un-buries more than one crease.

Multiply them out and the two are consistent. On the square twist, a buried cut takes the admitted count up by a factor of four (two vertices released) and the piece count down by a factor of eight — so the size of each piece goes up by thirty-two. A folder holding the cut sheet can reach thirty-two times as many letterings as before, of which four times as many exist. Cutting adds letterings and removes walls, and it removes walls faster.

Why it reaches so far #

The mechanism is short and it is entirely about the graph.

A crease is buried when both of its endpoints are interior vertices. Cutting a crease turns both of its endpoints into boundary vertices. Any other crease with an endpoint at either of those is therefore no longer buried, whatever else is true of it.

So the reach of a cut is the neighbourhood of its two endpoints in the crease graph, not the cut itself. On a degree-four vertex there are three other creases; two of them, on these patterns, were buried. On a degree-six vertex there would be five others, and more of them would be.

Which gives a rule a designer can use without any computation. The cost of a cut, in barriers removed, is the number of buried creases meeting either of its ends — and that is a count anybody can do on a drawing.

Every crease of the twist, one at a time #

The catalogue is worth reading whole rather than summarised, because its uniformity is the finding and a summary hides it.

The square twist’s twelve creases fall into two groups by inspection: the four sides of the central square, and the eight pleats running from that square out to the paper’s edge. Cut each in turn and recompute everything from the pattern that results.

All four central creases behave identically: two vertices released, three creases un-buried, sixteen pieces to two. All eight pleats behave identically: one vertex released, two creases un-buried, sixteen pieces to four. There is no crease anywhere on the pattern whose cut does something a third thing.

That uniformity is a property of this pattern’s symmetry rather than of cutting — the four central creases are equivalent under the pattern’s own group, and so are the eight pleats in pairs. On a pattern with less symmetry the catalogue would have more rows in it. What would not change is the rule the rows follow, which is a statement about the crease graph and not about the symmetry.

The sampled check on the same operation #

The pieces are a prediction, and predictions on this site are checked by landing in them.

Draw independent letterings of the cut pattern and count how many distinct pieces they land in. Before the cut, forty draws on the square twist land in fifteen of the sixteen pieces — which is what forty draws on sixteen boxes look like. After a cut along a buried crease, the same forty draws land in two, and after a cut along a pleat, in four.

The draws are not a proof of the piece count and the essay does not present them as one; a sample can show that at least this many pieces exist and never how many there are. What they rule out is the failure this measurement is most exposed to — a prediction from the graph that has drifted away from the object it is about — and they rule it out at every step of the catalogue.

What it means at the table #

Three things, and the first is the one that will surprise a folder.

A cut anywhere changes what can be done everywhere. There is no such thing as a local cut on a pattern with buried creases. Snipping one crease of a twist unit to make it easier to collapse has also made the whole unit re-letterable in ways it was not, including the one thing the handedness was supposed to have settled.

Which is why a cut twist can be turned the other way and an uncut one cannot. The direction a twist turns is written in its four buried creases, and cutting any of the twelve reduces those to one or two — so the sheet can now be pushed from a left twist to a right one without opening it out. That is a thing a reader can check with a pair of scissors in about a minute.

And a cut is not undone by tape. Rejoining the two panels restores the crease and restores the barriers, so the sheet goes back to sixteen pieces — but it goes back in whatever piece it is currently in. A cut, a re-letter and a repair is a route between two pieces that no local change provides, which is precisely why it is not a local change.

Where the arithmetic would change #

The two twists give identical answers because both are built entirely of degree-four vertices, and it is worth saying where a different pattern would give a different one.

At a degree-four vertex, three creases other than the cut one meet. On these twists, two of those three are buried and one runs out to an edge, which is why a buried cut un-buries three creases in total and a reaching cut un-buries two.

At a degree-six vertex there are five other creases. Releasing such a vertex un-buries every one of them that had its far end at another interior vertex, so a cut at a degree-six vertex could take as many as six creases out of the settled set — a factor of sixty-four rather than eight.

So the rule to carry away is not “a cut is worth eight”. It is a cut is worth two to the number of buried creases meeting either of its ends, plus itself if it was buried — which is a count on a drawing, and which reduces to eight and four on a pattern of degree-four vertices.

How many cuts make it one piece #

The catalogue above cuts one crease at a time, which leaves the obvious question unasked. If one cut takes sixteen pieces to two, what does it take to reach one — and is there a rule, or does it have to be searched for?

There is a rule, and it is exact.

A crease is buried when both of its ends are interior vertices, and cutting a crease makes both of its ends boundary vertices. So a set of cuts leaves nothing buried exactly when the cuts’ endpoints between them account for every interior vertex on the sheet. That is a familiar object under another name: a set of edges touching every vertex is an edge cover, and the smallest number of cuts that connects the whole lettering space is the smallest edge cover of the crease graph’s interior vertices.

Which has a formula. The minimum edge cover of a graph with no isolated vertex is the vertex count less the size of its largest matching — a set of edges no two of which share an end. Every interior vertex has creases at it, so the condition holds on any pattern here.

Run it on the square twist. Its four interior vertices sit on a four-cycle of buried creases, whose largest matching is two — a pair of opposite sides of the central square, sharing no corner. Four less two is two, so two cuts take sixteen pieces to one, and they are two opposite sides of the middle square rather than two adjacent ones. Cutting two adjacent sides covers only three of the four vertices and leaves a crease buried, and the sheet stays in two pieces.

The hexagon twist gives three: six interior vertices on a six-cycle, largest matching three, and three alternate sides of the central hexagon.

What that is worth, and what it is not #

Two readings of the number, and they pull in opposite directions.

At the table it is a small, checkable instruction. Two snips along opposite sides of a square twist’s middle square, and every lettering the pattern admits becomes reachable from every other without opening the sheet out — including the mirror-image twist, and including whatever the folder was told was settled when the pattern was drawn. Three snips do it for a hexagon twist. Neither of those is a large intervention, and neither removes any paper.

The other reading is a caution about what has been made reachable. The pieces are pieces of the locally admissible set: letterings that satisfy every condition at every interior vertex that still has conditions. A cut removes conditions, so the set grows as the walls come down, and a good deal of what is newly reachable is newly admitted rather than newly available. Reaching every lettering of a cut pattern is not the same achievement as reaching every lettering of the pattern that was drawn.

That is the honest form of the whole anchor’s result. Cutting buys movement, and it buys it by weakening the thing being moved through — which is why the count of cuts needed is a fact about the crease graph, and why the value of paying it is a question about what the folder wanted the barriers for.

What is not being claimed #

Two limits, and both are the site’s standing ones.

The letterings counted are the locally admissible ones. Developability, Kawasaki, Maekawa and the lemma at every interior vertex; whether any of them folds globally is NP-hard. The pieces are pieces of a candidate set, and cutting joins pieces of that set.

And the patterns cut are the small ones. Every crease of the two twists was cut and every consequence recomputed, which is exhaustive on those two. The Miura, the Yoshimura and the waterbomb tessellation have too many creases to enumerate their letterings, and the buried-crease prediction for them is a prediction — checked by the sampling instrument rather than by a list.