The corner that splits the shrink

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. The construction is a shrink: move every edge of the polygon inward at the same rate, and the traces the corners leave are the creases.

Hand it a corner that turns back and it produces nothing — not a worse pattern, nothing — and the difficulty moves backwards to whoever chose the polygons. That has been the position here since the design essays were written, and half of it has now changed.

The shrink at a reflex corner is computable. What it does there is the subject of this rung, and the reason the molecule still does not exist is not the reason it looked like.

What the corner does #

A convex corner moves inward along its bisector as the shape shrinks, and the only thing that ever happens is that some edge runs out of length and disappears.

A reflex corner is on the outside of its own bend. Both edges meeting there are moving inward, but the corner between them travels outward, along its bisector, and faster than the edges move in — the sharper the reflex, the faster. It is the one place on the outline where the shrinking region gains ground.

Where does it go? Into the shape, until it hits an edge on the far side. When it does, the shrinking region has touched itself: it becomes two regions, each with a copy of the edge that was hit, and each then goes on shrinking alone. That is the split event, and it is the whole content of the reflex case.

Solving for the moment #

The time is found rather than swept. An edge that has been shrinking for a time s is the set of points at signed distance s along its inward normal from where it started. The reflex corner at the same moment is at its own start plus s times its velocity. Setting the two equal is one linear equation in s with one root.

A root becomes an event only if three things hold: it is in the future, the edge still exists at that time, and the corner arrives between the two ends of the edge as they are at that moment rather than off the end of it. The third condition is the one that does the work, and it is why the count below is not the count anybody would guess.

A reflex corner does not guarantee a split #

Counted over seven outlines, the number of splits runs from nought to four, and it is never larger than the number of reflex corners and often smaller.

The dart has one reflex corner and one split. A comb with two teeth has two and two. A plus has four and three. A five-pointed star has five and four. An L has one — and no split at all: its reflex corner starts travelling outward, and before it reaches the far side one of the edges it was aiming at has already vanished in an ordinary edge event, taking the target away. The corner is consumed rather than arriving.

So a reflex corner is where a split can happen and not where one must. What decides is a race between the corner’s outward speed and the shrinking of everything in its path, and the winner depends on the proportions of the shape rather than on the angle at the corner.

How fast the corner travels #

The race can be given a speed, and the speed says why the essay’s conclusion — that the proportions decide it and not the angle — is the right one rather than a hedge.

A corner of interior angle $\theta$ moves along its bisector at $1/\sin(\theta/2)$ while its edges move inward at one. At $\theta = 180°$ that is exactly one, which is a straight edge sliding; past 180° it exceeds one and points outward, and it grows without bound as the corner sharpens toward a slit.

So a reflex corner at 190° travels at 1.004 and one at 270° — the corner of an L or a plus — travels at 1.414. A corner at 350°, which is very nearly a cut, travels at 11.5.

Set that against the far edge, which is approaching at one. If the initial gap is $h$, the two meet at

$$s = \frac{h}{1 + 1/\sin(\theta/2)}$$

Which the angle barely moves #

Now read the range. At 181° that expression is $0.500h$; at 270° it is $0.414h$; at 300° it is $0.366h$. Across every reflex corner in this collection the angle changes the split time by under twenty per cent.

The gap $h$ and the target edge’s own lifetime are set by the shape’s proportions and vary without limit — an outline can put its far edge a hair away or half the shape away, and can make that edge as short-lived or as long-lived as it likes.

So the angle contributes a factor of at most about 1.2 to the race and the proportions contribute any factor at all. That is the arithmetic behind “depends on the proportions rather than on the angle”, and it says the observation is structural rather than an accident of six outlines.

It also says which shapes to suspect. A split fails when the target edge dies first, so the outlines to watch are the ones whose reflex corner aims at a short edge — the L, whose notch aims at a side that vanishes early — rather than the ones whose corner is shallow. A deep reflex corner aimed at a long edge splits every time; a shallow one aimed at a short edge is the case a designer should expect to lose.

The skeleton survives and the event list does not #

One awkward finding is worth reporting rather than smoothing, because it says which of these quantities is real.

On a rectilinear outline everything happens at once. An L two units across collapses its notch, its short side and its far corner at the same instant, and whether that instant is recorded as a split or as an edge event depends on which the simulation takes first. Jitter the same L by a fifth of a millimetre and the tie breaks the other way: the split disappears and an extra node appears.

The arcs do not change under that jitter, up to nodes that were coincident separating by the jitter. The event list does. So the skeleton is a property of the outline and the sequence of events producing it is a property of the simulation, and only the first belongs in an essay. The counts above are therefore reported for the shapes as drawn, with the ties resolved by preferring splits — which is the resolution that gives an L the correct skeleton, and which was found by getting it wrong.

Why the molecule still does not exist #

The skeleton being available does not give the molecule back, and the reason is what a molecule is for.

A molecule is a crease pattern for one piece of paper — the leftover polygon between the discs of a packing — whose folded state puts the whole polygon’s boundary onto the axis. The shrink is not just a way of drawing lines in it; the shrinking polygon is the folded state at each moment, seen from above, and the creases are where the paper turned.

A split says that at some moment the folded state is two separate regions. There is no way to read that as one piece of paper folding: the paper did not divide, so the shrinking region ceasing to be connected means the shrink has stopped describing the fold. The construction has not become harder at a reflex corner; it has stopped being about the same object.

The shrink is the folded state, seen from above #

The sentence the argument turns on deserves more than an assertion, because it is the whole reason a connectivity failure matters.

A uniaxial base is a folded object all of whose flaps lie along one line, and the projection onto that line is the tree the design started from. Turn that round: at any moment during the fold, the paper’s shadow on the plane perpendicular to the axis is a region, and as the fold proceeds that region shrinks. The universal molecule’s construction is that shrink, run in reverse and used as a drawing rule — which is why the creases it produces are straight, why they are angle bisectors, and why the leftover polygon’s boundary ends up on the axis.

So each intermediate polygon in the shrink is a picture of the paper at one moment, and the paper is one connected piece throughout. A shrink that breaks in two is claiming the paper did as well.

That also explains why the failure is total rather than partial. It is not that the creases past the split are wrong or that the pattern needs an extra fold; it is that after the split the drawing has stopped being a picture of anything the paper does, so nothing downstream of it can be repaired.

What a designer actually meets #

None of this is a rare situation, and it is worth saying where in the design process it arrives.

The packing is the hard part: the flaps are discs, the discs are packed into the sheet, and what is left over between them is the paper that has to be creased into the connections. Those leftover regions are polygons whose corners are the disc contacts, and nothing in the packing search guarantees any of them is convex — a region between four discs where two of them nearly touch has a corner that turns back.

The design method’s answer has always been to keep going. Rivers and grafts add paper where the connections need it, which changes the leftover regions; the packing has a free parameter or two even after the radii are fixed, which moves the contacts. A designer nudges until the leftovers are polygons the molecule can fill, and that nudging is invisible in the finished pattern.

So the reflex corner is not an exotic obstruction. It is a routine one, handled by going back a step, and the reason it is worth an essay is that going back a step is exactly what a construction advertised as universal is supposed to make unnecessary.

The two obstructions are not the same #

It is worth separating this from the other reason a molecule can fail, because both are about a polygon and only one is about its shape.

The path condition is a requirement on the polygon’s distances: every pair of corners must be at least as far apart along the boundary as the tree says their flaps are. A polygon can be perfectly convex and fail it, and then the molecule does not exist either — for a reason that is about the design rather than about the geometry.

Convexity is a requirement on the polygon’s shape, and it is what this rung is about. A polygon can satisfy the path condition and be non-convex, and the molecule does not exist for it.

The two failures look the same from the outside — no pattern comes back — and they are repaired differently. A path-condition failure is repaired by adding paper, which is what a river is for. A convexity failure is repaired by changing which polygons there are, which means moving the packing. A designer who mistakes one for the other spends the wrong currency.

Why fold-and-cut is not affected #

The same skeleton, on the same non-convex outline, is exactly what the fold-and-cut construction needs — and it does reach a star and an L. The difference is worth stating precisely, because two constructions using one object with opposite results is the sort of thing that looks like an inconsistency.

Fold-and-cut wants the skeleton as a set of creases. The outline is drawn on a sheet, the skeleton’s arcs and the perpendiculars from its nodes are folded, and the outline lands on a line. Nothing in that asks the shrinking region to remain connected: it is a way of generating creases, and the creases are then checked against the theorems in the ordinary way.

The molecule wants the shrink as a folded state. Every intermediate shape is a claim about where the paper is at that moment, so the shrink’s connectivity is the paper’s connectivity.

One object, two uses, and only the second has a connectivity requirement. That is why implementing split events bought a new printable pattern and did not buy a new design method.

What would buy the molecule back #

The literature’s answer to a non-convex leftover polygon is not to fix the molecule; it is to avoid the polygon. Two routes, and both are decisions taken earlier.

Choose the packing so the leftovers are convex. The polygons a molecule has to fill come from the packing of discs, and a packing can be adjusted so that none of its leftover regions has a reflex corner. That moves the problem into the packing search, which is already the expensive part.

Split the polygon by hand and pay for it in paper. A non-convex polygon can be cut into convex pieces along a line and each piece filled with its own molecule, at the cost of the creases along the cut and whatever paper the extra structure needs. That is the practical answer and it is why a designer’s non-convex regions are not a crisis.

Neither is available to a construction that has to work on whatever it is handed, which is the property the universal molecule is prized for and the property it loses at a reflex corner.

The event that was never in the picture #

There is a reason the split is absent from every account of the molecule, including this collection’s own earlier ones, and it is not that anybody was hiding it.

The construction is always drawn on a convex polygon, because that is where it is defined. On a convex polygon the shrink has one kind of event and the picture of it — a nest of smaller and smaller copies, each one inside the last — is so orderly that the possibility of a second kind never comes up. Even the discontinuity that is there is easy to miss: slide one corner along its edge and the number of creases jumps from six to seven at an isolated shape, which is visible only if somebody sweeps the shape and counts.

So the reflex case has been described in this subject as the molecule does not exist there, full stop, and the sentence is true. What it omits is that the shrink still does something, that the something is computable, that it has a name in another field’s literature, and that it is the ingredient a different construction on this site needed. A gap in one method was a tool for another, and the two are three essays apart in this collection because nobody had run the shrink on a shape that turns back.

What this rung leaves #

The skeleton is general here now, and that is machinery rather than a result: it is used by fold-and-cut, it passes the equidistance assertion on every shape, and it agrees exactly with the older convex routine on the five convex outlines this collection folds.

The molecule is still convex-only, and the reason has moved. It is not that the skeleton cannot be computed at a reflex corner; it is that the shrink stops being one region, and a molecule’s shrink is its folded state.

And a reflex corner is a race rather than a condition. Whether it produces a split at all depends on whether it reaches the far side before the far side vanishes, and on the shapes here it does so four times in seven. That is the one number in this essay a designer could use directly: a non-convex leftover polygon is not automatically a polygon whose shrink divides, so a packing that produces one is not automatically a packing to reject — it is a packing to run the shrink on and look at.