The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases — hand it a reflex corner and it produces nothing rather than something worse, and the difficulty moves backwards to whoever chose the polygons.

Always is a statement about existence. It says nothing about how the answer depends on the question, and the answer does not depend on the question in the way anybody would assume.

What the molecule is #

The construction is one sentence. Shrink the polygon by sliding every edge inward along its own normal at the same speed, and trace the paths the corners take. Those paths are the creases.

Because every edge moves at the same rate, each corner moves along the bisector of its two edges, and the path is straight for as long as the two edges last. Two things can end a corner’s straight run. Two neighbouring corners can collide, which removes the edge between them; or — in a non-convex polygon — an edge can run into a corner from across the shape, which is the case the construction here declines to handle.

The object this produces is the polygon’s straight skeleton, and it is a construction with a life of its own outside origami. It is not the medial axis and it is not a Voronoi diagram — those are built from distances to points and are somebody else’s plane geometry — it is built by offsetting the polygon’s own edges, and the difference shows up exactly where a corner is sharp.

The events #

The skeleton is built by watching the shrinking polygon and recording events: the instants at which an edge vanishes because its two ends have met.

For a generic polygon the events happen one at a time. Five edges give a first collapse at one instant, then another, then another, and the skeleton has one node per event. The order in which the edges go is decided by the shape, and for most shapes it is decided robustly — a small change to the polygon changes when each event happens and not which happens first.

For some shapes it is not. At an isolated polygon, two events happen at the same instant, or three edges vanish together at a single point — and either side of that shape the events happen in a different order, so the skeleton is a different object.

The measurement #

Take a pentagon with four corners fixed and one sliding along its edge, and rebuild the skeleton at two hundred positions.

Nothing happens for most of the sweep. The corner moves, the skeleton’s nodes move with it, the creases lengthen and shorten, and there are six of them throughout.

At 0.2872 of the way along, three edges vanish at once and the count goes to seven, and stays at seven for the rest of the sweep. The count is an integer and it changes by one; there is no shape at which the molecule has six and a half creases, and no continuous deformation carrying the six-crease pattern to the seven-crease one.

Move the fixed proportions and the event moves. Move them further and it can leave the sweep entirely: at some proportions the corner can be slid from one end of its edge to the other without any event being crossed, and the molecule is combinatorially the same throughout.

It is worth having the mechanism at the shape itself. Just below the event, edges A and B collapse first and edge C collapses later, so the skeleton has a node for each and a crease running between them. Just above it, C goes first and the other two follow, and the node that used to sit between them is not there. Exactly at the event all three go together, the two nodes coincide, and the crease between them has length zero.

So the transition is a crease shrinking to nothing and being replaced by a different crease growing from nothing, and the count passes through the degenerate shape rather than through any intermediate. That is what makes it a change of kind rather than of degree.

What a designer is actually doing #

The tree method’s chain is: subject, skeleton of limbs, circles, packing, polygons, molecules, crease pattern. A designer adjusts the subject — a slightly longer neck, a slightly shorter tail — and expects a slightly different pattern.

Most of the time that is what happens, and it is why the expectation exists. The circles move a little, the packing shifts, the polygons deform, the molecules deform with them, and the pattern is recognisably the same pattern with different numbers on it.

Occasionally it is not. The adjustment carries one of the leftover polygons across an event, its molecule gains or loses a crease, and the pattern in that region is a different pattern — not a worse one or a better one, a different one, with a different number of creases meeting differently.

That is a discontinuity a designer meets as a surprise: two versions of a design that differ by a millimetre in the subject and by a visibly different set of folds in one corner of the sheet. It is not a bug in the software and it is not the designer’s mistake; it is the geometry.

There is a version of this that every designer has met without naming it. Two renderings of the same model, produced weeks apart from slightly different measurements, that “fold differently” in one region — and the usual diagnosis is a mistake somewhere in the chain. Sometimes it is. Sometimes the chain worked perfectly and the shape crossed an event.

Where the flatness comes from #

Once the packing is fixed, one number is left in the whole design — how far a leftover polygon can be shrunk before it stops being a polygon — and everything else about the crease pattern has already been decided. That framing is exactly right and it is worth seeing how this sits inside it.

The shrink distance is a continuous quantity, and it is continuous here too — the polygon shrinks smoothly whatever it is. What is not continuous is the sequence of events the shrinking passes through. So the last free parameter goes on being a single number and the object it parameterises changes kind at isolated values of everything else.

There is a second reading, and it is the one that connects this to the rest of the site. Nearly every discontinuity in this subject comes from a count being determined by a quantity: the number of foldable markings jumps when two sectors become equal; a twist tessellation loses its mountain-and-valley assignment entirely below a threshold angle while every angle condition goes on holding; a cut sheet is one piece and then six. The molecule’s crease count is another instance, and the family resemblance is that in each case the continuous quantity is what a designer controls and the integer is what they get.

Why it stays hidden #

Three reasons, and they compound.

The events are rare. A random polygon crossed with a random deformation meets an event with probability zero, in the same sense that almost every crease pattern fails to fold: the degenerate shapes are a set of measure zero in the space of shapes.

The designs that meet them are the symmetric ones. Three edges vanishing together is a coincidence, and coincidences in geometry are usually symmetries. A designer working with a symmetric subject — which is most subjects, since animals are bilaterally symmetric — is working exactly where the coincidences live. That is the same trap the vertices everybody folds fall into: the interesting degeneracies are not obscure, they are the ordinary practice.

Software does not report it. A molecule generator returns a crease pattern. It does not return “and this pattern has a different number of creases from the one produced a moment ago”, because nobody asked it to, and the difference is easy to miss in a pattern with several hundred creases.

What it is not #

Two nearby things this is not, because both would be easy to mistake it for.

It is not the molecule that does not exist. That is a failure of the construction on a polygon it cannot handle, and the answer there is a refusal. Every polygon in this essay has a molecule, and the molecules on either side of an event are both perfectly good.

It is not a numerical artefact. The event is a genuine coincidence of three edge collapses and it happens at a definite shape; a finer sweep locates it more precisely and does not make it go away, and the crease count on either side is an integer computed from a combinatorial structure rather than a number with a tolerance on it.

The third thing it is not is a reason to distrust the method. The tree method’s guarantee is that a molecule exists and wastes no paper, and that guarantee holds on both sides of every event. What does not hold is the unstated extra — that the answer varies smoothly with the question — which nobody claimed and everybody assumed.

Two designs on either side #

It is worth asking what the two patterns look like when set beside each other, because “different” is doing a lot of work above.

They fill the same polygon and they use the same paper. Both are valid molecules; both waste nothing; both would fold. What differs is the combinatorics: the six-crease version has two interior nodes and the seven-crease version has three, and the extra node is joined to the polygon by an extra crease. In a finished pattern with several hundred creases that is one short line appearing in one corner.

For a folder the practical difference is a fold that is there or is not. For a program comparing two versions of a design, it is a structural difference rather than a numerical one — the patterns cannot be aligned crease for crease, so any tool that reports what changed between two versions has to say “these are different” rather than “this moved by 0.4 mm”.

Where the model stops #

Convex polygons only. The construction here refuses a reflex corner, which means the events it can see are edge collapses and not the split events a non-convex polygon also has. A general polygon has more ways to be degenerate and they are not measured.

One family, one sweep. The measurement slides one corner of one pentagon at two proportions. It establishes that the discontinuity exists and where it is in that family; it does not say how often it is met in real designs, which would need a survey of real designs.

Creases are counted, not compared. The count jumping from six to seven is the signal. Whether the six-crease and seven-crease patterns are otherwise similar — whether one is nearly the other with a short crease added — is not analysed, and a designer would want to know.

Nothing here is about how hard it is to detect. Deciding whether a shape is at an event, or how near one it is, is a computation with a cost, and cost is not this subject’s to discuss.

What a folder would notice #

The short crease is the practical form of all of this, and it is worth putting in a folder’s units.

Near an event one crease of the molecule is very short. At a hundredth of the way from the event on the sweep above, that crease is a fraction of a millimetre on a 300 mm sheet — a mark rather than a fold. Folding it is impossible; ignoring it changes the pattern; and the pattern with it ignored is exactly the pattern from the other side of the event.

So in practice the event is not a point but a band, whose width is set by the paper’s own crease radius rather than by anything in the construction. Inside that band a designer has, without knowing it, a free choice between two patterns — and the honest thing for software to do would be to say so.

How wide the band is, and what sets it #

The section above says the event has a physical neighbourhood rather than being a point, and the width of that neighbourhood has a shape worth deriving even where the constant is not to hand.

At the event two skeleton nodes coincide and the crease between them has length nought. Move away from the event and they separate; the separation is linear in how far the shape has moved, because two points that meet at a parameter value generically part at a nonzero rate. So the vanishing crease’s length is proportional to the distance from the event, with a constant of proportionality that depends on the polygon and has the units of a length per unit of the design parameter.

A crease shorter than the paper’s own crease radius is not a crease. So the band in which the two molecules cannot be told apart in paper has a half-width of the crease radius divided by that rate — and since the rate scales with the sheet, the band’s width as a fraction of the parameter range scales with the crease radius divided by the size of the sheet.

That gives the practical rule and it runs the way a maker would want. A quarter-millimetre crease on a hundred-and-fifty-millimetre sheet is one part in six hundred; on a metre of paper it is one part in four thousand. The larger the model, the sharper the event. A big design has a genuine discontinuity at a definite shape, and a small one has a fuzzy region around it in which the mathematics distinguishes two patterns and the paper does not.

Which answers half of the last question #

The closing section asks whether either side of an event is better, and observes that efficiency does not choose. Inside the band it does.

Approaching from the seven-crease side, the extra crease is the one shrinking to nothing — so a shape just inside the band on that side has a pattern which is the six-crease pattern plus a mark too short to fold. A folder handed it would ignore the mark, and ignoring it produces exactly the pattern from the other side.

So within the band the two are not two designs at all: one of them is the other with an unfoldable line drawn on it, and the six-crease rendering is the honest one. That is a criterion, it is available, and it is the kind software could apply — drop any molecule crease shorter than the stated paper’s crease radius, and report that it was dropped.

Outside the band the question stands open and efficiency really does not choose. What can be said is that the two patterns differ by one node and one crease, so the shorter one is simpler to fold by exactly one fold, and there is no other axis on which they have been compared.

And how often a design meets one #

The estimate is rough and worth making anyway, because it decides whether this is a curiosity or a routine hazard.

A complex design has tens of leftover polygons, each with its own molecule and its own set of events. Each polygon is carried across its own parameter range as the subject is adjusted, and each has a band of the width above. If the bands are of order one part in a few hundred and there are fifty polygons, then an adjustment sweeping a per cent of the range gives roughly an even chance that some molecule somewhere on the sheet crosses an event.

That is an order of magnitude rather than a measurement, and every step of it is an assumption: that the events are spread evenly, that the polygons are independent, that fifty is the right count. What it establishes is only that the answer is not obviously small — which is enough to say that a designer who has never noticed this has probably met it and diagnosed it as something else.

Where the ladder goes next #

The obvious continuation is the near-event rather than the event. A polygon a hair away from a degenerate one has a molecule with a very short crease in it, and a very short crease is a fold nobody can make. So the practical boundary is not the event but a neighbourhood of it, and the width of that neighbourhood — in units a folder would recognise, like millimetres on a sheet — is a measurement worth having.

The other direction is the one the tree method’s own literature would ask. If crossing an event changes the pattern, is one side of it better? The two molecules fill the same polygon with the same paper and neither wastes any, so efficiency does not choose between them; something else would have to, and what that something is has never been written down.