The flap nobody holds

Once a subject is a set of circles, designing the model is fitting them into a square, and that step is where all the remaining difficulty in origami design now sits. The output of the step is an arrangement, and an arrangement is a list of positions — one point per flap, which the crease pattern is then built from.

For some numbers of flaps that list contains an entry the arrangement did not decide.

What “as tight as it will go” leaves behind #

Push the discs of an arrangement together until nothing can be pushed further and the radius stops growing. It does not follow that nothing can move.

A disc is held when the directions of its contacts surround it — when there is no direction it could go without pressing into something. That is a statement about the directions and not about the number of contacts: three contacts crowded into one side of a disc hold it no better than none, and four contacts pointing four ways hold it completely.

A disc that is not held is a rattler, and it can be moved. How far it can be moved is a length, measured by asking how far its centre may travel in each direction before something overlaps and taking the furthest.

The measurement, and why it can be trusted #

There is a difficulty in the way of taking that measurement, and it is worth naming because it decides whether the answer means anything.

This site’s packing search is an annealer: it wanders, and it stops with an amplitude of about a thousandth of the sheet, so its discs sit that far short of touching. On a packing like that every disc has a thousandth of room and the whole arrangement reads as loose. The measurement would be a measurement of when the search got bored.

So the arrangement is inflated first: grow the radius, push overlapping discs apart, repeat until nothing moves. That changes nothing about which discs are next to which — it is the same packing drawn tightly — and it has a check available that is as strong as any on this site. The inflated packings reproduce every published optimum below ten discs to about fourteen decimal places, and the inflation never consults one of them.

Two, three, four, five, six, seven, eight, nine: the classical values, several of them proved case by case and some computer-assisted, and this repository’s search plus a physical relaxation lands on all of them. That is a check the packings had not been given before, and passing it is what makes the rattler a fact about the optimum rather than about the search.

The census #

Once the packings are tight, the answer is unambiguous. A held disc can move nothing — the measurement returns 10⁻¹³ of a sheet-width, which is zero. A rattler can move a great deal. There is nothing in between.

Five, six, eight, nine, thirteen: every disc held. Seven: one rattler, with 0.1127 of the sheet to move in. Eleven: two rattlers, at 0.1125 and 0.068. Fourteen: one, at 0.0759.

The seven-disc case is the smallest and the cleanest. Its optimal radius is 0.174457630187009 — the published value, matched to the last digit the classical literature gives — and one of its seven discs is not doing anything to earn that number.

It is worth putting the seven-disc number in a designer’s units. On a sheet 30 cm square, the rattler’s centre may sit anywhere in a region about 3.4 cm across. That is not a tolerance and it is not a rounding: it is a third of the disc’s own diameter, and moving the flap across it changes which part of the paper that limb is made of while changing the model’s scale by nothing at all.

What that means for a design #

The tree method turns a subject into a set of circles, one per flap, with the radius of each being the length of that limb. The packing then dictates where the creases go: two families of crease, both determined by the packing, neither visible in the picture of the discs.

Both families are determined by the contacts. Where two discs touch, the crease pattern gets a ridge; where a disc touches a wall, it gets an edge relation. A disc with no contacts contributes nothing to either family, so a rattler is a flap whose position the construction has no opinion about.

That is a designer’s freedom, and it is one nobody is told they have. The arrangement handed over by the algorithm puts the rattler somewhere; the designer can put it anywhere in its region without losing a fraction of a per cent of efficiency; and the choice is a real one, because it decides where that flap’s paper comes from and therefore how the finished model hangs.

There is a sharper way to say it. How much paper a design wastes is the fraction of the sheet the flaps can claim, and the rattler’s freedom costs nothing on that measure — the efficiency is identical everywhere in the region. So the design has a parameter that the efficiency does not see, and every other consideration a designer might have is free to decide it.

A rattler is a shortfall in the contact count #

The test above asks of each disc whether its contact directions surround it, which is a per-disc question. There is a whole-packing version of the same question, it is a count, and it says how many rattlers to expect before any disc is examined.

Hold the radius at its optimal value. The arrangement is then $2n$ numbers — two coordinates per disc — and every contact is one equation on them: two discs touch when the distance between their centres is twice the radius, and a disc touches a wall when its centre is a radius from it. A packing whose contacts pin every disc leaves no freedom, so those equations must cut the $2n$ dimensions down to nothing.

A rattler is what happens when they do not. A disc free to move over a region contributes two dimensions the equations have not reached, so a packing with one rattler has contact equations spanning $2n - 2$ dimensions rather than $2n$, and one with two rattlers falls two short again.

For the seven-disc optimum that is a definite prediction: fourteen coordinates, one disc free in two dimensions, and therefore twelve independent contact equations rather than fourteen. The eleven-disc packing, with two rattlers, should fall four short of its twenty-two. Both are checkable by counting the contacts a tight packing has and comparing, and the counting costs nothing beyond what the census already computes.

Which makes the freedom findable without a search #

That is worth having because it turns the rattler from something to look for into something to count.

Detecting a rattler by the per-disc test means examining every disc, which is fine at seven and tiresome at fifty. Counting contacts is one pass, it needs no directions and no geometry beyond a comparison of distances, and the shortfall from $2n$ is the total dimension of freedom in the whole arrangement — which is the number a designer wants, since it says how many independent choices the optimiser silently made.

It also says what a shortfall of one would mean, and that case is worth naming because it is the one the per-disc test would report oddly. A single dimension of freedom is a disc that can slide along a line rather than wander in a region — pinned in one direction and loose in the other, which happens when its two contact directions are exactly opposed. The per-disc rule stated earlier calls such a disc held, and the count calls the packing one short, and the count is right: the disc really can move, along the one direction both contacts leave open.

So the two tests disagree in exactly one case, and the disagreement identifies it. A count that falls one short of $2n$ with every disc apparently held is a packing with a disc sliding on a line, and it is the case a census of regions would report as zero freedom while the arrangement quietly has a parameter in it.

Why a search will not tell anybody #

The rattler is invisible from the search’s own outputs, and it is worth seeing why, because the reason applies to a great many optimisers.

A search of this kind reports the best value it found and the configuration that achieved it. Both are true. What neither says is that the value was achieved on a set of configurations rather than at a point — that the objective is flat over a region, and the reported point is wherever the search happened to stop.

An annealer will in fact wander over the flat region while it runs, and finish somewhere arbitrary in it; a deterministic optimiser will stop at whatever its rule prefers. Neither has any way to say “and this coordinate is undetermined”, because neither was asked.

The rattler is therefore a small case of a general hazard in reading an optimiser’s answer: the answer to what is the best value and the answer to what achieves it are different questions, and software that is asked the first routinely reports something for the second without being asked at all.

There is a version of the same complaint the site has already made about a different search. A seeded search comes within a fifth of a per cent of the best radius anybody has proved, and proves nothing — which is an honest statement about the value. The rattler is the same honesty applied to the argument: even where the value is exactly right, the arrangement carrying it is one of many.

The other kind of freedom, and why it is not this one #

There is a nearby phenomenon that is genuinely different and worth separating.

Symmetry is a heuristic and not a theorem: design software reaches for symmetric arrangements because they make the search enormously easier, and how much that gives away can be measured — including the case where the optimum turns out to be symmetric with nobody imposing it. That is about a restriction on the search and what it costs.

A rattler is not a restriction and costs nothing. It is a dimension along which the optimum is constant, and it is present whether or not the search was restricted. The two can even interact: fixing a rattler at a symmetric position is free, which means a symmetric-looking optimal packing may be symmetric only because somebody chose to draw it that way.

What a folder can check by hand #

The seven-disc packing is small enough to be checked without any machinery, and doing so is the fastest way to believe the result.

Draw a square. Put four discs in the corners and two more against the middle of two opposite sides, all of radius a shade under 0.175. That is six of the seven, and every one of them is jammed: each touches two walls or a wall and two neighbours, and the directions surround it. The seventh goes in the gap in the middle, and the gap is bigger than it is.

That is the whole of the phenomenon. At seven, the arrangement that sets the radius is settled by six of the discs pressing on one another and on the walls; the seventh has to be accommodated, and accommodating it is what brings the radius down from the six-disc optimum of 0.1877 to 0.1745 — but once there is room for it, there is more room than it needs, and where in that room it sits is decided by nothing.

What the contacts actually decide #

It is worth being precise about which parts of a crease pattern a rattler leaves undetermined, because the answer is narrower than “the design”.

The base’s scale is decided: it is set by the radius, and the radius is what the held discs settle. The ridge creases between touching flaps are decided, since they come from contacts. What is not decided is where the rattler’s own circle sits, and therefore where its axial line lands and how the paper between it and its neighbours is divided.

So the freedom is local to that flap’s neighbourhood and it is real: two designs differing only in where the rattler sits have the same scale, the same efficiency, and visibly different crease patterns in one region of the sheet. Which of them is better is a question about the finished model, and it is a question nobody currently knows they are being asked.

Where the model stops #

Equal discs. Every packing here has flaps of equal length, because the proved optima that make the measurement trustworthy are for equal circles — and a subject reduced to a skeleton rarely has limbs of one length. A real subject has limbs of different lengths, the arrangement is a packing of unequal discs, and whether rattlers are commoner or rarer there is not measured.

Only two counts below ten are checkable against a proof. For eleven, thirteen and fourteen the arrangement is the best this search can find and not a proved optimum, so the rattlers reported there are rattlers of a very good packing rather than of the packing.

The room is a distance, not an area. What is reported is how far the centre may travel in the best direction. The region it may occupy is two-dimensional and its area would be a better measure of the freedom; it is not computed here.

Nothing about cost. How hard it is to find these packings, or to detect a rattler in one, is a question about algorithms and is not asked. What is measured is what the arrangements are.

And nothing about whether the base folds. A packing satisfying the separation condition through the tree yields a crease pattern, and whether that pattern folds is the question every local condition on this site leaves open. Moving a rattler does not change the answer, because it changes no contact.

Who found it, and when #

Rattlers are old and are not this subject’s discovery. They are standard vocabulary in the packing literature and in the physics of granular materials, where a rattler is a grain in a jammed packing that carries no force — and the seven-disc square packing is one of the textbook illustrations, precisely because it is the smallest case where the phenomenon is unmissable.

What is new here is the connection. The packing literature’s rattler is a curiosity about a configuration; in the tree method it is a flap, and a flap whose position is undetermined is a design decision that has been made by an optimiser’s stopping condition rather than by a designer.

Where the ladder goes next #

The natural continuation is unequal discs, where the design problem actually lives. The measurement above needs proved optima to be trustworthy and there are none for unequal packings, so the honest version would be a lower bound on freedom rather than a census: find rattlers where they can be found, and say nothing where they cannot.

There is a third direction and it is the one that would matter most to a designer working on a grid. Box pleating gives up the efficiency of a free packing to buy creases that land where they are supposed to, and a rattler’s region is exactly the kind of place a grid constraint could be satisfied for free — the flap can be slid onto a grid point without any loss at all, provided a grid point is in the region.

The other direction is to use the freedom. A rattler’s region is a set of positions all of which give the same base scale, so a designer could optimise something else over it — how the flap hangs, how much paper is wasted at its root, whether its creases land on a grid. That is a second objective on a set the first objective is indifferent to, which is the most comfortable kind of design freedom there is and one nobody currently exploits.