What the condition does not decide

Every pair, not every circle replaces a drawing of $n$ discs with a condition over $n(n-1)/2$ pairs, and a condition that grows quadratically sounds like one that decides everything. Twelve flaps give sixty-six requirements; it is hard to imagine an arrangement with any freedom left in it.

Almost all of them are slack. In the best arrangement the search can find, fewer than half the pairs are at their limit on any tree measured here, and the ones that are do a surprisingly complete job — but not a complete one.

Eight of twenty-one #

The seven-leaf tree — two antennae on a head, two legs on a thorax, two legs and a tip on an abdomen — has twenty-one pairs to separate. At the arrangement that maximises the scale, eight of them are exactly at their limit and thirteen have room to spare.

That number is the whole of the condition as far as the arrangement is concerned. Slack requirements are requirements that could be loosened without changing anything: the tree could be redrawn with those pairs further apart in it and the same arrangement would still be the best one. Only the tight pairs are paying for anything, and identifying them tells a designer which parts of the subject the sheet is actually fighting.

Four of the eight are measured through a body. That is out of proportion to their share of the pairs, and it is the earlier essay’s finding in a sharper form: the long-range requirements the circle picture cannot draw are not a minor correction sitting quietly in the background — they are half of what is binding.

Three constraints and four pinned leaves #

The five-leaf tree is stranger. Three tight pairs, and yet four of its five leaves cannot be moved at all.

Three pairwise constraints cannot pin four points on their own — there are eight coordinates and three equations. What does the rest of the pinning is the sheet’s own edges. A leaf pushed against a wall has nowhere further to go in that direction, and a leaf held by two tight pairs and a wall is held by three things whose directions span the plane around it, which is enough.

That is the same test the flap nobody holds applies to a circle packing — a disc with three contacts all inside a half plane is not held, whatever the count says — and it gives the same kind of answer here. Counting constraints is not the way to know whether an arrangement is decided; the way is to push each leaf in every direction and see whether any of them lets it move.

Where the pinning comes from #

Splitting the pinning between the pairs and the sheet is worth doing, because the two are not the same kind of constraint and they behave differently when anything changes.

A pair pins a leaf against one direction: the direction toward its partner. A leaf with two tight pairs is held against two directions, which leaves it free along the bisector unless the two directions are more than half a turn apart. Three tight pairs whose directions positively span the plane hold it completely.

A wall pins a leaf against every direction on one side at once, which is a much stronger constraint and a much cheaper one. A leaf in a corner is held by two walls and needs no pairs at all. That is why three tight pairs can pin four leaves: the walls are doing most of the work, and a base needs an edge to point at is the reason the leaves want to be at the edges in the first place — a flap’s point is made of boundary, so a leaf pressed against the sheet’s edge is a flap in the most natural place for one.

The distinction matters when the sheet changes. Rescale the sheet and the walls move while the requirements do not, so an arrangement pinned mostly by walls reorganises and one pinned mostly by pairs merely scales. A census that reported only the number of tight constraints would treat those two arrangements as the same, and they behave oppositely.

The same question, asked of circles #

The same question has already been asked of a circle packing, and the answers are worth setting side by side.

The flap nobody holds finds discs in an optimal packing that can be moved without any overlap and without the radius changing, and makes the point that a packing is reported as one arrangement rather than as the family it belongs to. Two packings, one radius is the same finding from the other side: two arrangements, the same scale, and nothing in the method to choose between them.

The pairwise condition does not remove that. It adds requirements — every pair across a body asks for more room — so one might expect it to pin more, and on the seven-leaf tree it nearly does: six of seven leaves held, and the seventh with half a per cent of room against the five-leaf tree’s twelve. But the extra requirements are mostly slack, and a slack requirement pins nothing. Thirteen of twenty-one pairs have room to spare, so they contribute exactly as much to the arrangement as if they were not there.

That is the general shape and it is worth stating plainly. Adding constraints to an optimisation does not add determinacy unless the new constraints bind, and whether they bind is decided at the optimum rather than when they are written down. A designer told that the real condition has sixty-six terms where the picture had twelve circles might reasonably expect the layout to be more determined; it is a little more determined, and mostly by the sheet’s edges rather than by the terms.

The leaf that is free #

Pushing every leaf in three hundred and sixty directions and binary-searching how far it goes before the scale drops gives a number per leaf, and the numbers are not all zero.

On the five-leaf tree, one leaf — a front leg — can be moved 11.9 per cent of the sheet in some direction without the scale falling by anything at all. It is in no tight pair and it is not against a wall; the condition simply has no opinion about where it goes.

On the seven-leaf tree the freedom is much smaller — half a per cent, on one hind leg — which is what the extra requirements buy. Twenty-one pairs against ten leaves the arrangement far more nearly determined, and the freedom that remains is a rounding rather than a choice.

So the amount the method hands back falls as the subject gets more complicated, which is the opposite of what a designer might hope. A simple subject leaves room to make the base look right; a complicated one does not.

Slack is not waste #

Thirteen slack pairs on a twenty-one-pair tree invites the thought that the tree is badly drawn — that a better subject would have all its requirements binding at once. It would not, and the reason is worth setting down.

A requirement binds when the two leaves are exactly as far apart as the tree says they must be. For every pair to bind at once, every pair of points would have to sit at exactly the distance its tree path prescribes, which is a set of $n(n-1)/2$ equations in $2n$ unknowns — hopelessly over-determined above four leaves. Slack is not a symptom of a poorly chosen subject; it is what a plane can do. A tree metric is a metric on a graph, the sheet is a metric on a plane, and a plane cannot realise an arbitrary graph metric exactly.

So the right reading of thirteen slack pairs is that thirteen of the subject’s separations are comfortable and eight are not. Which eight is a fact about the subject — a hind leg and an antenna are close in the tree and far apart on the sheet, or the other way about — and it is the sort of thing a designer knows by feel and cannot otherwise point at.

What a loose leaf is, and what it is not #

A leaf with room is not an error and it is not an inefficiency. It is a degree of freedom in the design, and the arrangement the algorithm reports is one point in a family of equally good ones.

That matters because the arrangement is not the end of the work. A packing becomes a crease pattern, the crease pattern becomes a base, and the base has to be foldable in practice: creases that meet at awkward angles, flaps that lie over one another, paper that has to travel a long way. None of those preferences appears in the scale, so a designer with a loose leaf has a free move available to satisfy them — and a designer who does not know the leaf is loose will not use it.

The reverse case is the one to watch for. An arrangement where every leaf is pinned is an arrangement with no such moves, and a subject whose base is awkward to fold in that configuration has no remedy short of changing the tree. The freedom census says in advance which situation a design is in.

What the free move is for #

A loose leaf is a move, and the moves worth making are decided by everything that happens after the arrangement.

What joins the flaps is where an arrangement becomes creases, and the creases depend on where the points sit: the ridges are determined by the tight pairs, and a loose leaf’s ridges are determined by nothing in particular. The molecule that does not exist is the reminder that not every region of a packing admits the filling the theory would like, so a free move that changes which regions appear is a move with consequences the scale cannot see.

There is also the grid. Spelling a tree on a grid prices what box pleating costs a design — every limb rounded to a whole number of units — and a loose leaf is exactly the slack such a rounding wants. A leaf with 11.9 per cent of the sheet to move in can very often be moved onto a grid line for nothing, and one that is pinned cannot. The freedom census is therefore a prediction about how expensive a design will be to put on a grid, made before any rounding is attempted.

That is the useful reading of the whole measurement. The scale says what the sheet can do; the freedom says how much of what the sheet can do is still available to be spent on something else.

What the census cannot show #

Everything above is measured on one arrangement per tree, and that arrangement is what a search found.

It cannot show that the arrangement is optimal. The search is a hill-climb from many starts and returns the best it saw; a better arrangement would have its own tight set, which might be larger or smaller. What the census does establish is that this arrangement, whose scale is the best known, is under-determined — and an arrangement can only become more determined by being improved, never less.

It cannot show whether the freedom is one-dimensional or two. The measurement finds the largest distance the leaf can travel in the best direction; whether the set of positions it can occupy is a segment, a wedge or a disc is a finer question the number does not answer.

And it cannot show what the freedom is worth. Moving a loose leaf changes the crease pattern that follows, and whether the change is an improvement depends on criteria this account does not have — from a packing to a crease pattern is where those would enter.

What the model assumes #

The leaves live in a unit square with hard edges. Walls do a large share of the pinning, so the shape of the sheet is part of the answer rather than a container for it. A round sheet or a rectangle would pin differently.

A pair counts as tight within two parts in a thousand. That tolerance is what separates a constraint at its limit from one that is merely close, and a looser tolerance would report more tight pairs on every row.

A leaf counts as pinned when no direction lets it move without the scale falling below its value by a part in a billion. That is a strict test, so the pinned counts here are lower bounds on how pinned the arrangement is in any practical sense.

And the scale is the only objective. Any preference beyond it — symmetry, foldability, the look of the base — would use the freedom and is not represented.

How the numbers were checked #

The tight set is recomputed from the arrangement rather than remembered from the search. Every pair’s ratio of separation to requirement is compared against the arrangement’s own scale, so a search that had converged to something else would report a different set rather than the same one.

The three kinds are required to account for every leaf: pinned plus loose must be the leaf count on every row, which would fail if the room measurement had silently skipped a leaf.

Fewer than half the pairs must be tight on every tree, which is the essay’s claim and is checked on the computed sets rather than read off the drawings.

And at least one tree must leave a leaf loose. A census in which everything was pinned would be a census with nothing to report, and the check says so.

Reading one arrangement #

It is worth walking the seven-leaf case once, because the numbers become a description of a subject rather than a table.

Eight pairs bind. Four of them run between flaps that share a node — the two antennae against each other, the two thoracic legs against each other — and those are the requirements the circle picture would have shown. The other four run through a body: an antenna against a hind leg, or the tip against something at the far end. Those are the ones the circles cannot draw and they are half of what is holding the design down.

Six leaves are pinned, five of them with a wall involved. The seventh, a hind leg, has half a per cent of the sheet to move in. So the description of this design is: the sheet’s corners and edges are carrying the arrangement, the long-range separations are carrying the rest, and there is almost nothing left over. A design like that is finished in the sense that matters — any further improvement has to come from changing the tree.

The five-leaf case reads differently. Three pairs bind, one of them across the body, and one leaf has a substantial free move. That design is not finished: it has a scale and it has a choice, and the choice is worth something to whoever folds it.

Still open: what the freedom costs #

The measurement above is a size and the useful question is a direction.

A loose leaf can be moved 11.9 per cent of the sheet, and nothing here says where it should go. The reason is that the scale is indifferent to it, so choosing needs a second objective — and the obvious second objective is the crease pattern that follows. A leaf’s position decides which ridge creases meet it and at what angles, and an arrangement can be more or less pleasant to fold at the same scale.

That is a computation with a definite shape: enumerate the positions the loose leaf may occupy, build the pattern for each, and measure something about it — the sharpest angle, the number of layers at the thickest point, the total crease length. Whether any of those varies enough across the freedom to be worth choosing on is unmeasured, and it decides whether the freedom is a design choice or a curiosity.

Sideways from here, the same question belongs to the tree rather than to the arrangement. A pair that is slack is a pair whose tree distance could be raised for nothing — which is to say the subject could have a longer limb there at no cost in scale. Which limbs are free is the designer’s version of this census, and it is one edge sensitivity away from the numbers here.

The habit worth carrying is about constraint counts. Count what binds, not what is written down. A condition with sixty-six terms in it may be doing the work of six, and knowing which six is the difference between a layout that is decided and one that is merely described.