Every pair, not every circle

Every flap on one axis states the restriction that makes the circle argument true: a uniaxial base’s flaps all lie along a single line, so a flap’s length is a distance measured on the sheet between the two pieces of boundary its point is made from. A flap costs a circle is the consequence everybody remembers — one disc per flap, radius the flap’s length, and the design problem becomes a packing problem.

The circles are not the condition. They are the condition with something left out, and what is left out is the subject’s own body.

The condition is about pairs #

Write the subject as a tree: a leaf for each extremity, an edge for each limb, and internal edges for whatever lies between them. Then the requirement on a set of leaf positions $p_i$ on the sheet, at scale $m$, is

$$|p_i - p_j| \ \ge\ m\, d_T(i, j)$$

for every pair of leaves, where $d_T$ is the distance from $i$ to $j$ measured through the tree. Two flaps that meet at one node have $d_T = \ell_i + \ell_j$, and requiring their points to be that far apart is exactly requiring two discs of radius $m\ell_i$ and $m\ell_j$ not to overlap. That is the circle picture, and it is correct for that pair.

Two flaps on opposite sides of a body are a different case. The path between them runs down one limb, along the body and up the other, so $d_T = \ell_i + \ell_j + b$ — and the circles ask only for $\ell_i + \ell_j$. The discs can be comfortably disjoint while the condition is broken, and nothing in the drawing says so.

What ignoring the body is worth #

A star — one hub with five equal limbs and no internal edge at all — gives the same scale both ways, which it must, because its leaf-to-leaf distances are the sums of its leaf edges. That is the case the circle picture was drawn for and it is exact there.

A five-leaf tree with a single body of 0.8 supports a scale of 0.2783 under the real condition and 0.3228 under the circles: the circles promise 16.0 per cent more. A seven-leaf tree with two internal edges supports 0.2112 against 0.2740, an overstatement of 29.7 per cent. Since the scale is what multiplies every flap, a designer who has packed circles into a square and read off a scale is being promised limbs a sixth or a third longer than the sheet will give.

The direction is not an accident and cannot reverse. Setting an internal edge to zero can only relax a requirement, never tighten one, so the circle condition is a relaxation and its answer is an upper bound. A relaxation is a perfectly respectable thing to compute — it is fast, it is drawable, and it certifies infeasibility — but it is not the condition, and the gap is not small.

What a body costs, swept #

Sweeping the internal edge from nothing upward turns the comparison into a curve, and the left-hand end of the curve is the circle answer. A body of zero is the circle condition — not approximately, exactly — and every increment after that is separation demanded of every pair that crosses it.

The fall is monotone and it has to be: lengthening an internal edge raises the requirement on every pair across it and lowers none, so no rearrangement can recover what it costs. By a body of two the scale is 28 per cent below its bodyless value. A subject with a long body is a subject the circle picture badly misdescribes, and the misdescription grows with exactly the feature that makes the subject interesting.

The same thing at seven leaves #

A second tree makes it clear that the gap is not a property of one arrangement.

Two antennae on a head, two legs on a thorax, two legs and a tip on an abdomen, with bodies of 0.6 and 0.9 between them. Twenty-one pairs, and only a handful of them are between flaps sharing a node — the antennae with each other, each pair of legs on its own segment. Every other pair crosses at least one body, and the pairs from an antenna to a hind leg cross both.

Sweeping one of its two bodies costs 22 per cent over the range drawn, with the other body still there the whole time. The two internal edges are not independent — the pairs crossing both feel the sum — so a subject with a segmented body accumulates the correction rather than paying it once.

That is the shape a designer should expect. A correction proportional to how much of the subject lies between its extremities is small for a spider and large for a caterpillar, and the circle picture is a good approximation exactly where the subject is mostly limbs.

A dozen circles, sixty-six requirements #

The counting is the other half of the difference and it is the part that decides whether the condition can be checked by eye.

Circles grow as the flap count and requirements as the pair count. Five flaps are five circles and ten requirements; twelve flaps are twelve circles and sixty-six. A drawing with a dozen discs in it is a drawing a designer can check by looking; sixty-six separations, most of them between flaps that are nowhere near each other, is not.

That asymmetry is why the circle picture survives its own inaccuracy. It shows the constraints that are easy to see — two discs overlapping is unmistakable — and hides the ones that are not, which are exactly the long-range ones that run across the body. The requirement a drawing cannot show is the requirement a designer will miss, and the count says there are many more of those than of the other kind.

Why the correction is invisible in the drawing #

The two conditions differ by a term that is easy to state and impossible to draw as a circle, and that asymmetry is worth pulling out because it is the reason the error survived.

A pair’s requirement is $m(\ell_i + \ell_j + b_{ij})$ where $b_{ij}$ is whatever lies between them in the tree. The first two terms belong to the two flaps individually and can be drawn as two radii, which is why the circle picture exists at all. The third belongs to neither flap: it is a property of the pair, it changes from pair to pair, and there is no object at either end that could carry it.

A designer wanting to see it has two choices and neither is a circle. Grow both radii by $b_{ij}/2$ — but then the same flap needs different radii against different partners, so it is not a disc any more. Or draw the requirement as a segment between the two points, as the figures here do — which is honest and produces a drawing with $n(n-1)/2$ lines in it.

That is the whole reason the relaxation is the one everybody draws. It is not that the body was forgotten; it is that the body is a pairwise quantity and the drawing is made of per-flap objects. The standard practice of putting a river between the flaps on either side of a body is the working designer’s repair, and it repairs the picture rather than the condition: a river is a strip of sheet reserved, which is a sufficient arrangement rather than the requirement itself.

The case where the picture is exact #

Drawing a star makes the boundary of the circle picture’s validity visible. Every pair’s requirement is the sum of its two leaf edges, every constraint is two circles touching, and nothing is being hidden. A designer working on a star is working with a complete drawing.

So the circle method is exactly right for subjects with no body, and the subjects it was devised for — insects, with long limbs and short bodies — are close to that case rather than in it. The method’s success and its error have the same source: a short body makes a small correction, and nothing in the drawing warns that a long one makes a large one.

What the drawings cannot show #

The arrangements here are found by a search, and every number is what that search found rather than what the condition allows.

The search maximises the smallest ratio of separation to required distance, from many random starts with a symmetric one among them. It is a hill-climb, so it finds a good arrangement and not provably the best; a better arrangement would raise the scale, which would lower the overstatement without changing its sign, since the same search is used for both conditions.

Nor do the drawings say what the flaps become. A set of leaf positions satisfying every pair is a packing, and turning one into a crease pattern is a further construction with its own choices — from a packing to a crease pattern is where that happens. Everything here is about whether the packing exists.

And nothing here draws a river. The condition is stated over leaf pairs, which is enough to decide feasibility; the picture designers actually draw puts a strip of the sheet between the flaps on either side of the body, and the relationship between that strip and these pairwise requirements is a construction this essay does not make.

A relaxation is still worth having #

None of this makes the circle picture a mistake, and saying why is worth a paragraph, because a relaxation has a job that the exact condition cannot do.

The circles’ answer is always at least the true answer. So if circles of a given size cannot be packed, the pairs certainly cannot be separated, and the design is impossible at that scale — a certificate that costs one packing instead of a search over a quadratic number of constraints. Packing is the hard part prices the search that the exact condition needs, and a cheap test that rejects hopeless cases before that search begins is exactly what an expensive search wants at its front.

What a relaxation cannot do is certify success, and that is the use it is habitually put to. A packing of disjoint circles is read as a design that works, and for any subject with a body it is a design that does not. The circles say no reliably and yes optimistically, which is the opposite way round from how they are read.

There is a third use, and it is the one this essay’s numbers are for: the ratio between the two answers is a measure of how much body a subject has. A star gives 1.00, the five-leaf tree 1.16, the seven-leaf tree 1.30. That number is computable before any design work and it says how far the familiar picture can be trusted for the subject in hand.

What the model assumes #

The base is uniaxial. Every flap along one line is what makes a flap’s length a sheet distance at all, and every flap on one axis is where that comes from. Nothing here applies to a base with flaps on several axes.

The sheet is a unit square and the leaves are points in it. No boundary condition is imposed beyond that, so a leaf may sit anywhere inside, and the requirement that a flap’s point come from the boundary — which a base needs an edge to point at establishes — is not enforced here.

The scale is the same for every flap. That is what makes a single number the answer; a design that stretched some limbs relative to others would be a different tree.

And the tree is given. Which extremities a subject has, and how long, is the modelling decision that precedes all of this, and a tree cannot argue is where that decision is examined.

What a designer does with the difference #

Three consequences follow for somebody actually laying out a base, and they do not all point the same way.

A packing found from circles has to be re-checked pair by pair. That is $n(n-1)/2$ distance comparisons, which is nothing to compute and impossible to do by eye at a dozen flaps. The check is cheap; what is expensive is having no drawing that makes its failures visible.

The scale has to come down, and by how much is not uniform. Sixteen per cent on one tree and thirty on another means there is no rule of thumb to apply — the correction is a property of the particular tree, and the only way to know it is to compute both numbers and divide.

And the shortfall is worst exactly where a subject is most characteristic. A design’s body is what makes it a lizard rather than a starfish, and it is the feature the circle picture cannot see. A method that is most accurate on the least distinctive subjects is a method whose errors correlate with the interest of the problem, which is an uncomfortable property for a design tool and is not usually stated.

How the numbers were checked #

The star is the calibration and it is checked rather than assumed. A tree with no internal edge must give the same scale under both conditions, to within a part in a billion, and the figure fails if it does not — which would mean the two conditions had been implemented as two different things rather than as one and its relaxation.

Every tree with a body is required to be overstated by the circles, by more than five per cent, so the comparison cannot pass by producing two numbers that happen to agree.

The sweep is required to be monotone at every step, which is the claim that a longer body can only cost. A single step upward would mean the search had found a better arrangement at a longer body than at a shorter one, which is a search failure rather than a fact.

And the metric is computed from the tree by walking it, rather than typed in as a table of distances, so a tree whose edges were changed would change its own requirements.

Still open: what the condition leaves undecided #

The arrangements above are the best the search found, and the interesting question is how much of each one the condition actually fixes.

A packing of circles is known to leave discs loose — the flap nobody holds measures exactly that, and finds discs that can be moved without any overlap and without the radius changing. Whether the pairwise condition leaves the same freedom, more of it or less, is a measurement nobody has made, and it matters because the freedom is precisely what a designer is choosing when the algorithm has finished. A condition that pins every leaf leaves no design; one that pins few leaves a great deal.

The measurement is a small one: push each leaf in every direction and find how far it goes before the scale drops. The answer is a number per leaf, and the sum of those numbers is the size of what the method hands back rather than decides.

Sideways from here, the overstatement has a use that this essay has not made. A relaxation whose answer is an upper bound is a bound, and a bound is worth having: if the circles cannot be packed at a given scale then neither can the pairs, so the cheap check certifies impossibility even though it cannot certify possibility. Packing is the hard part prices the search, and a cheap necessary condition is exactly what an expensive search wants at its front.

The habit worth carrying is about pictures that stand in for conditions. Ask which of the condition’s terms the picture can show, and assume the ones it cannot are where the errors are. Two overlapping discs are visible and a pair that is too close across a body is not, so a drawing that shows the first and hides the second will be trusted in exactly the cases where it is wrong.