The price of a limb is not its length

What the condition does not decide ends on a question about the tree rather than the arrangement: a slack pair is a pair whose tree distance could be raised for nothing, so some of a subject’s limbs must be free. Which ones, and what the others cost, is one finite difference away.

Lengthen each edge in turn, re-solve the arrangement, and see how far the scale falls. The answers are not what the drawing suggests. The costs do not follow the lengths, the body is not always dearest, and some edges cost nothing at all.

Three times the price for half the length #

The bird has wings of 1.6, legs of 0.8, a head of 0.9, a tail of 1.3 and a body of 0.5. Its wings are the longest things on it by a wide margin and they are nearly the cheapest to lengthen: 0.0264 of scale per unit. The head costs 0.0793 — three times as much, for an edge a little over half as long.

Nothing about an edge’s own length enters its price. What enters is how many pairs at their limit run through it, and how tightly. A wing is in five pairs like everything else, but in the best arrangement its pairs have room; the head’s do not. The price of an edge is a fact about the rest of the subject, and the edge itself cannot be examined to find it.

That inverts the natural move. A designer whose base will not fit looks at the longest limb, because the longest limb looks like the problem. On this subject the longest limb is very nearly free and the fifth-longest is the expensive one.

Sweeping the head from 0.5 to 2.1 takes 23 per cent off the scale, and it falls at every step because lengthening an edge can only raise requirements. A head this subject could have had at half the length would have bought it nearly a quarter more of everything else.

Two edges that are free #

The lizard’s two front legs can be lengthened by a fifth of a unit and the scale does not move. Not by a little; by nothing the measurement can see.

The reason is exactly the one the freedom census pointed at. An edge in no pair at its limit is an edge no constraint is measuring, and one of those legs was already the loose leaf — the one that could be moved 11.9 per cent of the sheet for nothing. A leaf with room to move also has room to grow, and it is the same slack seen from the other side.

That is a free lunch of a modest but real kind: this subject could have longer front legs at no cost whatever in scale, which is to say the base as designed is spending nothing on them. It is also a warning about reading a design as optimised. A tree with free edges is a tree that was not drawn against this condition, since anything free would have been spent. A base needs an edge to point at is the other constraint a tree is drawn against, and it is a different one: it says where a flap’s paper must come from rather than how long the flap may be.

Why the length does not enter #

The reason is worth stating in one line and then unpicking, because it is the same reason the tight set was small.

An edge’s cost is how much the scale must fall when its requirements rise. Lengthening edge $e$ by $\delta$ raises $d_T(i,j)$ by $\delta$ for exactly the pairs whose tree path runs through $e$, and leaves every other pair alone. So the cost is decided by the tightest of the pairs through $e$ — and if none of them is tight, the scale does not move.

Length enters nowhere in that. A long edge is in the same pairs a short one in its place would be in; its length changes the level of those requirements, not which of them bind. Two subjects identical except for a wing’s length have the same tight set as long as the arrangement does not reorganise, and therefore the same wing price.

This is also why the prices come in a small number of values. The lizard’s two hind legs cost exactly the same, because they are in symmetric positions; its two front legs cost exactly nothing, for the same reason. A tree’s symmetry shows up in its prices before it shows up anywhere else, and a price list with unexpected ties in it is a symmetry the drawing did not make obvious.

The body is only sometimes the culprit #

On the seven-leaf tree the dearest edge is internal — the short one between head and thorax, at 0.0593 — with a hind leg second and the other internal edge third. So there is no rule either way. An internal edge is in more pairs than a leaf edge, which tends to make it dear, and the pairs it is in may still have room, which makes it cheap. Both happen.

The pattern across the three trees is that the price is spread over a wide range and nothing visible in the tree predicts the order: not the edge’s length, not whether it is internal, not how many pairs it is in. The only thing that predicts it is which pairs are at their limit, and that is a property of the arrangement rather than of the tree.

What this makes possible #

A price per edge is what turns a design from a fixed problem into a negotiable one, and the negotiation is the part a scale alone cannot support.

A design that does not fit has a cheapest repair, and the table names it. Shortening the dearest edge by a tenth buys more scale than shortening any other edge by a tenth, and the ratio between the dearest and the cheapest is large — infinite where an edge is free. A designer working by eye would shorten whatever looks longest, which on the bird is the wing, and would get a third of the benefit. Packing is the hard part prices the search that finds an arrangement at all; this prices the tree the search is given, and the second is much the cheaper place to look for a fit.

A design that fits with room to spare has a cheapest improvement, which is the same table read the other way. That is the reading what the condition does not decide reached from the arrangement: the slack pairs are where the subject is comfortable, and the free edges are the same comfort expressed as lengths a designer may spend. Spend the slack on the free edges, where it costs nothing, before spending it on anything that is binding.

And the prices say what a subject’s real shape constraints are. A bird whose head must be long is a bird that will always be a small base for its sheet, and one whose wings must be long is not — the same total length of limb distributed differently gives a different answer, and nothing in the drawing says so.

What a price list is not #

Three misreadings are available and each is worth ruling out.

It is not a ranking of importance. A free edge is not unimportant to the subject; it is unconstrained by the sheet. A bird with no wings is not a bird, and the wings are nearly free. What the price measures is the sheet’s opinion, and the sheet has no view about what the subject is.

It is not additive. The costs are partial derivatives at one arrangement, so lengthening two edges together is not the sum of lengthening each. Where both are in the same tight pair the combined cost is more than the sum; where lengthening one moves the arrangement so that the other’s pairs go slack, it is less. Every pair, not every circle has the same warning at the level of the condition — a pairwise quantity does not decompose onto the objects.

And it is not stable under large changes. A free edge is free until it becomes tight, and it becomes tight when it has grown enough for one of its pairs to reach the limit. The lizard’s front legs are free at 1.0 and would not be free at 2.0, and nothing in the price says where the change happens. The sweeps are what answers that, one edge at a time.

What the circles would have said #

It is worth asking what a price list computed from the circles alone would have looked like, since that is the picture a designer has.

Under the circle condition an internal edge does not appear at all, so its price is exactly zero — the body is free, always, on every subject. That is not a small error in one number; it is the wrong sign of conclusion about the one edge a designer is most likely to want to shorten. On the bird the body is the second dearest of six edges and the circles would have called it free.

The leaf edges fare better, because a leaf edge is in the circle condition. But their prices would be computed at the wrong arrangement — every pair, not every circle finds the circles’ best arrangement to be one the sheet cannot actually hold — so the tight set would be the wrong tight set and the free edges would be the wrong free edges.

So the price list is not a refinement of something the circle picture already offered. It is a quantity that picture cannot produce at all, and the nearest thing it can produce is misleading about the edge that matters most.

What the pricing cannot show #

Every number here is a finite difference on a search, and each of those words is a limitation.

A finite difference over a fifth of a unit is not a derivative. Where the tight set changes within the step — a pair that was slack becoming tight as the edge grows — the measured cost is an average over two regimes rather than the slope at the start. The sweeps show the curves are close to straight over the ranges drawn, which is evidence that this is not happening badly, and not a proof that it never happens.

The search is a hill-climb, so a cost is the difference between two arrangements each of which is the best found rather than the best there is. Both being slightly short would largely cancel; one being short and the other not would not. The stability check is that the same six or nine numbers come back at three different search efforts, which is reassurance rather than a bound.

And nothing here says a free edge is free for the design. It is free for the scale. A longer front leg changes the crease pattern, the layer count where the flaps meet and how the base behaves when folded, none of which this measures. A tree cannot argue is the standing reminder that the tree is a modelling choice before it is an input.

What the model assumes #

Only the scale is being priced. The objective is the largest number by which every tree distance fits between the leaves, and every other property of a design is outside it.

One edge moves at a time. The costs are partial derivatives, so they do not add: lengthening two edges together may cost more or less than the sum, because the arrangement rearranges.

The step is a fifth of a unit on trees whose edges run from half a unit to one and six tenths, which is a large step chosen so that the search’s own noise is small beside the signal.

And the sheet is a unit square. The walls do much of the pinning, so the prices are prices on this sheet; a different sheet would move them and the comparison would have to be redone.

How the numbers were checked #

No edge may cost a negative amount. Lengthening an edge raises the requirement on every pair through it and lowers none, so the scale cannot rise; a negative cost would mean the search had done better on the harder problem, which is a search failure and is reported as one.

The prices must be unequal — the dearest more than about half again the middle one — which is the essay’s claim and is checked on the computed numbers rather than read off a drawing.

Each sweep must fall at every step, for the same reason, which catches a search failure at any one point of the curve rather than only at the ends.

And the measurement is run at three search efforts and required to give the same numbers. At forty restarts one of the lizard’s legs came out five times dearer than it is; at sixty, a hundred and twenty and two hundred it comes out the same, which is why sixty is the floor the figures use.

The three trees side by side #

Reading the three price lists together says something none of them says alone.

The bird has six edges and a nine-fold spread between its dearest and its cheapest: head 0.0793, body 0.0429, wings 0.0264 each, tail 0.0123, legs 0.0120. Its expensive edge is short and at the front; everything at the back is cheap.

The lizard has a six-fold spread among its priced edges and two that are free: tail 0.0451, body 0.0426, hind legs 0.0169 each, front legs nothing. Its expensive edges are the tail and the body, which are at the same end of the animal.

The insect has nine edges and no free ones, with a spread of two and a half: head–thorax 0.0593 down to a foreleg’s 0.0233. It is the most evenly priced of the three and it is also the one with the most pairs at their limit — eight of twenty-one against five of fifteen and three of ten.

The pattern is that a more constrained design is more evenly priced. A tree with few tight pairs has a few edges carrying all of them and the rest carrying none; a tree with many has the load spread. So the spread in the price list is itself a reading of how close a design is to its limit, available without computing anything else.

Still open: the tree that spends its slack #

The prices invite an optimisation this essay has not run.

Given a subject with some freedom in its proportions, which tree maximises the scale? The answer is not simply “shorten everything”: a design has limbs it needs. But within whatever latitude a subject allows — a head anywhere between 0.7 and 1.1, wings between 1.4 and 1.8 — the prices say immediately which way to move each edge, and the optimisation is a gradient ascent with the finite differences as its gradient.

What makes it interesting rather than routine is that the prices change as the tree moves. An edge that is free now becomes dear once the arrangement rearranges around it, and a designer who spends all the free length at once will find the freedom gone before the spending is. How far the prices hold before the tight set changes is the missing measurement, and it is the difference between a gradient worth following for one step and one worth following to a conclusion.

Sideways from here, the same prices are what a grid wants. Spelling a tree on a grid has every limb rounded to a whole number of units and measures the shape error that causes; rounding a free edge costs nothing and rounding the dearest edge costs the most, so a rounding that knew the prices would round in the cheap direction where it could. That is a better rounding than the one that minimises shape error alone, and nobody appears to have asked for it.

The habit worth carrying is about sensitivity in constrained problems. Price the inputs by moving them, not by looking at them. In a problem where a handful of constraints bind, the expensive input is the one those constraints run through, and it need bear no relation to the input that is largest, most prominent or most obviously the subject of the design.