A price holds until the arrangement moves

The price of a limb is not its length priced every edge of three subjects’ trees by lengthening it and re-solving the arrangement. The prices were far apart and followed nothing visible in the drawing: on the bird the head, at nine tenths of a unit, cost three times what a wing of 1.6 cost, and two of the lizard’s legs cost nothing at all. It ended with the optimisation those prices invite and a warning attached. The prices change as the tree moves, and how far they hold before the arrangement rearranges round the change is the measurement that decides whether a price list is a gradient worth following to a conclusion or for one step.

That measurement is straightforward to make. Hold the subject’s total length of limb fixed, move a little of it from the edge priced highest to the edge priced lowest, re-solve, re-price, and repeat.

The price list the spending starts from #

The first price list was made one edge at a time, lengthening each by a fifth of a unit. Pricing the wings together and the legs together changes the numbers a little and the order not at all: the head is dearest, the body second, the paired limbs cheap, and the tail, which no binding pair runs through, is nearly free. The list is a snapshot of one arrangement, and the arrangement is the thing about to move.

A price list of this kind is the obvious tool for the situation packing is the hard part describes, where finding any arrangement at all is expensive and a designer would rather adjust the subject than search again. It is also the obvious tool for a subject with a constraint of another kind: a base needs an edge to point at ties some flaps to the sheet’s boundary, and a list of prices says which of the others can be changed cheaply to make room. In both uses the list is read once and acted on, and what follows measures what acting on it does to the list.

Spending a fixed length #

The bird’s tree has a body of 0.5, two wings of 1.6, a head of 0.9, two legs of 0.8 and a tail of 1.3: 7.5 units of edge in all. Its arrangement on a square sheet supports a scale of 0.2651, meaning every distance through the tree fits between the matching leaves on the sheet when multiplied by that.

The wings and the legs are pairs, and a design that lengthened one wing and not the other would be a different subject, so each pair is priced and moved as one group. A group’s price is how much scale is lost per unit of length added across all its edges, measured by adding a tenth to each and re-solving. At the start the head is dearest at 0.092 and the tail cheapest at zero — the tail can be lengthened without moving the scale at all, which is what what the condition does not decide called slack: no pair at its limit runs through it.

So the first step moves a tenth from the head to the tail. The head becomes 0.8, the tail 1.4, the total is still 7.5, and the scale rises to 0.2753 — 3.8 per cent more for a subject no longer in total than before.

The free edge stops being free #

Re-pricing after that one step gives a different list, and not only at the two edges that moved.

The tail, free a step ago, now costs 0.044. The legs, which were not touched, have gone from 0.031 to 0.079 — the second-dearest group on the subject. The head is still dearest at 0.102, and the cheapest group is now the wings, at 0.034.

The tail was free for exactly one tenth of a unit. Its slack pairs had room, and the tenth used the room up. The pairs at their limit after the move are the same five as before — none of them through the tail — but the tail’s pairs are now so close to the limit that the next tenth brings one of them to it, and a price measured by adding a tenth sees that. The legs changed price the same way: the leaves shifted to accommodate a longer tail and a shorter head, and a pair through the legs that had room before is now one tenth from binding.

That second effect is the one a price list cannot warn about. An edge’s price depends on the other edges’ lengths, through the arrangement they jointly decide, so moving length between two edges re-prices a third. A designer following the list as a gradient is following it on the assumption that the pairs near their limit stay the same, and on the bird that assumption failed at the first step even though the pairs actually at their limit did not change.

The curve the spending traces #

The spending carries on with whatever the list says at each step. Step two moves length from the head to the wings, which were cheapest, and the scale barely moves — 0.2748, a hair below the step before, as the arrangement rearranges and a different set of pairs comes to the limit. Step three takes from the legs, step four from the body, both to the wings, and the scale reaches 0.2817: 6.3 per cent above the drawn subject, with the body at 0.4, the wings at 1.75, the head at 0.7, the legs at 0.75 and the tail at 1.4.

Then it stops. Step five takes from the head again and the scale falls to 0.2778; step six takes from the tail and it is 0.2780. The prices at step four have come together — every group between 0.054 and 0.081, against a spread from zero to 0.092 at the start — and when the dearest and cheapest prices are close, moving length between them buys less than the rearrangement it causes costs.

That is the condition an optimum of this kind has to meet: at the best distribution of a fixed total, every group that can still give or take length is priced the same, because otherwise moving length from the dear one to the cheap one would help. Step four is where the measured prices come nearest to that, and it is where the scale is highest.

The lizard, whose prices hold #

The lizard behaves differently, and the difference is the measurement’s point. Its tree has a body of 0.8, front legs of 1, back legs of 1 and a tail of 1.4, and it starts at a scale of 0.2783.

The back legs are dearest at 0.062 and the front legs cheapest at 0.022, and for four steps running they stay dearest and cheapest. Length moves from back legs to front legs a twentieth of a unit per leg per step; the back legs’ price creeps up, 0.062 to 0.077, the front legs’ holds at 0.022, and the scale rises at every step, to 0.2988. Throughout, the same three pairs are at their limit.

At step five the arrangement changes: five pairs come to the limit instead of three, the tail becomes the dearest group and the prices move together, 0.071 against 0.051. The scale gains almost nothing that step and a little the next, reaching 0.3011 — 8.2 per cent above the drawn lizard.

So a price holds for as long as the pairs at and near the limit stay the same. On the lizard that is four tenths of a unit of length moved; on the bird it is one. Nothing in either tree announces which, and the only way to find out is the re-solving the list was supposed to save.

Why a free edge fills so quickly #

The bird’s tail filled after a tenth because free is a statement about slack, and slack is finite. A leaf with room to move has a distance to the nearest arrangement in which one more pair would be tight, and lengthening its edge spends that distance.

What the condition does not decide measured those distances directly, as how far each leaf could be moved for nothing, and found them small — half a per cent of the sheet for the loosest leaf of a seven-leaf tree. A free edge is an edge whose slack has not been measured in the right unit: at a scale of about a quarter, a leaf with half a per cent of the sheet to spare can absorb about two hundredths of a unit of tree length before its pairs bind. The bird’s tail had more, but not much more than a tenth.

That gives a rule a designer can use without the re-solving. Price an edge, then ask how much slack its pairs have, and treat the price as good for that much length and no more. An edge that is free with a little slack is a small gift; an edge that is dear with a lot of slack in its neighbours’ pairs is a price that will hold.

What the spending assumes #

The total length of limb is what is held fixed. That is one budget among several a designer might hold. A subject whose head must be at least 0.7 and whose wings no more than 1.8 has a box of allowed lengths as well as a total, and spending within a box stops at its walls as well as where the prices meet.

Symmetric edges move together. The two wings and the two legs are priced and moved as pairs, which keeps the subject bilaterally symmetric; unpaired, the arrangement might prefer one wing longer, and that is a different design.

The arrangement is the best a search finds. Every scale here is from the same hill-climbing search over leaf positions, with sixty starts, and it is the best found rather than the best there is. The price of an edge is a difference of two such scales, and a difference of two approximations is noisier than either — which is visible in the bird’s head, priced at nothing at step five and 0.076 at step six.

And a step is a tenth of a unit. A smaller step would follow the prices more closely and take more re-solving; a larger one would jump over the changes in the tight set that this measurement is about.

What the measurement cannot show #

It does not find the optimum. Step four is the best distribution the spending reaches, and a spending that moved length between groups other than the dearest and cheapest, or in smaller amounts, might reach a better one. What it shows is that a greedy use of the prices climbs for a few steps and stops, not where it stops relative to the best possible.

It does not say what the subject is for. A bird with a head of 0.7 and wings of 1.75 folds larger than the bird as drawn, and whether it is still the bird a designer wanted is not a question the sheet can answer. Every pair, not every circle set up the condition these prices come from, and the condition measures what fits; it has no view about proportion.

The price depends on the size of the probe. Every price here is measured by adding a tenth of a unit and re-solving, so it counts a pair as binding if a tenth is enough to bring it to its limit. A probe of a hundredth would have called the bird’s tail free for longer and its legs cheap for longer, and a probe of half a unit would have called almost nothing free at all. That is not a defect of the measurement so much as its content: a price is always a price for a move of some size, and the size that matters is the size of the move a designer is about to make. A rounding to a grid moves limbs by a fraction of a unit, and a tenth is the right order for it.

And it does not separate the two effects of a move. When the legs nearly trebled in price after the first step, part of that was the arrangement moving and part the search finding a different arrangement of about the same scale. The two are indistinguishable in a single measurement and would need many searches per step to pull apart.

A price list is a local map #

The general statement is familiar from any optimisation with constraints, and it is worth saying in this subject’s terms because the drawing hides it.

The prices are the derivatives of the scale with respect to the edge lengths at the current set of binding pairs. Within that set they are exact and they hold; when a move changes which pairs bind, every derivative can change at once, including those of edges nobody moved. A uniaxial base is a solution to a problem with dozens of such constraints — twenty-one pairs for seven leaves — and only a handful bind at any one time, so the set is small and a small move can change it.

That is why the bird and the lizard differ. The lizard’s three binding pairs are far from the next pair that would bind, so its prices describe a region several tenths wide; the bird’s five binding pairs have neighbours close by, so its prices describe a region a tenth wide. The width of a price’s region is a property of the arrangement’s near-misses, and it can be read off the same arrangement the prices came from, by asking how much slack the loosest non-binding pair through each edge has.

Every flap on one axis described the tree method as turning a subject into a packing problem, and spelling a tree on a grid rounded the result onto a grid of whole units. The prices sit between the two: they say how the packing answers when the subject is changed, and they change the moment the packing does.

Still open: whether the prices can direct a rounding #

The spending moves length continuously, and a real design does not. Spelling a tree on a grid rounds every limb to a whole number of grid units and measures only how far the proportions drift; which way each limb rounds is chosen to keep the shape, not the scale.

A rounding is a small, discrete spending. Rounding the head down and the tail up is a move of length from a dear edge to a cheap one, of exactly the kind measured here, and the prices say which direction should gain scale — for as far as they hold, which a rounding of a tenth or two is likely to be within. Whether rounding every dear edge down and every cheap edge up gives a larger model than rounding to the nearest unit, and what it costs in shape, is a measurement the same re-solving can make.

Sideways from here, the spread of the prices is a single number that says how far a design is from its best distribution of length, and nothing in the method currently reports it. The last free parameter found one number left in a design once the packing is fixed; the spread is a number about whether the packing should have been fixed where it was.

The habit worth carrying is about sensitivities. A price is a derivative, and a derivative is a promise about a neighbourhood, not about a direction. Before following one, ask how wide the neighbourhood is — here, how far the pairs that set the price are from the pairs that would replace them — because the price will be wrong about everything past that edge, including edges nobody moved.