The trees drawn at no angle

The essay that derives the cube law draws a tree to illustrate it, and so does the essay that fits an exponent to junctions. Both pictures are correct in the quantity they are about. Both are wrong in a quantity nobody was looking at, and the second half of that sentence is what this essay is for.

Every junction in both trees takes its three radii from the cube law exactly. Every junction in both takes its angle from a number typed into the drawing routine — 30 degrees to a side in one, 32 in the other, because 32 is what the routine uses when nothing is passed. Neither number came out of anything. The cost that fixed the widths wants 37.4673 degrees to a side, and it has wanted that all along.

What the widths are built from #

The radii are not approximate and are not fitted. At every one of the 31 junctions in each tree, the cube of the parent’s radius equals the sum of the cubes of the two daughters’, to the precision of the arithmetic.

That is the whole content of the figures as they were drawn, and it is a real content. A picture in which the widths visibly obeyed something else would have been useless for the argument it accompanies, and this one does not.

What a tree of this size is #

Both figures draw five generations. A binary tree of that depth is 63 segments and 31 junctions, which is the number the widths are checked at; seven generations would be 63 junctions and is more than a panel can carry legibly.

Thirty-one is also worth stating because a segment count is not a junction count, and quoting one for the other is how a caption comes to claim twice the checking it did. Every number in this essay about how many junctions obey something is a junction count.

What “obeys” has to mean #

A picture obeys a rule when every quantity the rule constrains and the picture shows is taken from the rule. That is a stricter test than the one these figures passed, which was that the quantity the surrounding argument was about came out right.

The stricter test is the useful one because a figure is looked at whole. A reader cannot see which of its numbers were derived and which were typed, and has no way of knowing that the widths and the angles in one drawing came from two different places.

And what the angles are built from #

A constant. The routine that draws these trees takes a single half-angle and turns both daughters by it, generation after generation, and neither figure ever passed it a value derived from anything.

Thirty degrees is a round number. Thirty-two is the value the routine falls back on when a caller passes nothing, which is what the second figure does. So one angle in this collection is a choice and the other is a default, and the difference between them does not matter, because neither is an answer to a question.

They are not equally excusable, though, and the difference is instructive. A round number typed by hand is at least a decision somebody made and could have written down; a default inherited by passing nothing is a value that entered the picture without anyone choosing it at all. The second is the one that spreads, because every future figure drawn without an explicit angle gets it too.

Read as exponents #

An angle names an exponent, through the same closed form that names the angle. Read that way the two drawn totals say 2.5237 and 2.6239.

So each picture makes two incompatible statements about the same tree: its widths say the exponent is exactly three and its angles say it is about two and a half. Nothing in the drawing reconciles them, because nothing in the drawing knew they were the same quantity.

Where the drawn angles land on the curve #

Both sit on the steep part, below Murray’s value and well above the collapse at two. The curve climbs 23.030 degrees per unit of exponent at three, so the fourteen and eleven degrees the two pictures are narrow by are worth about half an exponent between them.

That is a large error in the units the surrounding essay is written in. An essay arguing that a network obeys a cube law, illustrated by a tree whose angles say two and a half, is illustrated by a counterexample to itself.

Half an exponent is also more than the whole of what the branching thread has spent its effort on distinguishing. Telling three from 2.9 needs thirty-two junctions measured to two per cent to establish, and the pictures are wrong by five times that gap in the direction nobody was watching.

Why nobody noticed #

Because it costs nothing. A total fork angle of 60 degrees costs 0.573 per cent more than the optimum, and 64 degrees costs 0.292 per cent.

Under a third of one per cent is not a departure a picture can show. The cost around a branch point is so nearly flat that a fork can be moved through more than forty degrees for one per cent of the network — which is a finding in its own right and, here, an explanation for a defect surviving several years of figures.

The miss, per daughter #

At a symmetric fork each daughter should leave at 37.4673 degrees. The two pictures draw them at 30 and at 32, so each daughter is 7.47 degrees out in one figure and 5.47 in the other.

Seven degrees is visible. Anybody comparing the two panels side by side sees a difference, and the reason nobody did is that there was nothing to compare against: the correct angle was not drawn anywhere on this site until it was computed.

That is the shape of the failure rather than an excuse for it. A quantity with no computed reference value is a quantity that cannot be seen to be wrong, however plainly it is drawn, and this collection has been caught the same way before — by a number nobody had varied, sitting inside a routine that was working perfectly.

The tree at the angle the cost asks for #

Drawing it is the cheapest part of the whole correction. The same routine, the same radii, the same five generations, with the half-angle set to what the closed form returns.

What changes in the picture #

Less than the arithmetic suggests, and that is worth admitting rather than hiding. The corrected tree is a little broader, its outer branches reach a little further across, and no reader would pick it out of a pair without the labels.

Which is the same fact as the flatness, seen from the eye rather than from the cost. A model that barely distinguishes two configurations produces two pictures that barely differ, and the picture is therefore not the instrument that catches this.

It is worth being clear about which direction that cuts. It does not make the correction unimportant: the difference is small to look at and large in the units of the argument, and those are two facts about the same pair of pictures rather than one fact stated twice.

The asymmetric case is worse, and grows #

Everything so far is the symmetric fork, which is the one case where a single angle for both daughters is at least the right shape of answer. It stops being that as soon as the daughters differ.

At a daughter ratio of 0.769 — one daughter about three-quarters the radius of the other — the cost wants 26.48 and 48.89 degrees, and the drawing gives the same number twice. The worst miss is 18.89 degrees. At a ratio of 0.5, where one daughter is half the radius of the other, the cost wants 13.04 and 64.53, and the worst miss is 34.53.

What the cost actually asks for there #

The large daughter should barely turn at all. Carrying most of the flow, it is nearly a continuation of the parent, and the thin one swings out past sixty degrees to reach its own destination.

The drawing does the opposite of that in the one respect that matters: it leans the thick daughter over as far as the thin one. A reader looking at that junction sees two branches of very different thickness leaving at identical angles, which is a statement about the junction, and it is false.

What a reader is entitled to assume #

A figure beside a derivation is read as an instance of it. That is the whole reason to draw one, and it is why an illustration is a claim rather than a decoration: it says this is what the rule produces, and everything visible in it is offered as part of that.

So a picture may leave a quantity out — a tree with no widths at all would have been honest — and it may not put a quantity in and take it from somewhere else without saying so. The angles here are the second case, and the caption made it worse by naming the law the widths obey.

One angle for two daughters #

The defect was structural rather than a bad value. The drawing routine had one angle parameter and applied it to both daughters, so there was no value of it that drew the lopsided fork correctly — 13.04 and 64.53 cannot both be one number.

That is why the asymmetric comparison above is not drawn by the tree routine. It is built from the junction’s own three radii and two angles, which was a statement of what was missing rather than a way around it, and which stays as it is now that the routine can do better: a figure whose job is to show what the drawing ought to be should not be drawn by the drawing.

And it propagates down the tree #

A tree whose daughter ratio is not one is a tree where every junction has the same defect, and the defect compounds: each generation inherits the direction its parent was drawn at, so a branch five generations down sits somewhere the accumulated angles put it rather than somewhere a cost put it.

The widths still come out exact at all 31 junctions, because the widths are computed per junction from the ratio. The shape is the only thing that is wrong, and the shape is what a picture is for.

A word that means two things #

There is a trap in the arithmetic above and it is worth naming, because it is exactly how a figure comes to be drawn at the reciprocal of what its caption says.

The drawing routine spells a daughter ratio larger over smaller: a ratio of two means one daughter twice the other. The cost’s own machinery spells it smaller over larger, as a number between nought and one, in agreement with the junction library the rest of the branching work is measured with. Two spellings of one word, and the figures here use only the second, converting at the single point where the tree is actually drawn.

The lopsided miss costs less, not more #

The natural expectation is that a bigger angular error costs more, and it does not. The excesses at the three daughter ratios run 0.573, 0.562 and 0.482 per cent for the thirty-degree tree and 0.292, 0.291 and 0.274 for the thirty-two-degree one.

The miss triples and the price falls. That is the flatness again: the one-per-cent band widens as the fork goes lopsided, faster than the optimum inside it moves, so the more wrong the drawing gets the less the cost minds.

Which is the wrong lesson to take from it #

Cheapness is not correctness. The cost is the reason the error survived, not a defence of it, and a figure accompanying a derivation is making a claim about that derivation rather than about a plant’s budget.

The claim a reader takes from a tree drawn beside the cube law is that this is what the cube law’s tree looks like. It is not. It is what the cube law’s widths look like, wearing angles from somewhere else.

The caption that has to be withdrawn #

A caption on this site said a drawn tree obeys the cube law. The widths do; the angles were never derived from anything, say about two and a half read as exponents, and are up to 34.53 degrees from where the same cost puts them.

So the sentence was true of the quantity it was pointing at and false as a description of the picture. That is the most durable kind of wrong caption, because every check that reads it finds the thing it names and finds it correct.

The same shape has appeared elsewhere here, on a claim that was not a caption: three consecutive samples of a grid were reported as a repeating pattern that a standing prediction had asked for, and every component of the routine that found them was right. A statement can be false as a description of what the reader is looking at while each of its parts checks out.

What made it invisible #

Nothing was measuring the angle. The branching thread measured radii, fitted exponents to radii, and asked which junctions carry information about the exponent — all of it about the three numbers at a junction and none of it about where the junction points.

An untested claim inside a picture is the same shape as an untested claim inside a code comment, which this collection has also had to correct: both survive because the machinery that would have contradicted them was never asked to run.

What is not wrong with the pictures #

Two things, and stating them is part of reporting the defect honestly.

The widths are exact, at every junction, in every figure. And the drawn angles are well inside what the geometry can produce: a branch point inside the symmetric configuration can reach total angles from 37.90 to 179.75 degrees, so 60 and 64 are ordinary interior values rather than something clipped or degenerate. The pictures are drawings of possible junctions that are not optimal junctions.

What the repair took #

One option, and it is now in place. The drawing routine accepts the two daughter angles in the order it already generates the daughters — smaller first, since that is the one the cost turns further — and defaults to the same angle twice, byte for byte, so that every existing caller is untouched. Nothing else changed: the radii, the shrink and the recursion were already right.

The default is the part worth defending. A correction that quietly moved every drawn tree to the optimum would have deleted the comparison this essay is about, and the symmetric tree is not a mistake to be tidied away — it is the thing the asymmetric one is measured against. So the figures above still show the fork the site draws beside the fork the cost wants, and what changed is that a figure can now be handed the second one as a tree rather than as a junction.

The gap was recorded rather than quietly carried, and stating it here is what made the repair a piece of work instead of something known only to whoever found it. It was made in a later change than this essay for a reason worth keeping: a correction and its own evidence should not arrive together. The figures above draw the junction geometry directly, so what the routine ought to produce was on the page and checkable before the routine was altered.

What this does not establish #

Nothing about trees. The cost’s angle is arithmetic about a model — one junction, in a plane, with three destinations held fixed — and a picture that reproduces a form is not an account of it in either direction.

In particular, none of this says a real fork sits at 74.93 degrees. The same flatness that hid the error means the cost never pushed anything to that angle in the first place, so a tree drawn at the optimum is a drawing of a model’s minimum rather than of a plant.

What would show the correction is wrong #

A derivation of a fork angle from the same cost that returns 60 or 64 degrees at a symmetric junction built at an exponent of three. The closed form has been checked against a direct minimisation over 281 networks and against two deliberately wrong arithmetics that both land more than ten degrees away, so the number is earned rather than quoted, and a competing value would have to break that.

Failing that, the honest record is the one above: two figures whose widths are exact, whose angles were never derived, and whose error is a third of a per cent of the quantity the whole argument is about. It is worth correcting for the same reason it was worth finding — a rule that cannot be caught being wrong is not doing any work.