One constant for every fork

The cost that fixes Murray’s exponent also fixes a fork’s angle, once it is minimised over where the branch point sits rather than over the radius. Under Da Vinci’s rule that minimisation goes wrong in an interesting way: area-conserving radii put the three segment weights on the boundary of the triangle inequality, and the constant left over walks the predicted total angle from a third of a degree to 120, through Murray’s own 74.93. No fork angle a person could measure contradicts the rule, because some value of the constant reproduces it.

That is true of one fork. It is not true of a tree, and the reason is what the constant is made of.

What the free constant is made of #

Under area conservation the pumping cost is the same in every tube however thick, so each segment’s weight in the network’s total cost is its cross-section plus a constant, β = a·k²/b. Here a and b are the two cost coefficients — the price of pumping and the price of upkeep — and k is the flow per unit of cross-section that area conservation fixes.

None of those is a property of one junction. The cost coefficients belong to the fluid and the tissue, and k is one number for a tree whose area is conserved at every fork. So a tree has one β, and every fork in it shares that value.

Why the pumping term is the same in every tube #

The constant appears for a reason worth following through. The cube law’s two costs are pumping, which per unit length goes as a·Q²/r⁴, and upkeep, which goes as b·r². Conserving area at every fork with a fixed flow per unit of cross-section makes Q = k·πr², so the pumping term is a·k²π²r⁴/r⁴ — the same in every tube, thick or thin.

A segment’s cost per unit length is then b·(r² + β) up to a constant factor, with β = a·k²/b collecting everything that is not the radius. Murray’s own sizing makes the pumping term exactly half the upkeep at the optimum and so leaves nothing additive; area conservation leaves the additive piece behind, and that piece is the whole of the free constant.

The constant is a radius axis #

At a fork whose larger daughter has radius r, the three weights are β + r₀², β + r₁² and β + r₂², with the radii proportional to r. Factoring out r² leaves the unit fork’s weights at β′ = β/r², and an angle depends only on the ratios of its weights. So a fork of size r in a tree with constant β has exactly the angle of a unit fork at β/r².

Checked directly, with a fork scaled by 0.2, 1 and 3.7 at β = 0.3 and a daughter ratio of 0.7, the scaled fork and the unit fork at β/r² agree to nine decimal places. The walk across every angle that made one fork unfalsifiable is, inside one tree, a walk across the sizes of its forks.

The sigmoid, read again #

The curve that made the rule predict everything runs from 0.36° at β = 10⁻⁵ to 119.99° at 10⁴. As a function of β it was a statement that the rule had a knob. As a function of fork size, with β′ = β/r², nine orders of magnitude in β are four and a half in radius, and the curve says that small forks open wide and large forks close.

Where it passes Murray’s angle is a choice of units, not a finding: the fork size at which β′ = 0.7024. Calibrate the tree’s constant so that one fork reproduces Murray’s 74.93°, and every fork larger than it is predicted narrower and every fork smaller predicted wider.

The weights, worked at an even fork #

At an even fork under area conservation the parent’s cross-section is the two daughters’ added, so with each daughter’s radius as the unit the weights are β′ + 2, β′ + 1 and β′ + 1. The angle each daughter makes with the parent’s line has cosine w₀/(2w₁), so the whole prediction is one line of arithmetic.

At β′ = 0.7024 the weights are 2.7024 and 1.7024, the cosine is 2.7024/3.4048 = 0.7937, each daughter leaves at 37.47° and the total is 74.93° — Murray’s angle, the one the drawn trees miss. At a fork √10 times larger, β′ = 0.07024, the weights are 2.0702 and 1.0702, the cosine is 0.9672, each daughter leaves at 14.72° and the total is 29.4°. As β′ falls to nothing the cosine reaches one and the fork closes; as β′ grows the weights become equal, the cosine reaches a half, and the fork opens to 120°.

Murray’s angle has no constant #

The cube law’s fork angle is the same at every size. Its weights are the three radii squared with nothing added, so scaling a fork scales every weight by the same factor and changes no ratio. An even fork opens at 74.93° whether it is the first fork of a trunk or the last fork before a leaf.

So the two rules differ in something a set of forks can show: one predicts a trend with size and the other predicts none. Neither prediction needs the value of any constant to be known — only whether the angle moves.

Angle against size #

With the constant set so that an even fork of relative size one opens at Murray’s angle, the rule predicts 119.1° at a tenth of that size, 101.7° at a half, 74.9° at one, 44.6° at twice and 9.6° at ten times. For forks whose daughters are in the ratio one to two the same sizes give 118.9°, 101.3°, 77.6°, 48.9° and 10.9°, against Murray’s 77.58° at every size.

The two shapes of fork agree closely. The trend is a property of the constant, not of how lopsided a fork is.

One constant, two shapes of fork #

The calibration does depend on shape, though, and a tree has only one constant to calibrate. An even fork reproduces Murray’s angle at β′ = 0.7024 and a fork with daughters in the ratio one to two at β′ = 0.3686. With one β shared by both, the lopsided fork reproduces Murray’s angle at a size √(0.3686/0.7024) = 0.72 of the size at which an even fork does.

So forks of different shapes sit on the same trend displaced along it, by about a third of a decade in radius between these two shapes. A test that pooled even and lopsided forks without recording their shapes would blur the trend it was looking for; one that recorded the daughter ratio beside the radius and the angle would not, because the displacement is computable from the shape.

Within one order of magnitude #

Read the other way, the curve says how far apart in size two forks must be for the rule to predict them at stated angles. From 100° down to 20° is a factor of 8.93 in radius at an even fork and 10.29 at a daughter ratio of a half. From 105° to 45° is ×4.48; from 90° to 60°, ×1.97; from 80° to 70°, ×1.25.

So most of the range of angles a fork could have is swept inside a single order of magnitude of radius, and a tree whose branches run from a trunk of ten centimetres to twigs of one spans more than that.

A hundred degrees a decade #

At the size where the rule reproduces Murray’s angle, the predicted total angle falls by 105 degrees for each factor of ten in radius at an even fork, and by 94 at a daughter ratio of a half. That is not a subtle slope.

Its size puts a number on how few forks it would take to see. With forks spread evenly over one decade of radius and scattered by twenty degrees about the trend, a straight-line fit resolves a slope of that size at two standard errors from two forks. No study should rest on two forks, and the arithmetic is idealised; the point is that the trend the rule predicts is steep enough that its absence would be conspicuous.

Across a whole tree #

A tree whose forks span a factor F in radius, centred on the size that reproduces Murray’s angle, must show a spread of predicted angle between its extremes. For even forks that spread is 46.9° across a factor of three, 82.1° across ten, 99.6° across thirty and 109.5° across a hundred. For forks in the ratio one to two it is 42.7°, 77.7°, 96.8° and 108.0°.

A tenfold even-forked tree runs from 29.4° at its biggest forks to 111.6° at its smallest. The rule that predicted every angle at one fork predicts a specific eighty-two-degree fan across a tree.

The flatness objection #

The obvious objection is the cost itself. One per cent of a network’s cost buys forty-three degrees of fork angle: the optimum is so shallow that a fork sitting anywhere from 55.9° to 99.2° costs less than one per cent more than the best. If a single fork is held that loosely, a predicted trend of forty or eighty degrees might sit inside the looseness and never show.

Which is larger #

Against the band, a tree spanning a factor of three shows a spread of 46.9° beside a band of 43.3° — barely more. A tree spanning ten shows 82.1°, nearly twice the band. A tree spanning a hundred shows 109.5°. Above a factor of about three in radius the trend the rule predicts is wider than the flatness any single fork enjoys.

But an angle spread and a cost band are different quantities, and setting one beside the other only suggests an answer. The direct question is what a tree would pay, under Da Vinci’s weights, to hide the trend by holding one angle at every size.

What hiding the trend costs #

At the biggest forks of a tenfold tree the rule’s optimum is 29.4°, one per cent of cost covers 14.8° to 61.5°, and holding Murray’s 74.9° there costs 1.69 per cent. At the smallest forks the optimum is 111.6°, one per cent covers 92.8° to 132.5°, and holding 74.9° costs 4.39 per cent.

So a Da Vinci tree that showed Murray’s angle at every size would pay more than the one-per-cent looseness at both ends, and more than four times it at the small forks. The flatness that hides the choice of angle at one fork does not hide a trend across a tree.

Which trees escape #

A tree can still hide the trend if its forks sit where the curve is flat — at the two ends of the sigmoid. A tree spanning a factor of three hides it only if its middle forks are predicted below 48.0° or above 86.7°. A tree spanning ten hides it only below 16.4° or above 112.1°; a tree spanning a hundred, only below 4.7° or above 119.2°.

Those are trees whose forks are all nearly straight or all near 120°. Every tree with ordinary forks across an ordinary range of sizes sits in the region where the rule’s trend is wider than the cost can absorb, and the escape narrows as the tree spans more.

A scan read the wrong way round #

That reading was first taken with the spread computed backwards — the big forks’ angle minus the small forks’ — which made every spread negative and every tree an evader, at every span alike. An answer that did not change when the span changed by a factor of thirty-three was the sign of it. The scan is corrected, and a check that a correct scan’s escape narrows as the span grows now runs with it.

Drawn #

The difference is visible without measuring anything. A Murray tree opens every fork at 74.9°, generation after generation. A Da Vinci tree whose trunk fork sits at a relative size of 3.2 opens its forks at 29.4°, 40.4°, 54.2°, 69.8°, 85.3° and 98.3° from the trunk outwards, because area-conserving radii shrink by the square root of two at each fork and each fork’s share of the constant grows by two.

The first picture is a tree that holds one shape at every scale. The second is a tree that closes at the trunk and spreads at the tips, and a photograph of a real crown can say which it looks more like.

A different kind of question #

The one-fork result was an undecidability of a structural kind: no precision of protractor helped, because the rule had no angle to be wrong about. Across a tree the question becomes an ordinary one. There is a predicted slope of angle against the logarithm of size, there is scatter about it, and a fit either finds the slope or finds none.

Ordinary questions have ordinary difficulties — how much scatter real forks have, how evenly a sample spans sizes, whether lopsided forks behave like even ones — and none of them is solved here. But they are the difficulties of a measurement rather than of a rule that cannot be asked anything, which is the change this essay is about.

What this corrects #

A rule that predicts everything concluded that no measured fork angle could refute Da Vinci’s rule. For one fork that stands. For a tree it does not: the free constant is one number for the tree, it enters every fork scaled by the inverse square of its size, and the rule therefore predicts a trend of fork angle with fork size that Murray’s rule does not. That conclusion has been qualified where it was drawn.

What it assumes #

That a fork’s angle minimises the same cost the angle derivation uses, given radii set by area conservation. Area-conserving radii are not the radii that minimise that cost — Murray’s are — so a Da Vinci tree has its sizing set by something else and its angles set by this cost. That combination was already accepted where the free constant was found, and it is carried unchanged rather than defended.

It also assumes k, the flow per unit of cross-section, is the same across the tree. Something mechanical, such as a lever arm sizing the branches, could set the radii while flow sets the cost — and then nothing requires the flow per unit area to be constant.

The refutation it supplies #

That second assumption is also where a real tree could escape honestly. A tree in which flow per unit of cross-section falls towards the tips would have a β that shrinks with size along with r², and the trend would weaken or reverse. So the test this essay proposes has a clean failure: measure k across a crown, and if it is not constant, the trend is not a prediction.

What this does not establish #

Nothing here measures a tree. It does not say that real forks open towards the tips, or that they do not; it says which rule predicts that they should and by how much. The forks drawn are symmetric or in one fixed ratio, and a real tree’s forks vary in both size and shape at once, which the band that decides the answer has shown is where a measurement’s sample starts to decide its result.

What would withdraw it #

A fork scaled by r whose angle is not the unit fork’s angle at β/r². A tenfold tree whose predicted spread is narrower than a single fork’s one-per-cent band. A cost scan on Murray’s weights that does not reproduce the finer scan’s band. Each is checked every time the measurement runs.

Angles on a tree sized by stress #

The next test puts the fork angle on the tree the lever-arm derivation sizes: radii set by equal bending stress, whose exponent depends on how branch lengths shrink. On such a tree the weights in the angle cost are neither Murray’s nor Da Vinci’s, and the test is whether the angle the cost predicts is fixed by the length ratio alone — a single angle at every size, as under Murray’s rule — or carries a trend with size, as under Da Vinci’s.