Phyllotaxis
the spirals are counted, not admired
Concept

Metabolic cost — where it appears

The energy of maintaining a tube and its contents, which rises with radius where the cost of pumping falls. The radius that minimises their sum is what Murray's law states, so the law is a prediction from a trade-off rather than a summary of measurements.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A branching tree in which every junction obeys the cube law. r₀³ = r₁³ + r₂³ at all 63 junctions, to 2e-16. The widths in the drawing are the radii the law gives, not widths chosen to look right.

The cube law

A branching network built to move fluid for the least work obeys one relation at every junction — the cube of the parent radius equals the sum of the cubes of the daughters. It is a minimisation result, it is checkable on a real tree, and it is the rare biological rule with a derivation.

branching · Murray The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

Which junctions say anything

Da Vinci's rule and Murray's law differ by 12% at an even fork and by a tenth of a per cent at a twig. So one even fork settles which is right, and two thousand twigs do not — a factor of two thousand across a tree, decided entirely by the shape of the junction and not by how carefully it is measured.

branching · Exponent The fork the cost chooses at a daughter ratio of 1: 37.47 and 37.47 degrees. Three ends held fixed, three weights fixed by the radii, and the branch point put where the total cost is least. At the optimal radius the pumping term is exactly half the upkeep term, so a segment's weight is its own cross-section and the three weights here are 1.5874, 1.0000, 1.0000. Minimising directly over the position — a 41×41 grid re-centred and shrunk 220 times, told nothing about any formula — puts the daughters at 37.4673° and 37.4673° from the parent's own forward direction, against the closed form's 37.4673° and 37.4673°.

The angle the cost chooses

Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.

branching · Fork angle Fork angles on crowns sized by the larger of flow and stress, handing over at six different generations. Thirteen-generation crowns at a length ratio of 0.707, every branch sized by the larger of flow and stress, twigs at the radius the transport cost prefers, with the two radii equal at generations 3 to 8. Handing over at 3, the forks from the trunk open at 38, 51, 67, 75°; handing over at 4, the forks from the trunk open at 27, 38, 51, 67, 75°; handing over at 5, the forks from the trunk open at 19, 27, 37, 51, 67, 75°; handing over at 6, the forks from the trunk open at 13, 18, 26, 36, 50, 66, 75°; handing over at 7, the forks from the trunk open at 8, 12, 17, 24, 35, 49, 66, 75°; handing over at 8, the forks from the trunk open at 4, 6, 10, 15, 22, 32, 46, 65, 75°. Every fork from the handover outward opens at Murray's 74.9°. No fork on any of the six opens wider than that, and none turns back: the angle rises through the stress-sized generations and stops.

A trend that stops at Murray's angle

A crown sized by whichever of flow and bending stress asks for the thicker branch is sized by stress at its trunk end and by flow at its twigs, and the twigs fix the constant that was free in the fork-angle prediction: a twig at the radius the transport cost prefers spends exactly half its upkeep on pumping. On such a crown the fork angle does not change sign at the handover. It rises through every stress-sized generation, meets Murray's 74.93° at the handover and stays there, and no fork anywhere opens wider. The trend turns back only when the twigs are thinner than the cost wants — past a pumping share of (λ^(−2/3) − 1)/(1 − λ^(4/3)), 0.70 at the planar crown and closing on one half as branches stop shortening.

branching · Fork angle

Named alongside it

The objects these essays reach for when they reach for this one.

Murray's lawOptimisationBranching exponentClaim testingClosed formDa Vinci's ruleFork angleAllometryBranchBranch pointBranching networkConvergence

All concepts