Phyllotaxis
the spirals are counted, not admired
Concept

Degeneracy — where it appears

A condition under which a rule stops distinguishing between the outcomes it was supposed to predict. Leonardo's rule is exactly degenerate: taking cross-section as the weight puts the three weights on the boundary of the triangle inequality, so the cosine comes back at one and every fork angle satisfies it.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The exponent sets the spacing, and only the exponent. The curve is φ^(2/p), drawn from the ladder's ratio of 1/φ² and nothing else. The dots are measured: each surface built, a blind counter walked up its axis, the places its answer changed recorded. A cone that elongates and a paraboloid that fills sit on the same point at p = 1 — so the spacing identifies neither the shape nor the way material arrives, only their product.

The shape and the law

A cone's transitions are a factor of φ² apart and a disc's a factor of φ, and the temptation is to read the ratio as the shape. It is not. Five surfaces built and counted show that the ratio measures one exponent, and that the exponent is the shape multiplied by the way material arrives.

cylinder · Shape exponent The three weights at an exponent of 2 and 3: one closes a triangle and one is a straight line. A branch point minimising a weighted sum of three lengths has an interior solution only when the three weights close a triangle, and the weight on a segment here is its own cross-section. At an exponent of 3 the three areas clear that condition by 0.4126, and the triangle they close is what the two fork angles are read off. At an exponent of 2 the parent's area is exactly the daughters' areas summed — that is what area conservation says — so the slack is -4.44e-16, the triangle collapses onto a line, and the fork closes to 0.0000 degrees. Leonardo's rule does not predict a different angle here; it predicts no angle.

A rule that predicts everything

Leonardo's rule says a fork conserves cross-section, and cross-section is exactly the weight the branch point is minimised against. So the three weights land on the boundary of the triangle inequality, the cosine comes out at one to the last bit, and the rule predicts no angle at all — and the free constant its own derivation leaves behind then walks the prediction across every angle a fork could have.

branching · Fork angle The fork angle Da Vinci's rule predicts, against the size of the fork, in one tree. A tree has one value of the constant Da Vinci's rule leaves free, so a fork's share of it falls as the square of the fork's size and the angle the rule predicts changes with size. For even forks it is 119.1° at a relative size of 0.1, 101.7° at a relative size of 0.5, 74.9° at a relative size of 1, 44.6° at a relative size of 2, 9.6° at a relative size of 10, where Murray's rule gives 74.93° at every size; for daughter ratio 0.5 it is 118.9° at a relative size of 0.1, 101.3° at a relative size of 0.5, 77.6° at a relative size of 1, 48.9° at a relative size of 2, 10.9° at a relative size of 10, where Murray's rule gives 77.58° at every size. From a total of 100° to 20° at an even fork is a factor of 8.93 in radius.

One constant for every fork

Da Vinci's rule leaves a free constant in the cost that sets a fork's angle, and running it over its range walks the predicted angle from nothing to 120 degrees, through Murray's 74.93. So no single fork can refute the rule. But the constant is one number for a whole tree, and a fork's share of it falls as the square of the fork's size — the constant is a radius axis. A tree spanning a factor of ten in radius must show forks from 29.4 degrees at its biggest to 111.6 at its smallest, a spread wider than one fork's flatness can hide, while Murray's angle is the same at every size.

branching · Fork angle The fork angle the transport cost predicts on a tree sized by stress, against the size of the fork. A tree sized so that equal loads on its tips bend every branch to one stress conserves rᵖ with p set by how much shorter each branch is than its parent, λ. On such a tree the cost that fixes a fork's angle carries one constant, entering a fork of size s as s^(2p − 6), so the predicted angle changes with size unless p is three. With the constant set so that a fork of relative size one opens at Murray's 74.9°: At λ = 0.707 (p = 2.000) an even fork opens at 119.1°, 111.6°, 74.9°, 29.4°, 9.6° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.74 (p = 2.091) an even fork opens at 114.5°, 106.4°, 74.9°, 41.1°, 30.1° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.794 (p = 2.250) an even fork opens at 106.3°, 97.6°, 74.9°, 53.9°, 46.4° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.87 (p = 2.498) an even fork opens at 92.7°, 85.5°, 74.9°, 66.0°, 61.5° at relative sizes of 0.1, 0.32, 1, 3.2 and 10; at λ = 0.95 (p = 2.793) an even fork opens at 78.4°, 76.6°, 74.9°, 73.4°, 72.2° at relative sizes of 0.1, 0.32, 1, 3.2 and 10. Murray's rule opens every fork at 74.9°.

Forks on a tree sized by stress

Da Vinci's rule predicts that a tree's forks open wider as they get smaller, because the constant in its angle cost is one number for a tree and enters each fork scaled by its size. A tree sized for equal bending stress conserves an exponent set by how much shorter each branch is than its parent, and on such a tree the same constant enters each fork as its radius to the power 2p − 6. The trend survives at every length ratio short of one and shrinks with it: 105 degrees a decade of radius for a crown filling a plane, 83 at a length ratio of 0.74, 53 for a crown filling a volume, 13 at 0.9. Only a crown shortening about as fast as a volume-filling one fans wider across a tenfold range than one fork's flatness can hide.

branching · Fork angle Three ways of sizing a crown, and the exponent each one conserves. Murray's flow rule sizes r³ against the tips a branch feeds and conserves three at every length ratio, reading nothing of the lengths at all. Equal bending stress conserves 3/(1 + ℓ) and equal deflection 4/(1 + 2ℓ), where ℓ = log₂(1/λ). So an exponent measured on a tree names a rule only with a length ratio beside it, and even then not everywhere: the stress and stiffness curves meet at λ = 0.7071, the stiffness curve passes three at λ = 0.8909, and the stress curve reaches three only as the branches stop shortening.

Three rules, one exponent

A measured branching exponent is quoted as evidence for a sizing rule, and it cannot be. Murray's flow rule conserves three at every length ratio and reads nothing of the lengths at all; equal stress conserves 3/(1 + l) and equal deflection 4/(1 + 2l), where l is log2(1/lambda). So an exponent names a rule only with a length ratio beside it, and even then not everywhere: of ninety-six length ratios between 0.3 and 0.99, thirteen have two rules within five hundredths of each other at a precision of 0.05, in three bands with three different reasons — stress against stiffness where they cross at the planar crown, stiffness against flow where the stiffness curve passes three at 0.8909, and stress against flow only as the branches stop shortening.

branching · Murray Each junction's exponent against its daughters' lengths, measured exactly and with a two per cent error in every radius. One thirteen-generation crown at a mean length ratio of 2^(−1/2), lengths spread by a half-width of 0.3 in log, its junctions five to ten generations above the tips — every third one drawn. Each junction appears twice, its exponent under equal stress and under equal bending, against the sum of its two daughters' log length ratios; the lines are the least-squares fits. Measured exactly, the stress exponents follow a slope of 0.298 and explain 62% of their scatter, the bending exponents a slope of 0.858 and 93%. With every radius read 2% wrong, the clouds swell — the stress scatter explained falls to 20% — and the slopes read 0.288 and 0.866. The error lands in the residual, and the slope is where the rule is.

The lengths that name the rule

A real crown has no single length ratio, and giving every fork a spread of daughter lengths does not blur what a sizing rule conserves: Murray's flow rule still conserves three at every junction, and the two mechanical rules keep their mean exponent, moved only as the square of the spread. What the spread adds is a second number. Each junction's exponent follows its daughters' summed log length with a slope of 0.30 under equal stress and 0.86 under equal bending at the planar crown, where the exponents are both two — and a two per cent error in every radius moves neither slope, while it swamps the scatter that looked like the obvious instrument.

branching · Murray Two lopsided crowns sized by the larger of flow and stress, drawn at the angles their radii predict. Two crowns of 9 orders whose every fork sends a thick daughter one order down and a thin daughter 2 orders down, so the crown has 89 twigs; each branch is 0.794 of its parent's length times the shortening of its order, and each takes the larger of the radius equal bending stress asks for and the radius flow asks for, the two equal at order 4. Every fork is drawn with each daughter at the angle the transport cost predicts from that fork's three radii and tips. On the left the twigs are the radii the cost itself prefers, and no daughter anywhere opens wider than the angle Murray's law gives its own fork. On the right the twigs spend twice their upkeep on pumping: the thin daughters open furthest past Murray's angle, by 12.5°, at the fork one order inside the handover, and the thick daughters furthest, by 10.4°, at a fork 1 order nearer the trunk. Branches sized by stress and branches sized by flow are drawn in two colours.

Two daughters cross Murray's angle together

On a crown sized by the larger of flow and stress whose every fork is lopsided, the thin daughter's subtree a few orders shallower than its sister's, the handover stops being one generation and becomes a band of forks one wider than the offset. Every fork past the band still opens each daughter at its own Murray angle, to a ten-trillionth of a degree. The thin daughter never overshoots alone: at the first fork whose daughters are both sized by flow, only the parent's weight differs from Murray's, and one side of a weight triangle passes its Murray length at a single value, so both daughters cross together, at a twig share within two hundredths of the even crown's. Past it they peak apart, the thin one by more.

branching · Fork angle

Named alongside it

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

Branching exponentClaim testingDa Vinci's ruleHonest limitsMurray's lawFork angleArea conservationClosed formFalsifiabilityFree parameterPredictionCriterion dependence

All concepts