Phyllotaxis
the spirals are counted, not admired
Concept

Daughter ratio — where it appears

The ratio of a fork's two daughter radii, one at an even fork and a hundredth at a twig hanging off a trunk. It is the axis every branching result here is read against, because the angle the cost predicts moves across it by fourteen and a half degrees while the cost itself barely moves at all.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

A tree built at 3, measured to 2%, reads 2.957 on the informative band. The exponent recovered from 100 junctions of a tree built at exactly 3, against the relative error placed independently on the parent and on both daughters — half a per cent is a machined section under a microscope, one to two per cent is callipers on a clean branch, five is a branch that is not round, ten is a radius read off a photograph. Each line is a band of daughter ratio and the whiskers are the central 90% of 300 replicate samples. Every band is displaced downward at every error and never upward: at 2% the informative band read 2.957 and the informative band read 2.957. Below, the same rows with the displacement and the spread drawn as separate bars, because only one of them falls when more junctions are measured.

The exponent an error moves

Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone.

branching · Exponent error One junction's bias moves 1.36-fold across the range its leverage moves 308,352-fold, at an exponent of 3. The bias a single junction of daughter ratio γ contributes to a least-squares fit, as a multiple of the squared measurement error, drawn against the leverage that junction carries — both computed from the expansion about a true exponent of 3 rather than fitted to anything. Across the whole range from an even fork to a twentieth the bias moves by a factor of 1.36 and the leverage by a factor of 308,352, and the uninformative junction contributes the larger share: 6.00σ² at γ = 0.05 against 4.50σ² at an even fork.

The fragile junctions are the informative ones

That is the obvious worry once the radii are uncertain, and it is false. Across the whole range of asymmetry a junction's contribution to the bias moves by a factor of 1.36 while its leverage moves by a factor of 308,352, so the junction that says nothing damages the answer as badly as the one that says everything — and a sample is spoiled by counting rather than by weight.

branching · Exponent error 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 What insisting on a fork angle costs: 43.3 degrees for one per cent of the network. The three ends are held where the optimum wants them, the branch point is moved everywhere inside them, and the cheapest network at each total angle is kept. The optimum sits at 74.93°. Everything within one per cent of the least cost runs 55.9° to 99.2° — a span of 43.3°, which read back as exponents covers 2.44 to 5.34 — and within a tenth of a per cent it still runs 13.6°. The prediction is steep in the exponent and the cost is nearly flat in the angle; they are the same curve read along its two axes.

An optimum too flat to reach

One per cent of a branching network's cost buys forty-three degrees of fork angle, covering exponents from 2.44 to 5.34, while the angle the theory predicts moves only fourteen and a half degrees across every daughter ratio there is. The prediction is steep and the cost is flat, and those are the same curve read along its two axes.

branching · Fork angle The two trees this site draws, at 30° and 32° to a side, against the cost's 37.47° and 37.47°. Two trees of 63 segments each, 5 generations deep and 31 junctions apiece, with every junction's radii taken from r₀³ = r₁³ + r₂³ exactly and every junction's angle taken from a constant. Read as an exponent through cos(θ/2) = 2^(2/p − 1), the drawn angles say 2.5237 and 2.6239, in pictures whose widths are built at exactly 3. The cost that fixed those widths wants 37.47° and 37.47° at this daughter ratio, 74.93° in total, and the misses cost 0.573% and 0.292% of the network — which is why a fixed angle can sit in a figure about a minimisation and never look wrong.

The trees drawn at no angle

Two branching figures in these essays set every junction's radii from the cube law exactly and every junction's angle from a constant nobody derived. Read as exponents the drawn angles say 2.52 and 2.62, in pictures whose widths say exactly three — and at a lopsided fork the drawing puts a daughter thirty-four degrees from where the same cost puts it.

branching · Fork angle What a swelling does to the exponent read from the informative band, by where the swelling is. 50 junctions of daughter ratio 0.6–1, built at exactly 3 and read with no random error, but with one or more radii measured fat. A parent read fat lowers the reading: 1% gives 2.864, 3% gives 2.631, 10% gives 2.075, and at 11.3% the tree reads Da Vinci's 2. Daughters read fat raise it: 1% gives 3.150, 3% gives 3.499; from 8% some junctions have a daughter measured wider than their parent, which no exponent balances, and the line stops. All three radii read fat by one factor return 3.000 at every swelling — the unswollen reading exactly.

A swelling at the fork

A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.

branching · Exponent error 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 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 Two crowns whose forks are lopsided by different amounts, drawn at the angles their radii predict. Two crowns of 9 orders whose every fork sends its thin daughter's subtree nought to three orders shallower than its sister's, the offset drawn at each fork; 76 twigs, each branch 0.794 of its parent's length per order, sized by the larger of flow and stress with the two equal at order 4 for the median branch of that order. Every daughter is drawn at the angle the transport cost predicts from its fork's three radii. On the left the twigs are the radii the cost prefers and no daughter opens past its Murray angle; on the right they spend twice their upkeep on pumping. This crown first turns back at a twig share of 0.512, at a fork whose parent is 1.008 times as thick as flow alone asks.

Mixed forks turn back at the cost's own share

On a crown sized by the larger of flow and stress, a fork whose two daughters are sized by flow opens both past their Murray angles exactly when the twigs spend more than q⁴/(1 + q²) of their upkeep on pumping, where q is how much thicker than flow its parent is — a closed form that is the even crown's threshold and every one-offset crown's at once. A crown lopsided by one amount at every fork has one q at its handover and a margin above the cost's own one half: 0.64 at a length ratio of 0.794. A crown lopsided by different amounts has a parent at every q, some barely thicker than flow, and turns back at 0.500 to 0.514. A branch's weight is its cost per length, the cost is flat at its own radius, and a slightly thickened parent with slightly thin twigs is lighter than its daughters expect.

branching · Fork angle Read at an opening of a degree, a blend turns the mixed crowns back later than the larger of flow and stress does. Twelve mixed crowns of fourteen orders and the even crown, every branch sized by the blend of flow and stress radii at the exponent on the axis, and in the last column by the larger of the two. For each, the twig share at which some daughter first opens more than 1° past its Murray angle. The bars are the twelve mixed crowns' range and the rings the even crown. At k = 2 the mixed crowns turn back at 1.122 to 1.151, at k = 4 at 0.817 to 0.821, at k = 8 at 0.706 to 0.754, and under the larger of the two at 0.603 to 0.672; the even crown at 1.084, 1.061, 0.943 and 0.724. The faint line is the same crowns read strictly, whose mean falls to 0.5014 at k = 8.

A blend turns the twigs back first

Size every branch of a mixed crown by a blend of what flow and stress ask, rather than by the larger, and no branch is exactly flow-sized any more. Read strictly, that removes every margin the crowns had: the forks of two twigs open past Murray's angles first, on a closed form of their own, and by a blend exponent of eight every crown does so within two thousandths of the cost's own one half, the even crown's 0.643 included. But those openings are thousandths of a degree. Read at a degree a protractor could see, a blend restores a margin at every exponent tried, 1.12 at two, 0.82 at four, 0.71 to 0.75 at eight, against 0.60 to 0.67 for the larger, and the fork that opens first climbs from the twigs to the handover as the blend sharpens.

branching · Fork angle

Named alongside it

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

Murray's lawHonest limitsBranching exponentFork angleClaim testingDa Vinci's ruleBiasMeasurement errorMetabolic costWeight triangleClosed formOptimisation

All concepts