Phyllotaxis
the spirals are counted, not admired
Concept

Fork angle — where it appears

The angle a branch's two daughters make, taken as a total or separately against the parent. A cost minimised over the branch point's position fixes it sharply — 281 networks agree with the closed form to under two ten-thousandths of a degree — while the cost around that optimum is nearly flat.

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

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 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 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 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 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 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 lawClaim testingHonest limitsBranching exponentDaughter ratioClosed formDa Vinci's ruleMetabolic costPredictionWeight triangleDegeneracyFree parameter

All concepts