Phyllotaxis
the spirals are counted, not admired
Concept

Closed form — where it appears

An exact expression for a quantity, as against a number a solver converged on. Two results here have one — the rises at which a cylindrical lattice is equilateral, and the fork angle a branching cost chooses — and both were found by inspecting a numerical answer that turned out to be rational.

Named by 26 essays across 4 fields — each of them below, with the objects they name alongside it.

The boundary located at 481 expansions, against D = 1/W. 481 expansions log-spaced across the range, each boundary found by bisecting the two drawn circles to two hundred steps rather than read off a grid. The hyperbola D = 1/W lies on the located curve to 2.220e-16 over the whole of it — the last bit of a double — so the textbook line is the boundary and not an approximation of it. The closed form and the bisection agree to 2.81e-15 at worst over all 3,200 located boundaries, which is what makes a residual of zero a reading.

The line was already exact

The boundary between shells whose whorls run into one another and shells whose whorls run free is quoted everywhere as D = 1/W, and a survey designed to measure how far off it sits found that it is not off at all. Located by bisecting the drawn circles at 481 expansions, the residual is 2.2 × 10⁻¹⁶ — the last bit a double holds, over the whole range.

shells · Morphospace The plane of stems: divergence across, rise up. Each shade is one parastichy pair. The marked points are the lattices where three families are equally short — the forks — and the Fibonacci ones run up the middle towards 137.51°.

The forks are exact

Where a stem's pattern has to choose between two futures, three spiral families are equally short and the lattice is exactly equilateral. A numerical solver found those points; the numbers it returned turned out to be rational, and chasing that gave a closed form — including the fact that every fork sits at a rational divergence, and the golden angle at none of them.

cylinder · Forks 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 An axial section of a spire at W = 2.4, D = 0.3, T = 1, with the angle that decides contact. The discs where the plane holding the axis cuts three whorls, on both sides of the axis. Every disc subtends the same half-angle from the apex, γ = 17.065°, about the line of the disc centres at β = 33.024° from the axis, so the envelope's apical angle is 100.178° whatever the expansion. Whorls with that angle touch below W = 1.8307, and at 2.4 they run free. Successive discs on one side are 2.4 times farther from the apex, and one disc gives back D = 0.3000 and T = 1.0000.

One angle decides contact

Seen from the apex of its coiling axis, every whorl of Raup's shell subtends the same half-angle, and two whorls touch exactly when the sine of that angle exceeds (W − 1)/(W + 1) — with no disagreement against the drawn discs at 400,000 random shells. Two of the three numbers enter only through the angle and the third only through the threshold, which decides which picture of a shell can answer the question: a spire's outline carries no W, a plan carries no T, and an axial section carries all three.

shells · Morphospace 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 An ellipse twice as tall as it is wide at W = 2.4, T = 0.5, beside a circle at T = 0.25. Each panel is an opening, in colour, and the same opening one whorl on, 2.4 times larger about the apex, drawn at the axis distance where the two just meet. An ellipse twice as tall as it is wide at a translation of 0.5 meets at D = 0.391257; a circle at a translation of 0.25 meets at D = 0.391257, the same number, because stretching the axis by 1/2 turns the ellipse into the circle and 0.5 into 0.25.

The fourth number divides the third

Every boundary on Raup's cube was located for a circular opening, and three essays ended on the same hedge: the numbers would move with a differently shaped aperture by an amount nothing had measured. Measured on the drawn outlines of eleven openings, the boundary with no translation does not move at all for any convex opening symmetric about the plane of coiling; an ellipse's height divides the translation and does nothing else; the square law in the translation belongs to a round tip; and a turned opening frees ground only in the D a plan reads.

shells · Morphospace One tree sized for flow and for stress, with each branch 2^(−1/2) the length of its parent. The same symmetric tree, 8 generations deep, each generation's branches 2^(−1/2) the length of the one before and turned 30° at every fork, sized two ways and drawn to one trunk width. On the left each branch's radius cubed is proportional to the tips it feeds — Murray's flow rule — and every junction conserves r³. On the right each branch is sized so that the same load on every tip bends it to the same stress at its base, radius cubed proportional to the sum of its lever arms to its tips; its trunk junction conserves r to the power 1.967, its outermost junctions 1.349, against a deep-tree limit of 2.000. The two trees thin at different rates from the same trunk.

A cube law with a lever arm

Murray's exponent of three comes from moving fluid for the least work, and Da Vinci's two has had no derivation here, only the name of the mechanical answer. Size every branch so that the same wind on every tip bends it to the same stress, and a junction conserves r to the power 3/(1 + log₂(1/λ)), where λ is how much shorter each branch is than its parent. A crown that fills a plane gives exactly two; halving lengths gives one and a half; no shortening gives three. Murray's flow rule gives three at every λ, so the lengths of a tree's branches say which mechanism sized it.

branching · Murray The ratio of two whorls' line counts is the growth factor raised to the clock's own power. Each curve is W^p for one law: flat for a clock that advances the angle at a constant rate, W for one that adds a constant length at the opening, W² for a constant area and W³ for a constant volume. At 3.20× per turn those are 1.00, 3.20, 10.24, 32.77. So a count of lines in two successive whorls, divided, and read back through a growth factor the curve already gives, names the law — and the four are further apart the faster the shell expands.

What the growth lines carry

A shell's curve says nothing about how fast the animal grew, and its growth lines say all of it. Under a law that holds the pth power of the radius constant per unit time, the time spent crossing one whorl is proportional to the change in that power across it — so the lines in successive whorls stand in the ratio of the growth factor raised to p, and each whorl holds the count the closed form predicts to within the one line rounding can move. Dividing two counts and taking the logarithm against a growth factor the curve already gives returns p: 17 of 20 readings name their own law, the furthest 0.012 from a whole number. The other three are not wrong, they are uncountable — at 4.5 per turn under a volume clock the inner whorl of the pair holds one line.

shells · Spiral A shell whose deposition law changes part way through, beside one that never does. Both panels are the same logarithmic spiral at 3.20 per turn, marked at equal intervals of time. On the left the animal holds a constant angular rate for the first 3.50 whorls and a constant length added after that; on the right it holds a constant length added throughout. The curves are identical to the last bit a double holds, because a curve records no clock at all. The counts per whorl are not: 56, 57, 56, 67, 191, 613, 1960 against the unchanged shell's, and the change is in where the marks crowd rather than in where the shell goes.

A shell that changed its law

An animal that grew as a juvenile under one deposition law and as an adult under another leaves a sequence of whorl ratios rather than one, and the sequence says where the change happened. The ratio across the change is a closed form that is neither law's — 6.72 between a length clock and an area clock at 3.2 per turn, exactly the average of 3.2 and 10.24 — and it is monotone in where inside its whorl the change sits, so it inverts. On a seven-whorl shell of 18,466 lines a change at 3.5 whorls comes back at 3.5001, in a band 0.027 whorls wide that holds the true position. The reading refuses a change in the outer three whorls or the inner three, because a plateau it will trust is two agreeing ratios and two ratios need three untouched whorls.

shells · Spiral 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 One planar crown sized for a load on its tips and for the weight of its own wood. The same symmetric crown, 9 generations deep, each branch 2^(−1/2) the length of its parent and turned 30° at every fork, sized so that every branch is bent to one stress, drawn to one trunk width. On the left the load is on the tips and the trunk junction conserves r to the power 1.980; on the right the load is the weight of the wood, found by iterating the radii until they stop moving, and the trunk junction conserves r to the power 0.969. A crown sized for its own weight thins much faster from the trunk, because a branch's weight grows with the square of its radius.

A crown that carries its own wood

Sizing every branch so that equal loads on the tips bend it to one stress gives a crown filling a plane Da Vinci's exponent of two. Move the load onto the wood and the sizing becomes a fixed point, because a branch's load now depends on the radii being solved for. Under the wind on its wood a planar crown still conserves two, but only as a limit its trunk is two tenths short of at fifteen generations. Under its own weight it conserves one — radius rather than area, the stress-similarity law that radius goes as length squared — and a crown carrying leaves and wood reads the leaves' two near its twigs and the wood's one at its trunk, with the handover set by how much of the trunk's load the wood carries.

branching · Murray Every law but one is pulled towards a length clock. The power a shell's outermost countable pair names, against how coarsely its lines can be told apart. A length clock sits flat at one however much of its record is lost. Every other law bends towards it: an angular clock reads 0.993 where the shell had 0 and a volume clock 1.083 where it had 3. The reason is a closed form: where the limit binds completely the surviving count in a whorl is that whorl's arc over the limit, and a logarithmic spiral's whorl arcs stand in the ratio of the growth factor exactly. So a shell too worn to read reports the law of its own geometry.

A count that is not exact

Reading a deposition law off two whorls' growth-line counts divides one by the other, so a miscount that is the same in both divides out: four lines in five missed at random moves the answer by five thousandths and costs only scatter. What biases it is a miscount that varies along the shell, and there is one that always does. The arc between successive lines rises or falls with the radius according to whether the law is shallower or steeper than a length clock, so a section's resolution limit eats the inner whorls of a shallow shell and the outer whorls of a steep one, and eats evenly at exactly p = 1. Where the limit binds, a whorl's surviving count is its arc over the limit, and whorl arcs stand in the ratio W — so a shell too worn to read reports a length clock whatever law it had.

shells · Spiral A shell section square on and seen 30 degrees off. The same three turns of a spiral built at 3.2 per turn, drawn as a camera normal to the coiling plane sees it and as one 30 degrees away from normal does. The tilted view is the plane compressed by 0.8660 along one direction. A fit to the first returns 3.200000 with a residual of 1.8e-15; a fit to the second returns 3.19527 with a residual of 0.07763. Nothing in the second picture says it is not a shell.

A section seen from the wrong angle

A photograph of a shell section taken off the normal is the coiling plane compressed along one direction by the cosine of the angle, and nothing in the picture says so. The fit that recovers a growth factor is moved by it — half a turn seen twenty degrees off gives a band of answers 23.7 per cent wide as the span's starting point moves round the shell, centred almost exactly on the right answer, so it is a spread and not a bias. The caliper measure is exactly immune at every tilt and every aim, because a projection scales all three points on a line through the centre by the same factor. And the fit's residual names the tilt to three decimal places, which makes this the rare error a section reports about itself.

shells · Spiral fit A stem is a thread at six rises of the sweep, a ribbon at 23 and a surface at 111. The 140 rises of the sweep, split by how many index families the tessellation carries. On 6 rises above 0.5010 a node has 2 walls and the strip is a thread; on 23 between 0.2187 and 0.4833 it has 4 and is a ribbon; on 111 below 0.2109 it has 6 and is a surface. The comparison means something different in each: two families leave 33.333 per cent in dispute on the surface and 0.00 on the ribbon, and three families leave 0.000 and 33.333.

The third family

On a seed head no threshold makes the counted contacts and the shared cell walls the same relation. On a stem they are the same relation exactly, at every one of a hundred and eleven rises and to three decimal places of nothing, provided the contact cut keeps three families where a count keeps two.

tissue · Neighbour definition The band of shells a measured growth factor cannot place. The boundary D = 1/W is exact — located by bisection to the last bit a double holds. A specimen is not: its growth factor arrives with an error, and carrying that error onto the line turns it into a band, running from 1/(W(1+b)) to 1/(W(1−b)). At an expansion of 3.2 and an error of 54.4% the band runs from 0.2024 to 0.6853 around an exact 0.3125. A shell inside it has whorls that a measurement cannot say are in contact or free.

The band nobody can be placed in

The boundary between shells whose whorls run into one another and shells whose whorls run free was located here to the last bit a double holds. A specimen is not a point on that line, it is a measurement with an error, and carrying the whole measured error budget onto the boundary turns the line into a band running from 1/(W(1+b)) to 1/(W(1−b)). At the budget with the dividers set aside that band covers 48.2 per cent of the box the morphospace figure here is drawn on, and its share runs from 6.7 to 59.5 per cent across the six boxes in use — the same box-dependence the contact region itself showed. The angle criterion carries the same error better above an expansion of 1 + √2 and worse below it, exactly.

shells · Morphospace The largest empty circle of a 2400-organ 137.508° golden head, ring by ring. Each dot is the largest empty circle in one ring of the head, on the interior's scale. The whole head's is 0.8437 at radius 0.871, at the centre; the rings peak at the radii where the lattice flips from one pair of neighbours to the next, and at the 4 flips beyond radius six the head reads within 0.0020 of the closed form. The dashed line is 1/√2 = 0.7071.

One over root two

On the interior's scale a golden head's largest empty circle is 0.8435 of a spacing at every size from 150 organs to 2,000, because one triangle at its centre decides it. Everywhere else it is 1/√2 — a square cell at every ring where the lattice flips — and a closed form in the angle's continued fraction says that only noble angles hold it there.

tissue · Gapgrowth Sizing for equal bending and sizing for equal stress cross at one length ratio. Equal stress holds r³ against the sum of a load's arms and gives 3/(1 + ℓ); equal deflection holds r⁴ against the sum of the arms squared and gives 4/(1 + 2ℓ), with ℓ = log₂(1/λ). Setting them equal gives 3(1 + 2ℓ) = 4(1 + ℓ), whose only root is ℓ = 1/2 — the crown that fills a plane, λ = 0.707107 — and there both are exactly two. Below that ratio the stiffness rule reads the lower exponent of the two and above it the higher, so the two criteria size the same crown at one length ratio in the whole family and it is the one Da Vinci's rule names.

A crown sized for how far it bends

Stress is one criterion for sizing a branch and stiffness is another. Holding every branch to the same deflection as a share of its own length sizes r to the fourth against the sum of each load's arm squared, where equal stress sized r cubed against the arm, and the junctions of a deep crown then conserve 4/(1 + 2·log2(1/λ)). A single cantilever under its own weight comes out at radius as length to the three halves — McMahon's elastic similarity, fitted here rather than assumed — against the square that equal stress asks for. And the two criteria agree at exactly one length ratio out of the whole family: λ = 2 to the minus a half, the crown that fills a plane, where both give exactly two.

branching · Murray How often a cell on a 2,400-organ golden head has a wall the contact cut gets wrong, by its distance from the nearest flip ring. The 1,680 interior cells of a 2,400-organ golden head, grouped by distance from the nearest flip ring in twentieths of the head's wall spacing, 1.92. The bars are the share of each group with at least one wall a three-family contact cut gets wrong; the dots are the share with five or seven sides. 353 cells have such a wall and the farthest is 0.638 spacings from a ring. Of the 1,292 cells 0.66 spacings or more from every ring, none has.

The empty interval is the rings

No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

tissue · Neighbour definition 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 Which tilts buckling sizes and which bending sizes, deep in the crown and near its tips. Crowns at the planar length ratio, forks turned 20°, sized by the larger of buckling and bending, at five balance angles. For each, the upper bar runs from vertical to the widest tilt at which buckling sizes a branch seven or more generations above the tips, and on from the narrowest tilt bending sizes; the lower bar is the same reading within six generations of the tips. Balanced at 30°, buckling sizes the deep crown to 20° and bending from 40°, and near the tips buckling reaches 60°; balanced at 40°, buckling sizes the deep crown to 40° and bending from 60°, and near the tips buckling reaches 60°; balanced at 50°, buckling sizes the deep crown to 40° and bending from 60°, and near the tips buckling reaches 60°; balanced at 60°, buckling sizes the deep crown to 60° and bending from 80°, and near the tips buckling reaches 60°; balanced at 70°, buckling sizes the deep crown to 60° and bending from 80°, and near the tips buckling reaches 60°. In the deep crown the two never overlap at any generation, so the criterion is chosen by direction; near the tips the arms have not converged, the bending term is smaller, and the upright core is wider.

A crown that would rather not buckle

A column held below the load at which it buckles and a cantilever held to a fixed deflection need the same radius at every length, because both hold the bending stiffness against a load times a length squared — so the three halves of elastic similarity is also the buckling law, and a crown whose loads all run along its branches conserves the same exponent under either. Gravity does not run along branches. It divides by the cosine of each branch's tilt, so a buckling junction's exponent is set by the direction its parent points, nothing past level is sized at all, and a crown sized by the larger of the two criteria splits by direction into an upright core and a spreading shell whose boundary junctions conserve more than either rule gives.

branching · Murray 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 The centre of a 900-organ Lucas head, with the band of two thirds of a spacing drawn round every flip ring. The organs of a 900-organ Lucas head out to a radius of 12.5, with a pale annulus 0.66 of a wall spacing either side of each flip ring, the rings keeping 1, 3, 4, 7, 11, 18. Warm: cells with five or seven sides; dark: six-sided cells a three-family contact cut gets wrong. Ringed: the one cell well inside a band that the cut reads exactly, organ 17 at radius 4.12, 0.23 of a spacing from its ring. Where two annuli overlap, a cell can sit well away from its nearest ring and still inside the next one's band.

The blur was at the centre

On a Lucas head the band of disputed cells round each flip ring looked blurred at its inner edge — exact cells as close as 0.23 of a wall spacing, disputed hexagons out to 0.59 where a golden head's stop at 0.43. Read a ring at a time, the two heads carry the same band on every resolved ring, to a hundredth: disputed hexagons within 0.16, exact cells from 0.64, a ring's own number of fives and of sevens and the number before it of hexagons. Every difference is inside a radius of six, where the Lucas rings of 4, 7 and 11 sit closer together than the band is wide, and the one exact cell is organ 17, which has no organ eighteen behind it.

tissue · Neighbour definition A crown sized against buckling on clamped bases, coloured by the share of that load each branch keeps on the base its parent really gives. A symmetric crown 9 generations deep, every branch 2^(−1/2) of its parent's length, turned 20° at every fork, with equal loads on the tips, sized against buckling as if every branch were clamped. Each branch is coloured by the share of its clamped buckling load it keeps once its base is the tip of a parent that bends, the parent alone: from 0.296 to 0.795 outside the trunk. A daughter turning back toward vertical off a steep parent keeps least, one turning outward most. Branches pointing level or down carry no compression and are drawn faint.

A base that gives

Every branch sized against buckling was a column clamped to a parent that does not move. A parent bends, and a column on a base that turns keeps only a share of its clamped load — the root of φ·tan φ = kL/EI — which no stiffening of the column itself lifts past the load at which a rigid rod on that base would tip. On the crown sized as if clamped, that share is 0.561 at every branch when the loads run along them, falls from the ground to 0.447 when the whole path is counted, and under gravity is a closed form in the turn from a parent's tilt to its daughter's: 0.795 for a daughter turning outward, 0.296 for one turning back upright off a steep parent. Resized for the bases they really have, the crown's junctions alternate about two from generation to generation, 226 branches of a thirteen-generation crown have no radius that holds them, and the cone of directions that divided buckling from bending no longer divides them.

branching · Murray A crown whose parents are sized for the bases their daughters need, coloured by the term that sizes each branch. A symmetric crown 9 generations deep, every branch 2^(−1/2) of its parent's length, turned 20° at every fork, under gravity, sized from the trunk outward by the largest of three terms: the radius its own column needs on the base its parent gives, the base term its daughters demand at a margin of 2.5 over the load at which a rigid rod would tip on that base, and the bending rule, balanced against buckling on the trunk at 50°. The base term sizes 249 branches, the column term 238 and bending 24; no branch is left without a radius.

A parent sized for its daughters

A branch whose compression times its length exceeds what its parent's tip could hold as a rigid rod cannot be saved by thickening the branch; the requirement is the parent's. Size every parent for the base its daughters need, at a stated margin over that ceiling, and nothing in a crown is left without a radius. With every load along its branches the junctions stop alternating at one margin, 2.043 at the planar ratio, and conserve exactly two above it. Under gravity a junction sized this way conserves the clamped crown's closed form read one fork further up. Joined with bending, the cone of directions comes back, wider. And the cheapest crown sits below the margin at which the alternation stops, at about 1.6 to 1.7, so the zigzag is not waste.

branching · Murray

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

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

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

Honest limitsClaim testingBranching exponentModel scopeMurray's lawFree parameterPredictionWhorlCriterion dependenceDa Vinci's ruleFork angleGrowth factor

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