Model scope — where it appears
Named by 32 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
What a mechanism would have to show
Every model of phyllotaxis comes with the caveat that reproducing a pattern is not explaining it. That is easy to repeat and hard to make precise. Here it is made precise — a list of what an account of phyllotaxis would have to establish, with each item marked according to whether the models drawn in these essays establish it.
What a spire buys
Translation along the coiling axis enters the contact boundary as its square, so a shell that has only just begun to walk along its axis has not moved the boundary at all. It is also strictly one-way, and it has a threshold above which no distance from the axis whatever puts the whorls in touch.
A growing organ is part of the rule
Every model in the earlier essays places primordia on a surface and then treats the surface as furniture. But the surface grows between one placement and the next, and that growth reaches the rule through the only channel it has — where the neighbours are. What looks like a boundary condition turns out to be a term in the model.
A boundary with no edge
Two continuous measures cross the line where a coiled shell's whorls begin to touch, and they disagree about whether it is sharp. One falls to zero as a straight line and makes the line a kink; the other leaves it as a three-halves power and makes it a tangency.
A comb is evidence of a rule
Build the same lattice kinematically — every node at an exact multiple of the divergence, an independent error on each azimuth, no feedback anywhere — and the spectrum is empty. The photograph is identical and the parastichy pair is identical. The comb is not a property of the arrangement.
A fraction of nothing
Six boxes differing only in where their edges were drawn give between 4.64 and 52.81 per cent for the same geometry, and sampling one axis geometrically rather than uniformly multiplies the answer by 4.60. The share tends to zero as the box widens, because the region under a hyperbola is a logarithm and a box is a line.
The residual is not the test
The fit that recovers a growth factor also hands back a residual, and that residual has been read as what separates a genuine logarithmic spiral from something that merely looks like one. Pointed at twelve curves it fits a circle exactly, accepts an Archimedean spiral, and refuses a golden one that is right to three decimal places.
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.
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.
One number for a shell that changes
An animal is under no obligation to grow at one rate from hatching to maturity, and the fit that recovers a shell's growth factor returns one number whatever it is given. Handed a shell whose expansion rises steadily from 2.8 to 3.6 a turn, it returns 3.17490 — the geometric mean of the two ends, exactly — with a residual it accepts. A change of sixty-four per cent over three and a half turns passes as one logarithmic spiral, and at the aperture, where contact is decided, the one number and the last whorl give opposite verdicts.
Three points on a diameter
Ammonoid workers measure a shell's expansion without a centre: two diameters half a volution apart, squared. On a logarithmic spiral that is exact, and the centre is needed only to aim the line. A quarter-radius aim error costs the fit 1.341 per cent and the diameters 0.0045, because the aim error enters as its square. Reading noise is another matter: at a thousandth of the outer radius the fit's four hundred points beat the calipers' three readings at every expansion up to the nautilus's, and which instrument is better depends on which error the section actually has.
A neighbourhood is a hypothesis
Every simulation of this kind stops summing somewhere. The earlier work found that where it stops decides what pattern comes out — so the stopping place is not a detail of the program but a claim about how far a primordium's influence reaches, and it should be written down as one.
What the septa count
A nautilus's chambers are each a scaled copy of the last, and an earlier essay gave their ratio as about 1.3 — what a growth factor of 3.2 gives over a third of a turn. It does not: a third of a turn at 3.2 is 1.474 in length. A ratio of 1.3 is 4.43 septa a whorl as a length, 8.87 as an area and 13.30 as a volume, so the dimension decides the count threefold. And the count is an exponent in any reading of the growth factor taken from one chamber to the next: one septum miscounted at thirteen moves it by 9.14 per cent. A chamber and the one a whorl out give W³ with no count at all.
A spiral with no clock
The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.
Two shapes, one threshold
Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.
The comb was never the rule
A control is only as strong as the alternative it builds, and the earlier work built one that varied the rule while holding the disturbance fixed at independence. Five rounds of the angle-sequence thread, with what each claimed and what still stands — and why the next evidence has to come from an intervention rather than from a longer stem.
A law that never stopped changing
A shell whose deposition law moved evenly from one end to the other gives a sequence of whorl ratios that is a straight ramp rather than a plateau, a crossing and a plateau, and the two are separated by more than the counts' own rounding on every shell holding three countable ratios. Each ratio on the ramp reads the law at the boundary it straddles — 0.3486, 0.6865, 1.0320, 1.3755, 1.7180 against 0.3333, 0.6667, 1.0000, 1.3333, 1.6667 — so the reading is local where a fit to the curve is global, and a fit handed the same shell returns the geometric mean of its ends with no warning. The reading that locates a single change refuses a drifting shell at every size, naming the number of ratios that agree with neither end.
The error budget for a nautilus
Every way a growth factor read off a shell section can be wrong has been priced here, one essay at a time. Added up they come to 332.6 per cent in the worst case and 279.6 in quadrature, against a golden-spiral claim that is 114.2 per cent away — so the budget does not refuse the claim at all. One entry decides it: the dividers, at 278.2 per cent on their own, and the dividers are the historical method and the only route measured that pushes a nautilus towards a golden spiral. Set them aside and the budget falls to 54.4 per cent and the claim is refused twice over. What the same budget cannot settle is anything smaller than half: 3.2 against 3.4 is inside it, and stays inside it until six of the seven sources are controlled.
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.
What a sample grid decides
The rule takes its minimum over 384 sampled azimuths, and that number has been a constant since this site's first commit. Tripling it changes nothing at the two rises the collection argues from — five runs of five, identical readings — and changes which answer appears at the one rise published as having no answer. A parameter of the program, measured rather than assumed.
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.
What the axis distance costs
Raup's contact boundary is a relation between two numbers and only one of them has ever been priced here. The second was expected to be the cheaper — a length against another length. It is not: an assumed centre costs it 79.3 per cent where the same centre costs the expansion 6.88, because a ratio of two distances is first order in the centre and a fitted rate is second. But a tilted camera costs it nothing at all, exactly, and averaging the reading round one whorl is free and worth a factor of 4.91. The two numbers fail at opposite ends, and they cross at 1.12 turns of arc.
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.
A floor no better fit can lift
The band of shells nobody can place was built from one of the two numbers the contact boundary relates, and the other has now been priced. Carried together they widen the band by a third and take 48.2 per cent of the morphospace box to 59.7. The number that matters is further down: with the whorl expansion measured perfectly, 12.6 per cent of the box is still undecidable, and at an expansion controlled to a hundredth 94 per cent of what remains belongs to the second number. And the two errors are not independent — they come out of one guessed centre, which traces a curve across the boundary rather than a rectangle around it.
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.
Three entries and one span
The error budget for a nautilus added seven ways a growth factor read off a section can be wrong, and asked whether they were independent. Three of them are not: the displaced centre, the span of arc and the oblique view are one error priced three ways, at two turns, over a turn and more, and at half a turn — 32.2 per cent together. A section has one span. Read together at one span, a quarter-radius centre and a ten-degree tilt come to 7.1 per cent at two turns and 188 at half a turn, and in their worst orientation they always add to more than their sum. So the budget refuses the golden spiral from three quarters of a turn of shell upward, and below that it cannot.
The dividers belong to the opening
The nautilus error budget's largest entry, dividers walked along the shell at 278 per cent, was set aside as the historical method. Priced at one span it turns out not to belong to the span at all. A person sets a pair of dividers to an opening and walks until the curve runs out, so the step count grows with the arc exactly as fast as the floor on it does: opened to less than the square root of the growth factor less one — 0.789 of the innermost radius for a nautilus — they read the factor exactly over every span from half a turn to six, and opened wider they read it too high over every span, least over the longest. The 278 per cent was nine steps along five turns, an opening of 37 innermost radii. With the dividers opened to anything up to five radii, the whole budget refuses the golden spiral from three quarters of a turn upward.
The rim sets the opening
A pair of dividers reads a nautilus's growth factor exactly when its opening is under 0.789 of the radius of the whorl it starts on — a quarter of a millimetre at the true centre of a real shell, which no hand can set. A real section starts where its whorls can be read, and a person sets the dividers against the shell in front of them. Measured that way, the rule becomes a span: dividers opened to a share f of the outer radius are exact over the last log((√k − 1)/f)/log k turns of any shell — 3.76 turns at a hundredth, 3.16 at a fiftieth — and the change-over falls exactly there at every opening tried. Held against the rim, their error grows with the span rather than falling, so a section with its centre broken away is read more exactly, not less; and the whole budget still refuses the golden spiral at every span from three quarters of a turn to six for any opening up to a fiftieth.
The outline finds its own centre
A worker with a sawn shell has an outline and nothing else: the centre, the expansion W and the distance from the axis D all have to come off it at once, and three numbers fitted to one outline can trade against each other. Fitted together, they do not trade where it matters. Started from a centre guessed a quarter of the innermost radius off, the drawing locates its own centre to about a thousandth of that radius, and over a turn of section drawn to a thousandth of the rim it returns W to 0.12 per cent and D to 0.045 — where reading the same drawing at the guessed centre gives 16.8 and 0.98. W's error rides the centre's; D's does not, and W and D do not trade against each other at all. The trade the round trip was set to watch for appears only where the model is wrong: a view five degrees off the section's plane moves W by three times its own noise error while the fit's residual stays within the noise, because the free centre moves to absorb the squash.
A wall and not a budget
Below a rise of about 0.005 this collection's stems stop settling onto a lattice, and the limit has been written up four times without anybody asking which kind of limit it is. Grown three times as long, the table is identical row for row: not one stem that failed to settle succeeds. The floor is a wall.
The twenty-first row
Recomputing the level moves one row out of the set the exchange sets aside and into the exchange itself. Its hop is four times larger than any the correction was fitted over, and the correction fails on it in the one way it had never failed.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsClaim testingWhorlGrowth factorMeasurement errorIdentifiabilityError propagationMorphospaceNegative resultInvoluteEvoluteLogarithmic spiral