Phyllotaxis
the spirals are counted, not admired

Almost everything said about spirals in plants is a claim about a count, and the count is almost never done.

A seed head is a set of points. How many spirals run through it clockwise, how many run the other way, whether those two numbers are consecutive Fibonacci numbers — all of that is checkable from the points alone, by machinery that is never told what angle built them. Doing it turns up things the captions do not mention: the counts change with where in the head you look, so “34 and 55 spirals” is a statement about one annulus rather than about a flower. And the famous angle is not a constant of nature. It is where a rule settles, over a range of one parameter, and outside that range the same rule settles somewhere else.

The two spiral families a counter finds between 0.43 and 0.67 of the radius. 21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 1 The two spiral families a counter finds in one head, drawn as the polylines joining every 21st and every 34th point in birth order. Nothing here is a curve fitted to a picture: a spiral is a fact about which points are neighbours. The essay carries the same figure with the counting band on a slider, and the counts change as it moves.

Start anywhere

twelve of 365

A stem unrolled: 28 nodes at 137.51° with a rise of 0.090 circumferences. The counter is shown these coordinates and the circumference, and finds 2 parastichies one way and 3 the other. The faint strips left and right are the same stem: a family leaving one edge re-enters at the other. Stems and cones

A stem is a cylinder

The sunflower is the photograph, and it is the hard case. Nearly all real phyllotaxis happens on a stem, where the geometry is a lattice on a cylinder with two parameters — and where the spiral counts, which on a disc change with radius, are the same the whole way up.

8 figures A logarithmic spiral growing by 3.20× per turn. Fitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r. Shells and growth

Growth as a rule

A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.

6 figures The rule, 26 steps in, at a growth of 0.40. The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle. Where the angle comes from

The angle is an output

137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart.

8 figures A branching tree in which every junction obeys the cube law. r₀³ = r₁³ + r₂³ at all 63 junctions, to 2e-16. The widths in the drawing are the radii the law gives, not widths chosen to look right. Branching and transport

The cube law

A branching network built to move fluid for the least work obeys one relation at every junction — the cube of the parent radius equals the sum of the cubes of the daughters. It is a minimisation result, it is checkable on a real tree, and it is the rare biological rule with a derivation.

6 figures Which patterns grow on a ring of circumference 1.40. Modes 4 to 15 have positive growth rates and mode 8 is fastest. Integrating the full equations from a disordered start gives 8 peaks. What a plant might be doing

Turing's last problem

Turing's final work was on phyllotaxis and it was unpublished when he died. Its core is a ring of cells and two diffusing substances, and the thing it does is select a number of peaks — which can be predicted from the equations before anything is integrated, and then counted from what the integration produces.

6 figures How many sides the cells actually have. The mean is 5.908, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch. Packing and tiling

Why the average cell has six sides

Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.

6 figures The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number. The claims, measured

Fibonacci is a branch, not a law

Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.

7 figures A head of 200 primordia at a divergence of 137.51°. Nothing is placed by hand: the nth point sits at n·137.51° and radius √n. The closest any two points come is 1.60 of the mean spacing. The pattern itself

A head is a set of points

The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work.

8 figures The angle between the first two peaks, for six starting disorders. The same equations, the same ring, six different starting perturbations: 177°, 47°, 109°, 151°, 47°, 133°. A spiral needs the angle between successive elements to be the same one each time, and a stationary ring does not supply that. What a plant might be doing

A ring cannot make a spiral

The peaks on a Turing ring do not all appear at once — there is a first and a second. But which two lead is decided by the starting disorder, so the angle between them comes out at 177°, then 47°, then 109°, then 151°. A divergence angle is a relationship that repeats, and this one does not.

8 figures The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle. The pattern itself

Counting the spirals

Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by a count that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.

8 figures The same counter, on a stem and on a disc. The stem returns 2 and 3 in all three bands. The disc returns 21/34, 34/55, 55/89 — three answers to one question, which is why a published count needs to say where it was taken. Stems and cones

Counting up the stem

The same counting machinery, pointed at a stem instead of a seed head, returns one answer three times where the head returned three answers. That contrast is a measurement rather than a preference, and it is the one the whole cylindrical argument rests on.

8 figures The exponent fitted from the junctions, rather than assumed. Sweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed. Branching and transport

Fitting the exponent

Assuming the exponent is three and reporting the error says how far the data is from that assumption. Fitting the exponent and reporting what it comes out as says what the network is doing — and an estimator has to be shown returning something other than three, or it is not a fit.

6 figures

All 365 essays, grouped by field · the 8 fields · the claims, with a number attached to each · what arrived most recently

Ideas with the most said about them

eight of 85 ladders

Ablation

  1. 1 The organ that was taken away
  2. 2 The experiment this site can specify
  3. 3 A rule that cannot heal a hole
  4. +24 more
27 rungs · What a plant might be doing

Rung interior

  1. 1 Where a handover sits
  2. 2 Two lines that cross once
  3. 3 A band that holds the angle still
  4. +22 more
25 rungs · Stems and cones

Damage shape

  1. 1 The damage has a period
  2. 2 One level and two exceptions
  3. 3 A step of one organ
  4. +13 more
16 rungs · What a plant might be doing

Settling

  1. 1 How long a stem takes to settle
  2. 2 A wall and not a budget
  3. 3 A steeper rule walls nowhere else
  4. +13 more
16 rungs · Stems and cones

Noise colour

  1. 1 A disturbance with a memory
  2. 2 A periodicity is not a lattice
  3. 3 The disturbance that travels
  4. +12 more
15 rungs · What a plant might be doing

Both walls

  1. 1 Both walls of the slot
  2. 2 A removal that changes nothing
  3. 3 The rung that two organs wreck
  4. +10 more
13 rungs · What a plant might be doing

Second statistic

  1. 1 What a summary throws away
  2. 2 The disorder is a staircase
  3. 3 A dip belongs to the head
  4. +10 more
13 rungs · The claims, measured

Attractor

  1. 2 The angle is an output
  2. 3 The pattern the cut leaves behind
  3. 4 The block is the count it was cut from
  4. +7 more
10 rungs · Where the angle comes from

All 85 ladders, deepest first · the objects each essay names

Threads running through

themes, not chapters

Counted, not admired

Every spiral count on this site is extracted from the point positions by machinery that is never told the divergence angle. A count of 34 is then evidence rather than a restatement of what the pattern was built from.

271 essays

The angle is an output

The golden angle is where a rule settles, not a number the rule contains. Sweeping the one parameter of that rule gives a diagram with a golden branch, a transition and a two-whorl regime — and the famous constant is one branch of it.

33 essays

State the claim, state the test

This subject arrives with more confident assertions attached than any other. Each is drawn as it is told, given a test it could fail, and reported as a number — including the one that survives, which does so in a sharper form than the version usually repeated.

323 essays

Form is not mechanism

A model that reproduces a pattern has not explained it. The same spirals come out of magnetised droplets with no biology in them at all, and the actual mechanism in plants — auxin moving between cells — was worked out long after the geometry.

69 essays

One parameter, one family

A divergence angle, a growth factor, a whorl expansion, a branch ratio. Each of these subjects turns out to be a one- or three-parameter family, and the knob is the explanation rather than a decoration on it.

124 essays

Two routes, one number

The strongest thing a figure can do is arrive at a quantity twice by arithmetic that shares nothing. A lattice on a cylinder predicts where a disc's spiral counts change; a dispersion relation predicts how many peaks an integration will produce; a branching tree predicts the angles a dynamical model settles on. Where the two routes meet, the number is not an artefact of either.

90 essays

The round trip

Build the pattern from a stated number, forget the number, recover it from the pattern alone, and compare. It works for the divergence angle and for the growth factor, and the agreement is the measurement.

19 essays