What's new
Essays arrive in groups rather than one at a time, and a group usually opens up a subject the collection had not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.
14 August 2026
12 essays onwhat a plant might be doing, the claims, measured, stems and cones and where the angle comes from
The noise that arrives through the neighbours
The two kinds of noise this site had were idealisations that bracket the rule's choice. The realistic disturbance is neither: a primordium is placed exactly, and then the organ grows, so by the time the next one forms its neighbours have moved. That is a third kind, and it is invisible in every measurement a plant offers.
What a plant might be doingA growing organ is part of the rule
Every model on this site 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.
The claims, measuredThe test a plant could settle
Every other open question in this collection is priced in tens of specimens, and one of them in a hundred and sixty. This one is priced in internodes on a single stem, and the number is fifty-six — because it is a statistic of one sequence rather than a share of a population.
The claims, measuredWhat a quiet plant is worth
Almost every measurement gets easier as the effect gets larger. This one gets harder — a stem's divergence sequence stops carrying information about its noise at precisely the scatter where the noise becomes obvious. The specimens worth measuring are the ones that look least interesting.
Stems and conesThe sequence has a memory
Every measurement this collection has made of a stem's divergence angles throws the order away. A spread is invariant to shuffling. Put the angles back in order and there is a large correlation between one and the next — 0.54 with no noise at all — which is the rule correcting itself, and which nothing had looked at.
Stems and conesWhat one angle says about the next
A tenth of a degree of placement noise moves a stem's divergence scatter from 0.50° to 0.62°, which nobody would report. It takes the correlation between consecutive angles from 0.54 to below zero. The other two kinds of noise, at scatters where no measurement can separate them, leave it at 0.6.
Where the angle comes fromA neighbourhood is a hypothesis
Every simulation of this kind stops summing somewhere. The previous phase 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.
Where the angle comes fromA hard edge is not a falloff
The prediction was that cutting the neighbourhood at three spacings would reproduce the pattern truncation had manufactured. It does — if the cut is smooth. A hard cut at the same distance produces no pattern at any width, and the reason is that it is the only one of the three whose neighbour set depends on where the candidate is.
Where the angle comes fromTwo 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 previous phase measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.
Where the angle comes fromThe fragility belonged to the window
A pattern that exists only because the rule cannot see far was expected to be held together by that cut, and to fall over when nudged. It does — while the cut is a loop bound. Written down as a falloff at the same range, the same rule keeps every run under the same nudge, at a scatter an inverse-cube rule cannot be told from.
Where the angle comes fromWhich minimum was chosen
The rule takes an argmin, so there are two completely different things noise can do to it: move the answer, or move the question. One of them can change what is chosen and the other cannot, ever — and the difference is exactly zero against one or two placements in a thousand, at amplitudes where every other measurement says the two are identical.
Where the angle comes fromThe boundary belongs to the pattern
Three kinds of noise, in three incommensurable units, destroy a lattice at the same place — about a degree and a half of divergence scatter. The previous phase measured that of two kinds and called it a scale rather than a constant. With a third it looks less like a coincidence and more like a property of what a lattice is.
Before that
Everything published earlier, newest first. Titles only — the cards are on the full listing.
12 August 2026
12 essays onthe claims, measured, stems and cones and where the angle comes from
- What a count is worth — the claims, measured
- How many plants would it take — the claims, measured
- The survey this site cannot do — the claims, measured
- An organ has no single exponent — stems and cones
- Noise is not a slow rate — where the angle comes from
- Two degrees of scatter — where the angle comes from
- What one exponent reports — stems and cones
- How much of a cone to measure — stems and cones
- Where the noise gets in — where the angle comes from
- How far a primordium reaches — where the angle comes from
- The exponent that barely matters — where the angle comes from
- A window that makes a pattern — where the angle comes from
10 August 2026
12 essays onthe claims, measured, the pattern itself, stems and cones and where the angle comes from
- What "whorled" was hiding — the claims, measured
- Two at a time — the pattern itself
- Counting without an index — the pattern itself
- A cone has a rise that falls — stems and cones
- A pattern with a rate — where the angle comes from
- Half the golden angle — the pattern itself
- The lag that is not there — where the angle comes from
- Transitions a factor of φ² apart — stems and cones
- The rate decides the branch — where the angle comes from
- The shape and the law — stems and cones
- Continuity from a coarse start — where the angle comes from
- Why a cone can be counted once — stems and cones
6–8 August 2026
34 essays onthe pattern itself, stems and cones, the claims, measured, shells and growth, where the angle comes from, branching and transport, what a plant might be doing and packing and tiling
- A head is a set of points — the pattern itself
- A stem is a cylinder — stems and cones
- Fibonacci is a branch, not a law — the claims, measured
- Growth as a rule — shells and growth
- The angle is an output — where the angle comes from
- The cube law — branching and transport
- Turing's last problem — what a plant might be doing
- Why the average cell has six sides — packing and tiling
- A ring cannot make a spiral — what a plant might be doing
- Counting the spirals — the pattern itself
- Counting up the stem — stems and cones
- Fitting the exponent — branching and transport
- Packing, measured four ways — packing and tiling
- Raup's three numbers — shells and growth
- The bifurcation diagram — where the angle comes from
- The claim that survives — the claims, measured
- A pump that works uphill — what a plant might be doing
- Droplets with no biology in them — where the angle comes from
- L-systems describe, they do not explain — branching and transport
- The counts change with radius — the pattern itself
- The gap that grows — packing and tiling
- The nautilus question — shells and growth
- Two numbers out of the points — stems and cones
- Lewis's law wants disorder — packing and tiling
- Recovering the angle from the counts — the pattern itself
- The Fibonacci ladder — stems and cones
- What a mechanism would have to show — what a plant might be doing
- Where the model stops — where the angle comes from
- A disc is a cylinder — stems and cones
- Two laws that want opposite tissue — packing and tiling
- How often is it Fibonacci — the claims, measured
- The forks are exact — stems and cones
- The tree and the attractor — where the angle comes from
- The angle is not the object — the claims, measured