Concept

The range of the interaction — where it appears

5 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
0501000.2500.5000.7501growth parameter Gspread of the last 30 steps (°) — 0 means settledfilled dark: convergedthe usable range is stated, not implied

Where the model stops

Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.

emergence · modellimit
00.2500.5000.7501012345distance from the candidate, in local spacingsweight the interaction is multiplied byhalf weightweight = f(d / 3√h)window runs to 4 widths

A 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.

emergence · interaction range
00.2500.5000.750100.2000.4000.6000.8001candidate azimuth, in turnsenergy the rule minimises, scaled to its own rangeexponentialhard — a step, not a sloperise 0.02 · p = 1two cut-offs, one lattice

A 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.

emergence · truncation
204012345range at which the interaction has halved, in local spacingsnear-shell contrast — the first shell's variation over the second'swhat the exponent sweep leavesp = 1 · shells 0–2 and 2–4 spacingscontrasts 6% apart, ranges 47%

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 previous phase measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

emergence · falloff exponent
loop cut at 3/√h, p = 133%27.78° of scatterexponential cut-off, p = 1100%0.68° of scatterexponential cut-off, p = 3100%0.82° of scatter6 runs each, separated by 0.2° of placement noisehalf-weight radius 1.5 spacings

The 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.

emergence · noise amplitude

Named alongside it

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

RepulsionCut offDivergence angleNeighbourhoodThe placement ruleTruncationArtefactDiscretisationLattice offsetEnsembleMeristemModel scope

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