STUDY #31 · 2026 · IN OBSERVATION

Chemotaxis — The 8π Gate

A model-driven visual study of self-attracting crowds.

MOVING IMAGE — THE IGNITION RELAY 12 clouds · m 30–430 vs gate (60, 130) · γ 0.004 · 512² · 30,500 steps

WHAT IS THIS

Bacteria and slime-mould cells secrete a chemical that attracts their own kind, then climb the gradient they made together. The crowd becomes a cloud with its own gravity. In two dimensions the outcome is a pure arithmetic of the total mass M: below 8π the cloud disperses, above 8π it collapses into a point in finite time — and exactly at 8π there is a single steady cloud, n(r) = 8/(1+r²)², that neither breathes out nor falls in.

The model is Patlak-Keller-Segel chemotaxis: a density that diffuses and drifts up the gradient of an attractant it secretes. This study runs a conservative lattice version of it in real time on the GPU — the sixth engine of the reaction-diffusion family here, and the first to carry an advection term: these patterns do not appear in place, they gather.

the devouring star field — clouds condensed to white-hot stars that now swallow each other clouds · χ 0.06 · γ 0.004 · ncap 4 · 512² · noctiluca

Motif Keller-Segel chemotaxis / 8π critical mass / chemotactic constellations

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation The fate of a self-attracting cloud reads off one number: below 8π it breathes out, above 8π it falls in, and the spread M₂(t) walks a straight line whose slope 4M − M²/2π is fixed the moment the mass is known. Spots born of attraction devour each other — the exact mirror of reaction-diffusion spots that divide.

Reference Evelyn F. Keller & Lee A. Segel, "Initiation of slime mold aggregation viewed as an instability," Journal of Theoretical Biology, vol.26, 399-415 (1970); Elena O. Budrene & Howard C. Berg, "Complex patterns formed by motile cells of Escherichia coli," Nature, vol.349, 630-633 (1991).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME FAMILY, TIME REVERSED

Gray-Scott's spots divide and multiply. These spots devour each other.

Study #02 — Gray-Scott Study #31 — Chemotaxis

Where the pattern comes from reaction and diffusion — the pattern is born in place advection — the crowd climbs its own gradient and gathers

What a spot does grows, splits, and colonises the plane swallows its neighbours — twelve clouds become eight stars

The arrow of time more spots, forever finer fewer spots, forever heavier — coarsening by cannibalism

What decides a fate local kinetics — no single number rules one number — the total mass, measured against 8π

PARAMETERS EXPLORED

param meaning effect on the image

M/8π total mass of the crowd (the fate dial) below 1 the cloud breathes out, above 1 it falls in; at 1 it holds its breath — the exact steady state

χ chemotactic pull — how hard the crowd climbs its own gradient speed of the gathering and the grain of the constellation

γ attractant decay — the territory radius √(Dc/γ) each star literally glows to the edge of its own chemical territory; large γ makes many small stars

g growth 0 = the theorem's world (nothing is created); >0 = the sea itself ignites and a star chart grows

σ initial cloud width the fate ignores it — that is this study's claim; only the mass is read at the gate

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

Two lines for the theorem, one line for the verdict. The attractant relaxes instantly in the theorem model; the artwork engine lets it diffuse and decay, and caps the crowd so stars saturate instead of diverging.

n_t = \nabla^2 n - \nabla\!\cdot\!\left(n\,\nabla c\right), \qquad -\nabla^2 c = n

The parabolic-elliptic Keller-Segel model (χ = 1): a density that diffuses while climbing the gradient of the attractant it secretes. The attractant is the free-space Green-function potential of the crowd itself — the "gravity" the crowd generates.

\frac{d M_2}{dt} = 4M - \frac{M^2}{2\pi}, \qquad M_2 = \int |x|^2\, n\, dx

The virial identity — exact, and the whole verdict. The spread changes at a rate fixed by the mass alone; it crosses zero at M = 8π. Measured here: M₂(t) straight to 7.3e-6; every collapse lands before the zero-crossing deadline (ratio ≈ 0.77 — Jäger-Luckhaus blow-up beats its own schedule).

n_s(r) = \frac{8}{(1+r^2)^2}, \qquad \int n_s = 8\pi

The held breath: the one steady cloud, carrying exactly the critical mass. Seeded into the engine it holds still (radial residual 3.8e-5) — the anchor of the fate ladder's 1.00 column.

n_t = D_n \nabla^2 n - \chi\,\nabla\!\cdot\!\left(n\left(1-\tfrac{n}{n_{cap}}\right)\nabla c\right) + g\,n^2(1-n), \qquad c_t = D_c \nabla^2 c + \alpha n - \gamma c

The artwork model (Mimura-Tsujikawa type): the attractant now diffuses and decays — giving each star a territory of radius √(Dc/γ) — and volume filling caps the collapse so a star is a saturated droplet, not a singularity. Its lattice fate gate was measured at (60, 130), consistent with the 8π mapping 8π·DnDc/(χα) ≈ 83.8 plus screening.

A visual interpretation of the model, not a claim of scientific precision. The dispersion of the artwork lattice was verified mode by mode against the exact quadratic — using the lattice symbol K² = 2(1 − cos k), which the GPU shader shares term for term with the Python engine (Pearson 1.000000).

SELECTED STILLS — 3

the devouring star field — the film's staging as a still clouds · χ 0.06 · γ 0.004 · ncap 4 · noctiluca

the constellation — a star lattice condensing out of a level sea, each grain a territory uniform n₀ 1 · γ 0.015 · exposure 0.4 · bloom 0.3 · noctiluca

the gate — same width, different mass: one cloud breathes out while the other has already become a star gate · m 55 vs 210 · σ 8 · γ 0.004 · noctiluca

PROCESS — PARAMETER SWEEPS

The fate ladder — the same cloud, its mass turned from 0.55 to 1.30 × 8π and nothing else changed. Up to 0.95 it slowly breathes out; the 1.00 column is the exact critical cloud holding its shape; 1.10 pulls inward; 1.30 has already collapsed to a point (marked *collapsed) before the second row. The whole story of this study in one contact sheet.

sheet 1 — the fate ladder free-space P-E · M/8π 0.55–1.30 · σ 1 · t 0–2.6

SIGNATURE — THE SCHEDULED LINE

The verdict is arithmetic. A cloud's spread walks a straight line whose slope is written the moment the mass is known.

For the free-space model there is an exact identity: the second moment of the crowd — its spread M₂ — changes at the rate dM₂/dt = 4M − M²/2π, a number that depends on nothing but the total mass. The right-hand side crosses zero exactly at M = 8π: lighter clouds must spread, heavier ones must contract to a point in finite time. π decides the fate of a crowd.

The engine was held to that line before any image was kept: the measured M₂(t) stays straight to 7.3e-6, the fate boundary bisects to 7.94π ± 0.09π, and every collapse arrives before the deadline the line schedules (the virial zero-crossing) — falling in is always ahead of its own arithmetic. In the Interactive Study below, that gold dashed line is drawn ahead of the white measured trace, and the trace walks onto it.

the needle and the gate — the exact parabola 4M − M²/2π with the measured slopes of M₂(t) sitting on it dM₂/dt vs M/8π · measured slopes · gate at 8π (gold)

COLOUR = DARK-FIELD BIOLUMINESCENCE

The palette is dark-field microscopy of a night sea: the void is deep indigo, thin crowds glow noctiluca teal — the bioluminescence of dinoflagellate blooms — and saturated cores go white-hot. Each star's halo has a physical radius: the screening length √(Dc/γ) of its own attractant, the edge of its chemical territory.

Variant palettes (luciferase amber, dark-field silver, GFP green, agar) re-image the same field the way different microscopes would. Gold is reserved for one thing only: the 8π gate.

noctiluca — the sea dims as it drains into the stars uniform n₀ 1 · γ 0.015 · territory √(Dc/γ) ≈ 8.2 · noctiluca

All colours are artistic approximations of these observation contexts, not measurements.

REFERENCES

Evelyn F. Keller, Lee A. Segel. "Initiation of slime mold aggregation viewed as an instability." Journal of Theoretical Biology, vol.26, 399-415 (1970).
Clifford S. Patlak. "Random walk with persistence and external bias." Bulletin of Mathematical Biophysics, vol.15, 311-338 (1953).
Elena O. Budrene, Howard C. Berg. "Complex patterns formed by motile cells of Escherichia coli." Nature, vol.349, 630-633 (1991).
W. Jäger, S. Luckhaus. "On explosions of solutions to a system of partial differential equations modelling chemotaxis." Transactions of the American Mathematical Society, vol.329, 819-824 (1992).
M. Mimura, T. Tsujikawa. "Aggregating pattern dynamics in a chemotaxis model including growth." Physica A, vol.230, 499-543 (1996).

INTERACTIVE STUDY

Pour mass into a cloud and read its future on a needle. This panel runs the exact radially symmetric reduction of Keller-Segel — the same equation the study used to verify the 8π dichotomy, so the gate here is arithmetic, not staging: hold the cloud until the needle crosses the gold tick and the same cloud that was breathing out starts falling in, exactly as the gold dashed line schedules. It is a deliberately simplified instrument, capped in resolution with a few curated knobs and no export, separate from the full GPU engine used to author the finished works.

SIMPLIFIED INSTRUMENT · EXACT RADIAL REDUCTIONKELLER-SEGEL · χ = 1 · HOLD TO POUR

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.