§ 14 · Instrument Filed 2026.05 · Reference SL · 26 · 014

Local rules, arbitrary worlds.

Three families of cellular automata in one page. Conway's Life on a crisp grid, Wolfram's eight-bit rules drawn as a triangle of evolving rows, and Gray–Scott reaction–diffusion as smooth analog organic. Same idea, three lenses on emergence — the fact that nothing but a neighbourhood rule, run many times, can produce patterns indistinguishable from animal markings, weather, or computation itself.

§ Life 120 × 72

Click or drag to paint · shift-drag to erase · pick a pattern then click to stamp

Generation 0 Live 0 Density 0.0%

Rule B3/S23 Stamp Free draw

Speed

14 gen/s

Pattern

Seed

Run

Three lenses on emergence

§ 02 · Notes

Conway's Life has just three rules: a live cell with two or three neighbours survives, a dead cell with exactly three is born, everything else dies. That's enough to build a Turing-complete machine inside the grid.

A glider is the canonical traveller — five cells in a shape that re-prints itself one cell diagonally every four generations. The Gosper gun emits a new glider every thirty.

Wolfram's elementary rules are the simplest non-trivial automata possible: a one-dimensional row of cells whose next state depends on three neighbours. Eight inputs, one bit each output — a number from 0 to 255.

Rules group into four classes. Class 1 fades to uniform. Class 2 freezes into stripes. Class 3 looks random — rule 30 is so chaotic it was once used as a random number generator in Mathematica. Class 4 — rule 110 — is provably universal.

Gray–Scott is a continuous cousin: two chemicals U and V, each diffusing at its own rate, with V eating U in an autocatalytic reaction. The PDE has two scalar knobs — feed and kill — and a fairyland of attractors hiding in their plane.

Pulled the right way, the same equation paints leopard spots, zebra stripes, fingerprint mazes, and replicating drifters known as U-skates. Alan Turing predicted in 1952 that reaction–diffusion was how animals get their markings; he was, as usual, correct.

Implementation notes — rules, neighbourhoods, and time-stepping

Life is a 120 × 72 toroidal grid of bytes, double-buffered. Each step counts the eight Moore neighbours per cell and applies B3/S23. The canvas is filled by an ImageData buffer at native cell resolution then scaled up with image-rendering: pixelated so the grid stays crisp.

Wolfram elementary rules treat the row as a 1-D toroidal lattice. For each cell, the triple (left, self, right) indexes one of eight rule bits — the rule number's binary expansion. We draw the entire triangle row-by-row into a single ImageData the size of the canvas; the whole image is one DOM blit.

Gray–Scott integrates ∂U/∂t = Dᵤ ∇²U − UV² + F(1−U) and ∂V/∂t = Dᵥ ∇²V + UV² − (F+k)V with the discrete five-point Laplacian (weight 1 for orthogonal neighbours, 0.05 diagonal — actually a nine-point stencil), explicit Euler at Δt = 1. Several substeps run per animation frame so the chemistry looks fluid at sixty hertz.

Every preset is just a pair of feed and kill values: spots at (0.037, 0.065), stripes at (0.040, 0.060), mitosis at (0.0367, 0.0649), coral at (0.054, 0.062), maze at (0.029, 0.057), and the famous U-skate world at (0.062, 0.0609).