An interactive essay · eight plates
Emergence
Nothing on the following pages was drawn. Every image is computed, live, sixty times a second, from a rule short enough to fit on one line — and every one of them is yours to interfere with.
v = (cos θ, sin θ), θ = noise(x, y)
Move your cursor across the field. Press → to begin.
Plate I · Flocking
A flock has no leader
Each bird can see only its nearest neighbours, and obeys three urges: don't crowd them, match their heading, stay near them. No bird knows the shape of the flock. The shape happens anyway.
v ← v + a·separate + b·align + c·cohere
Drag on the field — the flock treats your cursor as a hawk.
Plate II · Cellular automata
Two numbers, one universe
A cell survives with two or three living neighbours and is born with exactly three. That is the entire physics. Out of it come gliders, oscillators, and guns — machines that build machines, in a universe with no other laws.
B3 / S23
Draw living cells straight onto the grid — click and drag.
Plate III · Harmonic analysis
Every drawing is a chord
Chain enough spinning circles tip to tip and they will trace any closed curve you care to name. Draw one yourself: the machine listens to your line and tells you which circles it was made of, largest first.
f(t) = Σ cₖ · e^(2πikt)
Draw a closed shape anywhere on the field and release.
Plate IV · Optimal packing
The angle every plant agreed on
Place each seed one fixed turn from the last. Nearly every angle wastes the disc in spokes and gaps. One does not — 137.507°, the golden angle. Move it by a hundredth of a degree and the sunflower falls apart.
θₙ = n·137.507°, rₙ = c·√n
Click the angle slider, then tap ← → for one-thousandth of a degree.
Plate V · Iteration
An infinite coastline in six symbols
Square a number, add where you started, repeat forever. Keep the points that never run away. The boundary you get is endless: zoom for a thousand years and you will still be finding new rooms.
z ← z² + c
Click to magnify ×2 · shift-click to pull back.
Plate VI · Superposition
Where two waves meet
Waves do not collide; they add. Crest onto crest doubles, crest onto trough cancels to nothing. From that single fact comes the interference fringe — the pattern that first proved light was a wave, in a darkened room in 1801.
ψ = Σ sin(kr − ωt) / √r
Drag a source to move it. Two is all Young needed.
Plate VII · Deterministic chaos
A forecast, forever wrong
Three equations for a rolling convection cell. The path never repeats and never escapes; it winds around two hollow wings forever. Start two runs a millionth apart and within a minute they disagree about everything.
ẋ=σ(y−x) ẏ=x(ρ−z)−y ż=xy−βz
Drag to turn the attractor in space. Separation shown live.
Plate VIII · Rewriting systems
How to grow a tree from a sentence
Start with a stem. Replace every stem with a stem that forks left and right, then apply the same replacement to the forks. Do it nine times over. The grammar is eleven characters long; the result has bark, and it moves in the wind.
F → F[+F][−F]
Every regrowth is a new tree. Hover to part the branches.
Coda
Eight rules. No artwork.
Everything you just watched was reassembled from a handful of arithmetic, one frame at a time — no images, no libraries, nothing stored. Which is, more or less, the trick the universe is using too.
one page · zero assets
Push the letters apart with your cursor. They reassemble.