Wiz's Lab

How a flock flies with no leader, made hands-on and WIZ-narrated. This lab has a run of experiments that hand you a rule and let you watch what it does on its own: the Game of Life with a grid, Collatz with one number, Turing Patterns with two chemicals. This one does it with three local rules and a few hundred birds. The rule is Craig Reynolds' boids model from 1986: every bird avoids the ones too close, steers to match the heading of its neighbors, and drifts toward their average position. Three cheap rules, no leader, no blueprint, and out of them a murmuration falls: a single fluid body that splits around a predator, ripples, and knits back together. Drag three dials to tune separation, alignment, and cohesion, and watch the flock shift between a loose murmuration, a tight polarized school, a midge-like rotating swarm, and a gas of loners who want nothing to do with each other. Switch perception from a fixed radius to the seven nearest neighbors, the real algorithm the STARFLAG project recovered from tracking thousands of starlings over Rome in 2006, and feel how the flock changes: tighter, more responsive, and self-healing in a way the radius mode cannot match. Then move your cursor in: you are the falcon, the flock pours around you, and the alarm wave propagates through the seven-nearest-neighbor chain far faster than any bird could see you and decide. The reframe is the point: coordination at scale nearly always turns out to be this, local rules and no global plan, and the shape falls out. The same principle runs fish schools, wildebeest stampedes, traffic jams, and stock market crashes, each a flock obeying a different local signal with no one deciding the global shape. A live simulation running hundreds of agents a frame in your browser, and the cleanest proof that coordination needs no coordinator.

How a leopard gets its spots, made hands-on and WIZ-narrated. This lab has a small run of experiments that hand you a rule and let you watch what it does on its own: the Game of Life with a grid, Collatz with one number, the Golden Angle with one angle, Chladni with one frequency, the Logistic Map with one growth rate. This one does it with two invisible chemicals on a dish and two knobs. Every cell of the dish holds two substances, U and V. Four things happen to them over and over: both spread into their neighbors with U diffusing twice as fast as V, wherever they meet two V plus one U react to make three V so V eats U and copies itself, U is fed in from outside at a feed rate, and V is removed at a kill rate. That is the whole chemistry, and out of it, from a flat almost featureless start, the dish paints spots, stripes, a maze, branching coral, or dots that swell and split in two like dividing cells. The only thing that decides which animal you get is two numbers, the feed rate and the kill rate. Drag them across the map and a leopard becomes a labyrinth becomes a turbulent boil that never holds still, and WIZ names the region you are in, measures how much of the dish V has covered, and tells you whether the field has locked into place or is still alive and churning. You can even paint your own seed onto the dish, but the rule erases it: it settles into its pattern, not yours, decided before you touched it. The reframe is the point. A leopard starts as a single cell that divides into a ball of identical cells, all carrying the same DNA, with nothing labelled spot here or stripe there, and yet it grows the same markings in roughly the same places every time with no architect and no blueprint. Turing showed the pattern does not need a painter: take two reacting chemicals where one spreads faster than the other and a flat even mix is secretly unstable, so the tiniest wobble gets amplified into peaks and valleys at a fixed spacing set entirely by the reaction rates, and that spacing is the pattern. Wide spacing on a small animal gives spots, the same chemistry on a long thin tail gives rings, which is exactly why spotted cats so often have striped tails and no striped cat has a spotted one. The same math turns up in real chemistry in the Belousov-Zhabotinsky reaction, in the ridges of your own fingerprints, in the spacing of hair follicles, and in the stripes a zebrafish actually grows and rearranges as it gets bigger. Alan Turing, the man who cracked Enigma and laid the foundations of computing, wrote it down in 1952 in The Chemical Basis of Morphogenesis, his last major paper before he died in 1954, working the equations by hand and on one of the first computers he had helped build. It is a live Gray-Scott reaction-diffusion simulation running thousands of cells a frame in your browser, no image loaded and nothing drawn, just two numbers and a rule. A simple-rules machine, a wonder toy, and a live demonstration that structure was never something life had to design: sometimes it just has to let go and let the chemistry fall into shape.

Deterministic chaos, made hands-on and WIZ-narrated. This lab has a small run of experiments that hand you a rule and let you watch what it does on its own: the Game of Life with a grid, Collatz with one number, the Prime Spiral with the primes, the Golden Angle with one angle, Chladni with one frequency. This one does it with a single line of arithmetic and one knob. Picture a population as a fraction x between 0 (extinct) and 1 (packed full). Next year's population is r times x times (1 minus x): it grows in proportion to how many there are, and is held back in proportion to how little room is left. The only knob is r, the growth rate, and it decides everything. Turn it up slowly. For a while the population just settles to a single steady value and holds it, perfectly predictable and almost boring. Then at r = 3 it refuses to settle and starts flipping between two values; at 3.449 between four; then eight, sixteen, thirty-two, the splittings arriving faster and faster, until at r = 3.56995 the period becomes infinite and the thing goes chaotic: an equation with no randomness in it anywhere that you could never predict, where two orbits started a billionth apart fly to opposite ends of the interval in a handful of steps. And folded inside the chaos are islands of perfect order, the period-3 window the most famous, clean cycles sitting in the middle of the storm with chaos on both shores. Drag the dial and watch the whole bifurcation diagram, the fig tree, light up where the orbit lives, while a live cobweb plot beside it spirals into a point, loops in a cycle, or fills the box chaotically. WIZ names the regime, reads out the period and the Lyapunov exponent (negative for order, positive for chaos), and measures how fast two near-identical orbits tear apart. The reframe is the point: there is no randomness here, the unpredictability is pure, and it comes from any tiny error in the start doubling and doubling until it swamps the answer. The route in is universal too: the ratio of one doubling window to the next converges to Feigenbaum's constant 4.6692016, and the same number turns up in a dripping faucet, a fibrillating heart, and a convecting fluid that share nothing with populations. Pierre-Francois Verhulst wrote the equation down around 1838 for constrained population growth; Robert May showed in 1976 it goes chaotic and helped found chaos theory; Mitchell Feigenbaum found the universal constant at Los Alamos with a pocket calculator; Li and Yorke named the field in 1975 with Period Three Implies Chaos, building on Sharkovskii (1964) and Lorenz's 1963 butterfly effect. A simple-rules machine, a wonder toy, and the cleanest door into the discovery that determinism never promised predictability.

Cymatics, made hands-on and WIZ-narrated. This lab has a small run of experiments that hand you a rule and let you watch what it does on its own: the Game of Life with a grid, Collatz with one number, the Prime Spiral with the primes, the Golden Angle with one angle on a dial. This one does it with a vibrating metal plate and a single number, a frequency. Scatter fine sand on a square plate and shake it. At almost any frequency the plate flexes in a messy, lopsided way and the sand just buzzes around and stays scattered. But every plate has a set of special frequencies where it flexes into a clean standing wave: some lines on the surface barely move at all, the nodes, while the regions between them slam up and down, the antinodes. Sand thrown by an antinode skitters away from it and comes to rest on the nearest still line, so after a second or two every grain has fled the shaking surface and piled up along the nodal lines, tracing the standing wave as a figure you can see: a cross, a star, a flower, a lattice, a cathedral window. The hook is the same one the Golden Angle had: a single number decides whether you get noise or geometry, and the targets are narrow. Slide the dial through the dead zone between two resonances and the sand will not settle no matter how long you wait, because there is no still line for it to find. Hit a resonance dead on and a mandala you never drew assembles itself in front of you. Drag the frequency, hunt for the notes that ring, jump straight to a named figure, or hit sweep and watch the plate fall in and out of pattern as the pitch climbs. WIZ names the figure, reports the mode and how close to resonance you are, and tells you which way to nudge. The plate is modeled with the classic square-plate superposition that Chladni's own figures obey, so the patterns are not canned pictures, they are the real nodal lines of a standing wave, and the sand finds them the same way real sand does, by being unable to rest anywhere it is still being thrown. The reframe is the point: the order was never in the sand, it was in the frequency, waiting for you to match it, and the same standing-wave math sets the resonances of a guitar body, the note a wine glass shatters at, and the modes a star rings in. Ernst Chladni drew these in 1787 with a violin bow on a sand-strewn brass plate, toured Europe with them, and performed for Napoleon in 1809, who funded a prize that Sophie Germain won in 1816 with the elasticity theory underneath this math; Faraday studied the finer crispations, Hans Jenny coined the word cymatics in 1967, and violin makers still sprinkle glitter on their tops today to read the same patterns and tune a plate before they string it. A simple-rules machine, a wonder toy, and a live demonstration that you can watch an invisible rule reach into the world and arrange matter into a shape, but only at the exact right pitch.

Phyllotaxis, made hands-on and WIZ-narrated. This lab has a small run of experiments that hand you a rule and let you watch what it does on its own. The Game of Life did it with a grid, Collatz with one number, the Prime Spiral with the primes. This one does it with a growing plant and a single angle. A sunflower head, a pinecone, a cactus all build the same way: a new seed is born at the center, drifts outward as the next one appears behind it, and each seed is turned by the same fixed angle from the one before. Helmut Vogel wrote the model down in 1979, seed n at a radius proportional to the square root of n, turned by n times the divergence angle. That divergence angle is the only knob, and it decides everything. Set it to 137.5077 degrees, which is the full circle divided by the golden ratio, and hundreds of seeds pack into a flawless rosette where the spiral arms you can count come out as consecutive Fibonacci numbers, 34 winding one way and 55 the other, with not a sliver of wasted room. Nudge that angle by a single hundredth of a degree and the whole head falls apart into coarse spirals and bald wedges. Drag the dial, drop in a famous value like 90 or 120 or 144, and WIZ counts the spiral arms, tells you exactly how far off perfect you are, and narrates what your angle did. The reframe is the point: the golden angle works because the golden ratio is the most irrational number there is, the hardest of all to approximate with any fraction, so no two seeds ever line up on a ray and they pack tighter than any other angle on the dial can manage. Sunflowers, pinecones, pineapples, cacti and romanesco all found it with no math at all, because it is simply the arrangement that fits the most seeds in the least space, and a plant that grows by pushing each new seed away from the crowded center falls into it on its own. Douady and Couder proved in 1992 that it is pure physics by reproducing the same spirals with magnetized drops of oil and nothing alive in the room. A simple-rules machine, a wonder toy, and a live demonstration that the most beautiful packing in nature is exactly one irrational number wide.