https://thoughttoys.com/ A cabinet of explorable explanations — abstract ideas turned into little worlds you can poke at until they click. Built in public, a little every day. en Tue, 16 Jun 2026 14:10:00 +0000 https://thoughttoys.com/exhibits/13-buffons-needle.html exhibit-13-buffons-needle Tue, 16 Jun 2026 13:55:00 +0000 Rule a floor with evenly spaced lines, scatter matchsticks at random, count the fraction that cross a line — and that fraction, flipped, hands you π, with no circle anywhere. Rain down thousands of sticks and watch the estimate converge on 3.14159: the gentlest possible introduction to Monte-Carlo estimation. Verified to land on 3.141–3.142 across needle lengths. https://thoughttoys.com/exhibits/12-fourier-epicycles.html exhibit-12-fourier-epicycles Tue, 16 Jun 2026 13:45:00 +0000 Stack spinning circles tip-to-centre, hold a pen at the end, and the combined wobble draws any shape — a square, a star, a heart. Add circles one at a time and watch a blob sharpen into the exact outline: a Fourier series assembling itself, big slow circles for the gist and tiny fast ones for the detail. The circle sizes come straight from the shape via the Fourier transform. https://thoughttoys.com/exhibits/11-percolation.html exhibit-11-percolation Tue, 16 Jun 2026 13:35:00 +0000 Open a slab's pores at random and pour water on top. Nothing gets through — until, near 59% open, one more pore opens a channel all the way down. The spanning probability stiffens into a near-vertical step as the grid grows: a genuine phase transition, the same knife-edge behind forest fires and epidemics. Threshold p_c ≈ 0.593 verified by Monte-Carlo sweep. https://thoughttoys.com/exhibits/10-evolution-of-trust.html exhibit-10-evolution-of-trust Tue, 16 Jun 2026 13:25:00 +0000 Cheating always pays more in the moment, so why does trust exist? Fill a world with simple strategies — copycats, cooperators, cheaters, grudgers — and let the winners breed. Meet once and the cheaters devour everyone; meet again and again and the cheaters go extinct while copycats and grudgers take over. The repeated prisoner's dilemma, evolving in fast-forward; one-shot collapse and many-round cooperation both verified. https://thoughttoys.com/exhibits/09-bayes-theorem.html exhibit-09-bayes-theorem Tue, 16 Jun 2026 12:55:00 +0000 A disease hits 1 in 100; a "90% accurate" test comes back positive — and your real odds of being sick are about 9%, not 90%. Count a thousand people and the paradox dissolves: the few false alarms among the many healthy people swamp the true cases. Base-rate neglect, made visible, and why evidence updates a prior rather than replacing it. https://thoughttoys.com/exhibits/08-logistic-map.html exhibit-08-logistic-map Tue, 16 Jun 2026 12:45:00 +0000 One tidy equation for a population with a single dial. Turn it up and a steady number splits into a 2-cycle, then 4, then 8, then shatters into chaos near r ≈ 3.57 — with a calm 3-year window hidden inside the storm. The famous bifurcation tree you can sweep, with the order/chaos verdict decided by the Lyapunov exponent. Period-doublings verified against the known Feigenbaum cascade. https://thoughttoys.com/exhibits/07-game-of-life.html exhibit-07-game-of-life Tue, 16 Jun 2026 12:35:00 +0000 A grid of cells, four plain rules about loneliness and crowding, no player and no randomness — and out crawl gliders, blinkers, and a gun that fires spaceships forever. Draw your own cells or drop a glider gun and watch complexity assemble itself. The classic proof that simple rules are not the same as limited ones; patterns verified in code before shipping. https://thoughttoys.com/exhibits/06-monty-hall.html exhibit-06-monty-hall Tue, 16 Jun 2026 11:00:00 +0000 Three doors, one prize. The host opens a loser and offers the swap — and switching wins twice as often, because he's quietly funnelling all the rejected probability onto the one door he leaves shut. Play it, run a thousand rounds to watch 2/3 appear, then push it to 100 doors and the answer turns obvious: switching wins (N−1)/N of the time. https://thoughttoys.com/exhibits/05-galton-board.html exhibit-05-galton-board Tue, 16 Jun 2026 10:50:00 +0000 A coin-flip at every pin, so no one can call where a single bead lands — yet a few thousand beads always stack into the same bell curve, the very one the maths drew before the first bead fell. Wild one bead at a time, dead reliable by the thousand: the binomial converging on the normal, the original central limit theorem you can watch. https://thoughttoys.com/exhibits/04-predator-prey.html exhibit-04-predator-prey Tue, 16 Jun 2026 10:00:00 +0000 The Lotka–Volterra model of rabbits and foxes. Neither side ever wins: the numbers swing forever, with the foxes always cresting a quarter-turn after the rabbits, riding one closed loop around a knife-edge balance point. Two lines of arithmetic, the oldest rhythm in ecology. https://thoughttoys.com/exhibits/03-double-pendulum.html exhibit-03-double-pendulum Tue, 16 Jun 2026 09:00:00 +0000 Two joined arms, no randomness, one exact rule — yet a path you can never repeat. Release a fan of near-identical pendulums and watch a difference too small to see grow until they fly apart. Sensitive dependence on initial conditions, the technical heart of chaos. Lift them gently and they stay in step. https://thoughttoys.com/exhibits/02-schelling-segregation.html exhibit-02-schelling-segregation Mon, 15 Jun 2026 09:00:00 +0000 Schelling's segregation model: everyone is easygoing, yet a wish as mild as "I'd just like a third of my neighbors to be like me" still tears the whole city into solid blocks. The segregation is far sharper than the preference, and nobody intended it. https://thoughttoys.com/exhibits/01-phantom-traffic-jam.html exhibit-01-phantom-traffic-jam Mon, 15 Jun 2026 08:00:00 +0000 A loop of cars, each following one rule. Slow their reactions a touch and a jam assembles itself out of nothing and crawls backward around the loop — which is why a highway can stop dead for no reason at all.