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 11:30:00 +0000 https://thoughttoys.com/exhibits/06-monty-hall.html knurl-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 knurl-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 knurl-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 knurl-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 knurl-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 knurl-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.