whatA lattice of spins, each a little tick that wants to agree with its four neighbours; heat makes them fickle. Metropolis dynamics, one temperature knob, and Onsager's famous number waiting in the middle of the slider: T_c = 2/ln(1+sqrt 2) = 2.269.

whyThe claim: the critical point looks like nothing else. Hot, the field is salt-and-pepper; cold, it freezes into great provinces. At T_c the system cannot decide -- domains inside domains, flickering at every scale at once, the majority's territory a fractal (box-counting dimension printed live, near 1.9, not 2). Quench cold and the noise coarsens: provinces swallow provinces, the mean domain growing with the patience of a square-root law -- grain growth's cousin. Click flips a patch; the cold majority heals it, the hot one shrugs.

howRandom-site Metropolis with a seeded flip stream (deterministic given the same slider history); domains by union-find, dimension by box-counting on the largest domain; the auto-story cools through T_c, holds at the flicker, quenches, and reheats. One integer seeds every flip.