Thought Toys · Exhibit 11
Take a slab of rock and open its pores at random — a few percent, then more. Pour water on the top. For a long time it just soaks into dead ends and stops. Then, at one astonishingly sharp cut-off, one more pore tips the balance and a channel suddenly runs all the way through. Coffee, forest fires, epidemics, the moment a rumour goes everywhere — same knife-edge.
Water poured on the top edge — amber is wet, grey is open-but-dry, dark is solid
Chance a slab lets water through, as openness rises — and how the jump sharpens with size
Every cell is a tiny pore that's either open or solid. The Open pores slider sets the chance each one is open — but the slab's randomness is fixed, so as you slide up you're opening more and more of the same material, never reshuffling it. Water starts along the top edge and seeps into every open pore it can reach by stepping up, down, left or right. Wet pores glow amber; open-but-dry ones stay grey; solid rock is dark.
Drag the slider slowly from the left. At low openness the amber barely sinks in — isolated puddles, going nowhere. Keep going and the wet region grows raggedly… and then, somewhere around 0.59, almost without warning, the amber lunges down and touches the bottom. The slab has percolated. Nudge back a hair and the channel breaks; nudge forward and it slams shut again. There's no gentle ramp — connectivity arrives as a threshold.
The lower chart is why this is famous. It plots how often a slab lets water through against how open it is. For a tiny grid the curve is a lazy S; for a big one it stiffens into a near-vertical step at the same spot. In the limit of an infinite slab it becomes a true jump — a phase transition, as abrupt as water turning to ice. Below the critical openness: essentially never. Above it: essentially always. That sudden switch is the same math behind a forest fire that either fizzles or consumes everything, and a disease that either dies out or sweeps the population.
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