A pile of rice, seen from the side. Nobody sets the angle it settles at — and yet it always settles at the same one, right at the edge of collapse.
Each dot: how often a slip of that size happened. Both axes are logarithmic.
Click the pile to drop a grain
show me → skip to free play →
You stopped at .
The Oslo ricepile model: Christensen, Corral, Frette, Feder & Jøssang (1996), “Tracer dispersion in a self-organized critical system”, Phys. Rev. Lett. 77, 107. 10.1103/PhysRevLett.77.107 Each site's tolerance is re-drawn from {1,2} after every slip. That re-draw is the whole model — remove it and the pile is not critical, which is what chapter 8 lets you do.
Bak, Tang & Wiesenfeld (1987), “Self-organized criticality: an explanation of 1/f noise”, Phys. Rev. Lett. 59, 381. 10.1103/PhysRevLett.59.381
Frette, Christensen, Malthe-Sørenssen, Feder, Jøssang & Meakin (1996), “Avalanche dynamics in a pile of rice”, Nature 379, 49 — power-law avalanches for elongated grains, none for rounder ones. 10.1038/379049a0 Six years earlier, Held, Solina, Keane, Haag, Horn & Grinstein (1990), “Experimental study of critical-mass fluctuations in an evolving sandpile”, Phys. Rev. Lett. 65, 1120, had failed to find any scaling in real sand. 10.1103/PhysRevLett.65.1120
Zipf (1949), Human Behavior and the Principle of Least Effort — the rank–frequency law, first for words and then for cities. The line for US cities has held roughly the same slope for two centuries while the ranking churned; Krugman (1996), The Self-Organizing Economy, calls that regularity “spooky” and an embarrassment for economic theory, and Gabaix (1999), “Zipf's law for cities: an explanation”, Q. J. Econ. 114, 739, is the standard attempt to answer it. 10.1162/003355399556133 A caution this page takes seriously: the pile is not Zipf's law in the strict sense. Zipf means a rank–size slope near 1, which is roughly what cities and words do; this pile is close to 1.9, and that difference is chapter 10's whole point. Same family, different steepness.
Cities — the 30 largest US cities at the 2020 census, sorted by population. Ranks change every decade; the line does not. Earthquakes — USGS earthquake statistics, averaged over 2000–2021: about 1.1 events a year at magnitude 8 or above, 14.7 at 7 or above, 150 at 6 or above and 1 811 at 5 or above. Those cumulative counts are ranks. Size is plotted as energy rather than magnitude, because magnitude is already a logarithm — one step up the scale is about 32× the energy. The constant factor per magnitude step is the Gutenberg–Richter law (1944). usgs.gov Words — not quoted from anywhere: the words of this page, counted in your browser from the prose you have just read. Zipf's own dataset, and the only one here you can check by eye.
Deliberately suppressing large avalanches makes the exceptional ones more likely, while deliberately triggering them lowers the average size — discussed, with the prescribed-burn comparison, in Gang, Jia & Herniter (2022), Int. J. Wildland Fire 31, 1. 10.1071/WF21092 This used to be a chapter. It was cut because nothing you can do in the cell demonstrates it: “biggest so far” only ever climbs, there is no second pile running alongside as a control, and so the lesson arrived as a citation rather than as something you saw. The tool stayed; the claim moved here.
The settled mean slope (~1.72), the time to reach it, the avalanche-size distribution, the three betting brackets, and the “times rarer per ten-fold step” figure are all measured by running this page's own model, not quoted. tests/run.js re-measures them against the shipped file on every run, so a change to the simulation that breaks a sentence fails the tests.
tests/run.js