essays on complexity & emergence
Interactive laboratories for building intuition about the models that shape our understanding of the world.
Pólya's urn (Eggenberger & Pólya, 1923) starts with one red ball and one blue ball. Draw one at random, put it back, and add another of the same color. This single rule produces path dependence, preferential attachment, and a surprising result: every long-run proportion is equally likely.
The Girvan-Newman algorithm (2002) discovers hidden groups by iteratively removing the edges that carry the most shortest-path traffic. It reveals the principle that boundaries between communities are where the information flows.
Schelling's model (1971) starts with a city of individually tolerant residents — each willing to be in the minority. A single round of moves later, the city is deeply divided. No bigotry required.
Craig Reynolds' Boids (1987) starts with birds that can only see their neighbors. Three local rules — don't crowd, match heading, steer together — produce the swirling, splitting, regrouping dance of a real flock. No leader required.
Deming's Red Bead Experiment (Out of the Crisis, 1982) puts six workers on a factory floor with a bowl of 4,000 beads. The defect rate is fixed by the system. But management keeps finding individuals to blame.
Leontief's input-output model (1936) tracks the web of purchases between sectors of an economy. A change in demand for one sector ripples through all the others — and the total effect is always larger than the direct effect.
Conway's Game of Life (Gardner, Scientific American, 1970) starts with a grid of dead cells and four rules. From this simplicity: stable structures, oscillating clocks, traveling gliders, self-replicating guns, and a system powerful enough to simulate any computation.
Axelrod's tournament (The Evolution of Cooperation, 1984) pitted strategies against each other in a repeated prisoner's dilemma. The winner was the simplest entry: Tit for Tat — cooperate first, then mirror. No cleverness required.
The Lorenz attractor (1963) starts with three deterministic equations and no randomness. Two trajectories beginning a thousandth apart end up in completely different places. The butterfly effect: prediction is impossible not because the rules are unknown, but because initial conditions can never be known precisely enough.
Galton's quincunx (Natural Inheritance, 1889) drops balls through a triangular field of pegs. Each ball bounces left or right at random — a coin flip at every row. Drop enough, and they pile into a perfect bell curve. The central limit theorem, made physical.
Kriging (Krige, 1951; Matheron, 1963) uses spatial correlation to produce optimal predictions from scattered samples. Unlike simpler methods, it quantifies its own uncertainty — you get both a map and a measure of how much to trust it.
Nash equilibrium (Nash, 1950) is the point where every player is doing the best they can, given what everyone else is doing. Often self-interest and the common good align — and when they don't, understanding the equilibrium reveals exactly how to redesign the rules so that they do.
Sen's critique ("Rational Fools," 1977) dismantles the orthodox assumption that choice, preference, and welfare are one and the same. A single ordering cannot capture commitment, adaptive desires, or the liberal paradox — and treating it as if it can produces policies that mistake deprivation for satisfaction.
Ostrom's governance framework (Governing the Commons, 1990) dismantles the assumption that shared resources inevitably collapse. Communities worldwide have sustained fisheries, forests, and pastures for centuries — not through privatization or state control, but through institutional design: communication, monitoring, and graduated sanctions.