The map groups these phenomena into seven families by mechanism — what kind of thing each one is. That is a useful sorting, but it hides the better story. The same handful of deep ideas keep reappearing in systems that look utterly unrelated: magnets and opinions obey one law; sandpiles and stock markets another; snowflakes and cities a third. This page is the cross-index. Pick an idea and see everywhere it shows up. Most entries appear under more than one theme — that is exactly the point. The ideas interlace, and an entry's place in this list is a map of which currents run through it.
Phase transitions & critical points
Turn one knob slowly and, for a while, nothing much changes — then at a single sharp threshold the whole system flips from one regime to a completely different one. The transition is abrupt even though the knob moved smoothly, and right at the threshold the system is at its most delicate and most interesting.
- The Ising Model — order appears spontaneously below a critical temperature.
- Percolation — a spanning cluster snaps into being at a critical density.
- The Vicsek Model — flocking switches on as the noise drops past a threshold.
- The Kuramoto Model — oscillators lock once coupling crosses a critical value.
- Opinion Dynamics — consensus gives way to fragmentation as open-mindedness falls.
- Random Boolean Networks — order yields to chaos at a critical connectivity.
Power laws & the absence of scale
Events of every size, with no typical size at all. There is no "average" avalanche or "average" fire to speak of; the rare giant event and the common tiny one are not different kinds of thing but two ends of the same straight line on a log-log plot.
- The Abelian Sandpile — avalanches span every size, from a single grain to the whole pile.
- The Forest-Fire Model — fires range from a sputtered spark to a continent-burner.
- Preferential Attachment — a few giant hubs tower over a sea of small nodes.
- Global Cascades — most shocks fizzle unnoticed; rare ones sweep the entire network.
Self-organized criticality
A critical point is normally a knife-edge you have to tune to by hand. Some systems cheat: driven slowly, they tune themselves to the edge of a phase transition and hover there, with no one touching a knob. The critical point becomes not a place you have to find but an attractor the system falls into on its own.
- The Abelian Sandpile — grains pile up to the angle of repose and then hover exactly there.
- The Forest-Fire Model — trees regrow to precisely the density where fire just barely percolates.
Symmetry breaking
A setup with perfect symmetry — no built-in preference for one outcome over an equivalent other — nonetheless has to pick. The laws don't favour up over down, or this group over that one, yet the system commits to a single choice, and the symmetry of the rules is nowhere to be seen in the result.
- The Ising Model — the magnet must choose all-up or all-down, though nothing favours either.
- Chimera States — two identical groups, yet one synchronizes and the other will not.
- Random Boolean Networks — a frozen core crystallizes out of random, unbiased wiring.
The edge of chaos
Between frozen order and boiling chaos lies a thin, fertile seam where complexity and computation live — too rigid on one side to change, too noisy on the other to remember anything. The most interesting behaviour clusters in that narrow band. (For the full story, see the essay The Edge of Chaos.)
- Random Boolean Networks — at the critical connectivity, a perturbation neither dies out nor explodes.
- Elementary Cellular Automata — Rule 110's gliders compute on the boundary between orderly Rule 90 and chaotic Rule 30.
- Lenia — its creatures persist only inside a narrow band of parameter space.
Rich-get-richer (positive feedback)
A small early lead is not just an advantage but a compounding one: getting ahead makes it easier to get further ahead. Tiny initial differences, amplified by their own success, run away into lopsided, winner-take-most dominance.
- Preferential Attachment — already well-connected nodes attract still more links.
- Wealth Condensation — fair coin-flip trades drain all the money to a single agent.
- Global Cascades — each new adopter makes the next adoption more likely.
- Slime Mould Networks — trails that get used get reinforced, thicken, and pull in still more traffic.
Diffusion-limited growth
When a structure grows by gathering something that wanders in from far away, its exposed tips reach further into the supply than its sheltered valleys do — so tips grow faster, stick out more, and branch. The result is a feathery, fractal form that is mostly edge.
- Diffusion-Limited Aggregation — random walkers stick on contact and freeze into a feathery fractal.
- The Snowflake — vapour pulled from the air builds a six-fold branching dendrite.
Order from local avoidance
Global pattern doesn't always come from agents copying one another. Sometimes it comes from the opposite — each agent merely getting out of the others' way. Pure avoidance, repeated everywhere, settles into surprisingly orderly arrangements.
- Pedestrian Lanes — opposing crowds, each just dodging, sort themselves into smooth one-way lanes.
- Phyllotaxis — each bud grows into the biggest gap left by the others, yielding the golden angle.
- The Vicsek Model — alignment plus a little spacing condenses scattered particles into a flock.
Cyclic dominance & travelling waves
A beats B, B beats C, and C beats A — so there is no overall winner and the contest never resolves. Instead it rolls on forever, and in space it organizes into chasing fronts and rotating spirals.
- Spatial Rock-Paper-Scissors — three species chase one another into endless spirals, and all three survive.
- Excitable Media — a refractory pause after firing turns a sweeping wave into a self-pumping spiral.
Coarsening & the drift to consensus
Like clings to like: matching regions merge, small domains get swallowed by big ones, and the boundaries between them slowly straighten and disappear. The system simplifies over time, drifting toward uniformity.
- The Ising Model — aligned domains coarsen and grow below the critical temperature.
- The Voter Model — agents copy neighbours until one opinion fills the world.
- Schelling's Segregation Model — a mild preference sorts a mixed city into solid blocks.
Stigmergy & swarm intelligence
Many simple agents, none of them clever, leave marks in a shared environment and react to the marks others left. The environment becomes a kind of external memory, and through it the swarm solves problems no single agent could even pose.
- Ant Colony Foraging — evaporating pheromone trails converge on the shortest path to food.
- Slime Mould Networks — a body of brainless agents grows an efficient transport network.
- Boids — three local rules, read off each other's positions, make a whole flock.
Memory as attractor landscapes
Imagine the dynamics as a ball rolling on a hilly surface: wherever it starts, it rolls downhill into the nearest valley and stops. If you carve the valleys deliberately, the resting places are the stored information — and recalling a memory is just letting the system fall into the right basin.
- The Hopfield Network — a corrupted pattern relaxes downhill to the nearest stored memory.
- Random Boolean Networks — the tangle settles into a small handful of cyclic attractors.
Read the Atlas this way for a while and the family boundaries start to dissolve. A sandpile and a stock-market crash are the same power law; a magnet and a segregating city are the same coarsening; a Hopfield memory and a frozen gene network are the same landscape. The families tell you what each system is made of. The themes tell you what they are all secretly doing.