Teaching objectives
How does an ant colony organize itself without anyone giving orders? Why do spirals appear in a chemical mixture, or separate neighbourhoods in a city nobody planned? This secondary-school lab explores cellular automata and other local-rule models to show that a simple rule, repeated over a grid, is enough to bring order out of randomness. This lab is worth highlighting in particular for its first room, where spirals come out of pure noise; the third, with the Game of Life; the fifteenth, where Schelling's segregation appears; and the seventeenth, where ants find food and start carrying it back to the nest. Each simulation is followed by a room where students apply the same rule by hand on a small grid. It comes after the Order from chaos lab (flocks and synchrony), although it can be done on its own.
What you'll learn
Students discover that self-organization requires no central intelligence: it is enough for each cell to look at its neighbours and follow a simple rule. The journey alternates "wow" simulations with pencil-and-paper exercises.
- Cellular automata (BZ spirals, Conway's Game of Life, a forest struck by lightning, a sand pile that topples) are studied as grids where each cell applies the same neighbourhood rule step by step.
- The self-organized criticality of the sand pile illustrates why complex systems spontaneously settle at the edge of chaos.
- The majority rule separates a mixture into growing domains, like oil and water.
- The percolation transition reveals that there is a critical density threshold p at which a path crossing the entire network suddenly appears.
- The voter model and the Schelling model take the same ideas to society: a tolerance of just 33 % is enough for spatial segregation to emerge even when nobody individually desires it.
- Ants with pheromones and diffusion-limited aggregation (DLA) close the lab with a path nobody has drawn and with fractal patterns of non-integer dimension generated by random walks.
Key mathematical ideas
- Local rules → global behaviour: macroscopic complexity is not encoded in any single cell.
- Cellular automaton: discrete grid + neighbourhood function + iteration; B/S notation for the Game of Life.
- Self-organized criticality: the sand pile reaches a critical state where the distribution of avalanches follows a power law.
- Percolation threshold: phase transition at p ≈ 0.593 for square grids; no percolating path exists below it, and one almost always exists above it.
- Probability in the voter model: the probability of adopting a colour is the fraction of neighbours that have it.
- DLA and fractal dimension: the branching structure has a Hausdorff dimension ≈ 1.71, arising from diffusion and randomness.
- Pheromone as distributed memory: a local concentration gradient replaces a global map.
Room-by-room contents
Room 1 · Spirals from nothing
Cyclic automaton (analogous to the Belousov-Zhabotinsky reaction): each cell "eats" the neighbour that is one step behind in the cycle. From pure noise, rotating spirals emerge.
Student tasks
- Observe the formation of spirals from the random initial state.
- Reset several times and compare what patterns appear each time.
Room 2 · Grid and neighbours
Formal introduction to cellular automata: the grid, the Moore neighbourhood (8 neighbours), and a manual application of the "rule of 5" on a 3×3 mini-grid.
Student tasks
- Identify the 8 neighbours of a cell in the grid.
- Calculate the next state of each cell by applying the cyclic automaton rule.
Room 3 · Soup that invents creatures
Conway's Game of Life: from a random "soup", gliders, oscillators and stable structures emerge using only two rules (birth with 3 neighbours; survival with 2 or 3).
Student tasks
- Observe which stable or periodic structures emerge spontaneously.
- Edit the grid to place a glider and watch it travel.
Room 4 · Apply B3/S23 by hand
Cell-by-cell quiz: the student marks the next state of a Game of Life configuration using the notations B3 (born with 3) and S23 (survives with 2 or 3).
Student tasks
- Count the live neighbours of each cell.
- Apply B3/S23 and mark the new state of the grid.
Room 5 · Forest, rain and lightning
Forest fire model: trees grow at random, lightning strikes sporadically and fire spreads through connected trees. A dynamic equilibrium emerges between density and catastrophe.
Student tasks
- Observe the balance between growth and destruction.
- Vary the lightning rate to see how the average forest density changes.
Room 6 · Burn this forest by hand
Quiz: given a grid with trees and fire, the student calculates the next state by applying the rules (tree with burning neighbour → burning; burning → empty) with no growth or lightning.
Student tasks
- Identify which trees have at least one burning neighbour.
- Mark the resulting state of the entire grid after one step.
Room 7 · Sand pile on the edge
Model of self-organised criticality (Bak-Tang-Wiesenfeld): grains are added one by one; when a column exceeds 4, it collapses and distributes grains to its neighbours, triggering avalanches of all sizes.
Student tasks
- Add grains and observe small and large avalanches.
- Verify that the system always returns to the edge of instability.
Room 8 · Solve this avalanche
Quiz: given a grid with numerical values, the student applies the collapse rule (cell ≥ 4 loses 4 and shares 1 with each neighbour) until no cell exceeds the threshold.
Student tasks
- Locate the cells that trigger the initial collapse.
- Propagate the avalanche step by step and calculate the final state.
Room 9 · Oil and water that separate
Ising/spinodal-decomposition model by majority rule: each cell copies the majority colour of its 4 neighbours. From a random mixture, growing domains emerge that recall phase separation.
Student tasks
- Observe how small domains disappear and large ones grow.
- Compare the random initial state with the state after 50–100 steps.
Room 10 · What does the majority say?
Quiz on the majority rule: the student determines the next colour of each cell based on whether at least 3 of its 4 neighbours (north, south, east, west) share the same colour.
Student tasks
- Count the neighbours of each colour for each cell.
- Mark the new state of the grid by applying the majority rule.
Room 11 · The threshold that connects everything
Percolation transition: cells are filled at random with probability p. The student explores from which threshold a cluster appears that crosses the whole screen from top to bottom (p ≈ 0.593, not 0.5).
Student tasks
- Vary p and observe when the first continuous path appears.
- Record the approximate value of p at which the transition occurs.
Room 12 · Follow the path
Connectivity quiz: given a partially filled grid, the student determines whether there is a path of black cells running from the top row to the bottom row using only up/down/left/right movements.
Student tasks
- Trace the possible path (or prove its non-existence) on the grid.
- Answer yes/no with justification.
Room 13 · Voters that spread ideas
Voter model: each cell periodically picks a random neighbour and adopts its colour. The student observes whether diversity of opinions survives or whether the system converges to a single colour.
Student tasks
- Observe the time to convergence for different grid sizes.
- Compare with the majority model from Room 16.
Room 14 · Calculate the probabilities
Quantitative quiz: a cell with 4 neighbours of given colours picks one at random to copy. The student calculates the probability of the cell adopting each possible colour.
Student tasks
- Count the neighbours of each colour.
- Express the probability of adopting each colour as a fraction.
Room 15 · Schelling's segregation
Schelling model (1971): each agent moves away if fewer than T% of its neighbours belong to its group. Even with very high tolerances, ghettos emerge, showing how individual preference produces unexpected global segregation.
Student tasks
- Adjust the threshold T and observe when segregation appears.
- Reflect on the difference between individual preference and collective outcome.
Room 16 · Does this one move?
Quiz applying the Schelling rule with T = 33 %: the student analyses a neighbourhood and marks which agents will move because fewer than 1/3 of their neighbours belong to their group.
Student tasks
- Calculate the fraction of same-group neighbours for each agent.
- Mark the dissatisfied agents who move.
Room 17 · Ants that find the way
Stigmergy model with pheromones: each ant lays a trail and prefers paths with more pheromone. Without a map or a leader, an optimal route between nest and food emerges. The explanation of the rule "choose the neighbour with the most pheromone" is integrated into this room.
Student tasks
- Observe how the shortest path is progressively reinforced.
- Block the active path and see whether the ants find an alternative route.
Room 18 · Fractal from random walks
Diffusion-limited aggregation (DLA): particles wandering randomly stick on touching the central cluster. From a single seed a branching fractal structure of non-integer dimension (~1.71) grows.
Student tasks
- Observe the step-by-step formation of the fractal.
- Notice why the arms grow faster than the interior gaps.
Room 19 · Does it stick here?
Closing quiz: given a grid with a DLA cluster already formed, the student marks all the cells where a random walker would get stuck if it stepped on that square (i.e., all cells adjacent to the cluster).
Student tasks
- Identify the outer perimeter of the cluster.
- Mark exactly the empty cells adjacent to at least one cluster cell.
Rooms to project
The most striking ones to show and discuss in class.