Teaching objectives
Why does traffic sometimes stop dead and then flow again a while later, with no crash, roadworks or traffic light? The answer isn't an external cause: the jam organises itself. This secondary-school lab (with extensions for upper secondary) follows the phenomenon from the famous Sugiyama experiment to the mathematical models that explain it, showing that a jam can emerge purely from too many cars and the small human imperfections of driving.
What you learn
The core idea of self-organisation: simple local rules (accelerate, keep your distance, brake a little when distracted) produce a collective phenomenon — the jam — that nobody programmed or caused.
- Experience the phenomenon (rooms 1-2). You watch a jam form by itself on a circular track and recreate the real Sugiyama experiment (22 cars), discovering with a slider the density threshold: below it traffic flows, above it the phantom jam appears.
- The toy model (rooms 3-6). The Nagel-Schreckenberg cellular automaton: first applying the rule by hand (building the next state), then local rules in simulation, critical density ρ_c = 1/6, the role of human randomness and the cascade of braking that propagates the jam backwards.
- Quantify (rooms 7-8). The flow-density fundamental diagram (a bell) and its counterintuitive consequence: past the peak, more cars = less flow (and why variable speed limits are used).
- Refine and generalise (rooms 9-11). The VDR variant (the stopped restart late → phase coexistence and hysteresis), the multilane model (platoons) and the 2D Biham-Middleton-Levine urban model with its abrupt gridlock transition.
- Apply (rooms 12-13). The self-driving cars that dissolve the jam with a small fraction (Stern, 2017) and a final synthesis.
Key mathematical ideas
- Self-organisation and emergence: a global pattern (the jam) that is in none of the drivers' individual rules.
- Optimal-velocity model (Bando, 1995): each car adjusts its speed towards an ideal speed that depends on the gap to the car ahead. It is the framework Sugiyama used to analyse his experiment.
- Threshold / phase transition: below a certain density the uniform flow is stable; above it, any small perturbation amplifies into a stopping wave.
- Backward-travelling waves: the jam moves opposite to the cars, while each car crosses it braking and re-accelerating.
Room-by-room contents
Room 1 · A jam nobody caused
Hook with an emergent explanation: three timed sentences pose the question (have you been in a jam with no crash or lane closure?) and then the animation appears: cars on a circular track, no obstacles. Colour encodes speed (green→red) and a red zone forms on its own and drifts backwards.
Student tasks
- Read the three intro sentences.
- Spot the red zone and see it move against the cars.
Room 2 · The Sugiyama experiment (2008)
Emergent intro (three sentences on black) telling the real experiment: 22 cars, instruction to drive at 30 km/h, nothing external… and yet a jam appears. Then the student recreates it with a density slider and a "Free flow / Phantom jam" badge. Ends with a conceptual question.
Student tasks
- Follow the experiment sentences.
- Find with the slider the number of cars at which the jam appears.
- Answer what causes it: nothing external, only the excess of cars.
Room 3 · What is a cellular automaton?
A three-row space-time diagram: the road is cells and time advances row by row. State 0 and State 1 (already computed with the rules) are given, and the student builds State 2 by placing each car at its new position after applying the rules by hand (one brakes, one accelerates, one keeps speed). The correct set is validated, with kind correction and a hint after two misses.
Student tasks
- Read the rules and understand the worked example (State 0 → State 1).
- Place the State 2 cars by applying the rules to each one.
Room 4 · The toy model
The Nagel-Schreckenberg automaton in simulation: the road is cells and each car follows three rules. With no randomness and few cars all reach top speed (critical density ρ_c = 1/6). Average speed and flow are shown in a time diagram (more readable than a flickering number).
Student tasks
- Add cars and confirm that below the critical density there are no jams.
- Read in the time diagram how average speed and flow change.
Room 5 · Add the randomness
The fourth rule is added: with probability p a car brakes for no reason (a lapse). One slider controls p and another the density. Now jams appear even with plenty of room: human randomness is enough.
Student tasks
- Raise p and watch jams emerge.
- Compare with the previous room (no randomness).
Room 6 · A cascade of braking
Traffic flows almost smoothly. Two buttons: "Brake a little" (the car stops for an instant) and "Slam the brakes" (it stays stuck for several steps, far more powerful). The braking forces the one behind to brake and a red wave propagates backwards, like a chain reaction.
Student tasks
- Try both brake types and compare the size of the cascade.
- Discuss why one brake can trigger a jam.
Room 7 · The fundamental diagram
The flow-density curve J(ρ) is built by sampling the NaSch model: it rises, peaks and falls (a bell). A slider moves the dot along the curve with the live road below. Conceptual question on the descending branch.
Student tasks
- Trace the curve with the slider and find the peak.
- Answer what happens to the flow past the peak.
Room 8 · More cars, less flow
The counterintuitive consequence: past the peak, adding more cars REDUCES the flow through a point. A big counter makes it obvious. Question on variable speed-limit signs at rush hour.
Student tasks
- Raise the density and watch the counter of passing cars fall.
- Answer why the speed limit is sometimes lowered at rush hour.
Room 9 · The stopped restart late (VDR)
VDR variant: a stopped car takes longer to set off than a moving one. Phase coexistence appears (free zone + compact jam) plus hysteresis: the jam "remembers" itself. Question on why the jam persists.
Student tasks
- Vary the density and observe the sharp, compact jam.
- Answer why, once formed, it is slow to clear.
Room 10 · Two lanes — do jams disappear?
Multilane model: two lanes with lane changes when the next lane is freer. Platoons (groups travelling together) appear, and at high density jams return all the same.
Student tasks
- Watch the lane changes and the platoons.
- Add cars and see that more lanes do not remove jams.
Room 11 · The grid city (BML)
2D Biham-Middleton-Levine model: red cars east and blue north, taking turns, on a Manhattan grid. Sharp transition: below a density everything flows; above it, total diagonal gridlock. Question on the transition.
Student tasks
- Raise the density and find the locking point.
- Answer how the transition is (abrupt, not gradual).
Room 12 · Self-driving cars
The jammed ring returns, but some cars (in cyan) are autonomous: they drive smoothly and do not amplify braking. A slider raises the percentage and, with very few autonomous cars (~10%), the jam dissolves, as in the real Stern (2017) experiment.
Student tasks
- Raise the percentage of autonomous cars and watch the jam clear.
- Estimate the minimum percentage needed for traffic to flow again.
Room 13 · What we have learned
Synthesis with the four key ideas (jam with no external cause, density threshold, more cars = less flow, solutions) and a final question checking the central idea of the lab.
Student tasks
- Review the four ideas.
- Answer the synthesis question.
Rooms to project
The most striking ones to show and discuss in class.