This Traffic Jam Untangles Itself! 1% more and it doesn't!

This is the Biham-Middleton-Levine traffic model, introduced by the physicists Ofer Biham, A. Alan Middleton and Dov Levine in 1992. Cars sit on a 256 by 256 grid whose edges wrap around, so a car that drives off the right edge comes back on the left, and one that drives off the bottom comes back at the top. The two colours take turns: ➡️ All the yellow cars step one square right ⬇️ Then all the blue cars step one square down A car only moves if the square in front of it is empty. Otherwise it waits for its next turn. ✨ 30% full: we scatter 19,705 cars at random. Little jams form at once: a stuck car holds up the car behind it, and that one the next. For thousands of ticks they drift, dissolve and form again, while the yellow and blue cars slowly gather into diagonal bands. In the end a single car keeps getting stuck, and from tick 8,888 on, every car moves on every tick, forever: the traffic has sorted itself into stripes that slide through each other without a single car ever being blocked again. Nobody planned it. No car knows anything but the square in front of it. 38% full: the jams spread instead. On tick 1,682 one last car moves, and after that nothing ever moves again: one giant jam wraps around the whole grid. 32% full: neither. Jams and free flow live side by side, about two thirds of the cars moving, still after 120,000 ticks. On bigger grids the tipping point creeps down. On a 512 by 512 grid, 2 of our 8 starts at 30% had not untangled after 800,000 ticks; on 1024 by 1024, 2 of 6 (one froze solid). At 31%, none of 14 untangled. But here's the strange part: Mathematicians have proven that the traffic always untangles itself if there are fewer cars than half the width of the grid (Tim Austin and Itai Benjamini, 2006). On this grid, that is fewer than 128 cars. We used 19,705. They have also proven that if cars are packed in tightly enough, traffic jams for good (Omer Angel, Alexander Holroyd and James Martin, 2005). In between, simulations find strange states where jams and free flow live side by side (Raissa D'Souza, 2005), and the tipping point shifts as the grid gets bigger. But for a huge grid, nobody can prove that the traffic ever flows freely, not with 30% of the squares filled, not even with a few percent. #Math #Mathematics #BMLTrafficModel #CellularAutomata #Traffic #SelfOrganization #Emergence #PhaseTransition #STEM #UnsolvedMath #MathAnimation #Marimba Chapters 0:00 Three grids, one rule 0:10 Building the grid 0:22 The rules 0:41 Let it run 1:06 38% full: gridlock 1:18 32% full: it never settles 1:31 180 grids, and an open question