Toys
Flocking: the Vicsek Model
Each dot is a self-propelled particle moving at constant speed. Every step it just turns to match the average heading of its neighbors, plus a little random noise. Out of that one rule, flocks appear. Slide the noise up and watch the order melt.
What you are looking at
This is the Vicsek model
(1995), the simplest model that shows a flocking phase transition. There is no leader and
no goal. The order parameter phi is the length of the average velocity vector,
normalized to [0, 1]: it sits near 0 when the swarm is a disordered gas and rises
toward 1 when everyone points the same way. Push the noise slider past the critical value and
the plot collapses; pull it back and order re-condenses.
Metric mode: a particle aligns with everyone inside a fixed radius
r. Topological mode: it instead aligns with its k
nearest neighbors regardless of distance. Real starling flocks were found to be topological
(each bird tracks ~7 others), which keeps cohesion as the flock stretches and thins.
The box has periodic boundaries, which means it is really a flat torus: leave one edge and you re-enter on the opposite side. Hit View: Torus to glue those edges up for real and watch the flock circulate on the donut (drag to orbit). This is the same torus renderer used in the Brownian Studio.