At the meadow
Can a flock find its way
without a leader?
Give every arrow the same local rule: turn toward nearby headings. See what the group can do together.
The picture is unavailable. Use Step and the numerical readouts to explore.
Gold arrow: the focal individual. Dark green arrows: its current neighbors. The circle shows its interaction radius. Edges join: leaving one side returns on the opposite side. Split circles show the same neighborhood across an edge.
Starts paused. Play shows 12 model steps per second; this is not a real bird’s flight speed. You can explore entirely with Step.
All arrows use the same radius and include their own heading. At zero, each listens only to itself unless another occupies exactly the same point.
A 20° range adds an independent turn between −10° and +10° after averaging. 360° permits any direction.
Restart rewinds positions and random turns while keeping radius and noise. Compare runs from the same seed, then try another.
What the group is doing
- Heading alignment · 0 to 1
- 0
- Focal neighbors, including self
- 0
- Focal heading · from right, clockwise
- 0°
1 means every heading agrees. Near 0 means headings mostly balance; the arrows still move. This measures direction, not how tightly they cluster.
Local rules, collective motion
Every arrow moves at the same fixed speed. At each step it averages the directions inside its neighborhood, then adds a random turn. No arrow knows the whole group’s direction. Try a quiet run for 200 steps, restart, and compare with a noisy run. One short run is an example, not a statistical result.
The model and its boundaries
This is a small Vicsek model, using distance-based neighbors and angular noise. Updates are simultaneous: positions move using the previous heading; new headings use the previous neighborhood. Direction is a circular average of unit vectors, not an arithmetic average of angles. If those vectors cancel exactly, the old heading is retained before adding noise.
There is no attraction, collision avoidance, leader, predator, wind, or three-dimensional flight. Arrows can overlap. The wraparound square has no real park boundary. These choices isolate alignment; this is not Boids or a calibrated bird simulation, and does not reproduce a murmuration.
A study of starling flocks found evidence for interactions based on a number of nearest neighbors rather than a fixed distance. The radius rule here is a simplifying model choice, not a universal rule for birds.
Sources and further exploration
- Vicsek and colleagues, 1995: self-driven particles — the local-alignment model, periodic square, angular noise, and heading order parameter.
- Ballerini and colleagues, 2008: starling field study — evidence about which neighbors matter in observed flocks.
Original implementation for Hidden World. Model distances and steps are not meters or seconds. Finite groups fluctuate; this page does not measure a phase transition.