Murmuration

Seven birds

Seven birds follow three rules: keep apart, stay close, turn like the birds next to you1. Real starlings turn slowly, one after another3. Each bird bends the grid as it flies, and leaves a line of code behind it. Your cursor is a hawk. The birds run from it.

Neighbours

N_{i} = {j \neq i : the k nearest to i}, k = 6

The six nearest, near or far: what starlings do2. With seven birds, each sees all the others.

Separation

s_{i} = - \sum_{j \in N_{i}, d_{ij} < r_{s}} \frac{x_{j} - x_{i}}{d_{ij}^{2}}, d_{ij} = ‖ x_{j} - x_{i} ‖

Keep apart. A neighbour inside the radius pushes, harder the closer it is1.

r_{s}

spacing

Cohesion

c_{i} = \frac{1}{k} \sum_{j \in N_{i}} x_{j} - x_{i}

Stay together: a pull toward the neighbours' middle.

w_{c}

pull

The hawk

e_{i} = \frac{x_{i} - x_{h}}{d_{ih}^{2}} [d_{ih} < r_{e}]

The cursor. A bird inside the radius flees it, harder the closer it comes. A cursor at rest for three seconds is no hawk; a finger is one while it touches.

The roost

b_{i} = - \frac{max (0, ρ_{i} - ρ_{0})}{1 - ρ_{0}} {\hat{ρ}}_{i}

ρ_{i} = ‖ (\frac{x_{i} - x_{c}}{W / 2}, \frac{y_{i} - y_{c}}{H / 2}) ‖

The window is a roost. Past its middle, a pull back toward the centre that grows to the edge, as a starling model holds a flock over its roost4. The words are not in its way: the birds fly free and dim behind them.

The force

F_{i} = {clip}_{F_{\max}} (w_{s} s_{i} + w_{c} c_{i} + w_{e} e_{i} + w_{b} b_{i})

The four pulls, weighted, cut to a ceiling1. It does not move the bird: it asks for a turn and a change of pace.

The turn

\frac{d ω_{i}}{d t} = η (\frac{F_{i} \cdot {\hat{n}}_{i}}{u_{i}} - ω_{i}) + \frac{J}{k} \sum_{j \in N_{i}} \sin (φ_{j} - φ_{i}) + σ ξ_{i}, \frac{d φ_{i}}{d t} = ω_{i}

The turn rate has inertia. The sideways force asks for a rate and the rate eases toward it; the neighbours' headings pull on it too, so a turn crosses the flock as a wave, as measured on starlings3. The noise is on the rate, so the heading stays smooth.

η

turn response

J

alignment

σ

noise

The pace

\frac{d u_{i}}{d t} = F_{i} \cdot {\hat{t}}_{i} - γ (u_{i} - u_{0}), {\hat{t}}_{i} = (\cos φ_{i}, \sin φ_{i}), {\hat{n}}_{i} = (- \sin φ_{i}, \cos φ_{i})

The forward force speeds the bird or slows it; a spring pulls the pace back to cruise4. Nothing is clipped. Every second on the dials is two on the clock: the flock flies these equations at half speed.

u_{0}

cruise speed

The step

ω_{i} \leftarrow ω_{i} + {\dot{ω}}_{i} Δ t + σ ξ_{i} \sqrt{Δ t}, φ_{i} \leftarrow φ_{i} + ω_{i} Δ t

u_{i} \leftarrow u_{i} + {\dot{u}}_{i} Δ t, x_{i} \leftarrow x_{i} + u_{i} {\hat{t}}_{i} Δ t

Once a frame, in this order, each from the value just set. The noise is a unit normal draw times the root of the step, so the flock is the same at any frame rate.

The fold

The grid is a sheet seen from above. A weight dents it: every line bends toward it. The weights are the seven birds and the hawk.

The pull

u (p) = \sum_{i} A s_{i} \frac{R^{2} (x_{i} - p)}{{‖ x_{i} - p ‖}^{2} + R^{2}}, p \mapsto p + u (p)

A grid point is drawn where every weight's pull moves it. The pull peaks one radius out and vanishes at the centre, and its slope never reaches one, so lines squeeze and never cross.

The trace

A cell a bird crosses keeps a character for six seconds. The characters come in order, one bird one line, so a flight writes its program across the grid. Over words the pen lifts.

1N = 6 NEAREST 2S = -SUM (XJ-XI)/D^2 3C = MEAN(XJ) - XI 4F = WS*S + WC*C + WE*E + WB*B 5DW = ETA*(F.N/U - W) + J*MEAN SIN(PJ-PI) 6DU = F.T - GAMMA*(U - U0) 7X = X + U*(COS P, SIN P)*DT

01Reynolds, 1987: the boids, separation and cohesion as steering forces, the sum cut to a ceiling.
02Ballerini et al., 2008: a starling watches its six or seven nearest neighbours, however far.
03Attanasi et al., 2014: turns cross a flock as a wave; the inertial spin model.
04Hildenbrandt et al., 2010: a starling model with a cruise speed and a pull toward the roost.