A live 4D scene, followed from beside the subject: a peregrine falcon computed from equations flies over a night city of points and dives along a golden spiral around a tower mast, with digits of φ falling like rain behind it. Every earlier moment of the flight stays drawn as a green falcon along its path; the present moment is highlighted. The flight loops on its own. Scrolling down draws the camera back until the whole spiral is in view, then carries the flight to its last frame, the falcon perched on the mast, before the page moves on; in that last frame a crosshair marks the falcon and a detail view shows it close up. Scrolling up reverses it. A pinch, or the plus and minus keys, bring the camera closer or farther. The scene is synthetic.

The golden stoop.

A peregrine falcon computed from equations, diving on a golden spiral through a city of points. Synthetic

00:00:00 HOLD 0.00× frame 000

4D.OS · falcon-phi

Computing the falcon…

0%

A ratio that keeps its shape.

φ=1.6180339887…

Cut a square off a golden rectangle and what is left is the same rectangle, turned and shrunk by φ. Join the corners of the squares and the curve grows by φ every quarter turn: r(θ) = r0 · φ−2θ/π.
Turn each seed 360° ÷ φ² = 137.508° from the last and they never line up. The plaza under the tower is paved with this disk.

φ = (1 + √5) ÷ 2 is the only number that becomes its own square by adding one: φ² = φ + 1. It is why the rectangle keeps its shape and why the seeds never repeat. It is also a spiral a falcon can fly.

Measuring the spiral on the points of this pack…

The spiral, from above.

A peregrine does not dive straight at its prey. The sharpest part of its eye looks sideways, so to keep the prey in view without turning its head it flies a logarithmic spiral, closing in by a constant ratio every turn. Tucker, 2000, Journal of Experimental Biology 203:3745.

This falcon flies one member of that family, the golden spiral, and every moment of the flight is still there.

θ
0.000 rad
r(θ − π/2) ÷ r(θ)
— first quarter turn
phase
—

Twelve a second.

In 1882 Étienne-Jules Marey built a photographic gun to catch birds in flight: twelve exposures a second on one turning plate. In 1887 he cast the successive positions of a gull’s wingbeat as a sculpture. Here the wingbeat is an equation, and every stroke stays on the plate.

Against black, like the shed Marey photographed in: the city is switched off. This plate keeps one exposure every three frames, ten a second, so that each stroke stands apart; the gun’s twelve are below.

The downstroke takes 40% of each beat. The beats average 4.4 Hz and vary by 5% from one to the next, like a living rhythm, not a loop.
    One second through Marey’s gun: twelve source frames of the same flight.

    From equations to points.

    In a real capture the video comes first. Here the equations come first, and everything else is computed from them, in this order.

    1. Equations

      Path, funnel and wingbeat are functions of time. The page reads the same functions.

    2. Skeleton

      Body, three-segment wings, primaries and tail are posed for every frame. No model, no capture.

    3. Points

      ··· points per frame, taken once and kept on the same place of the body in every frame.

    4. Frame

      A drone camera renders the frame a real video would have given, to stand in for it.

    5. Screen

      Your browser draws every moment at once and dithers it to sixteen colors, every frame.

    What the pack holds.

    Frames
    ···
    Duration
    ···
    Frame rate
    ···
    Moving points
    ···
    Points per frame
    ···
    Same points in every frame
    ···
    Still points
    ···
    Source frames
    ···
    Weight on disk
    ···
    Scene
    ···

    Read from the pack loaded right now, not typed by hand.

    Where the bytes go