Permutations

a hundred and fifty points on concentric circles, the first making one turn in a cycle, the second two, the hundred and fiftieth a hundred and fifty. Nothing is random, and nothing is tuned by hand: the whole picture is that one rule, so the fan of points falls into a star at every simple fraction of the cycle and comes back to a single spoke at the end of it. points sets how many turn, cycle how long the slowest takes, step the whole number between one rate and the next, and dotSize the marks. An original Ollin interpretation, written from his own statement of the rule; no code was ported, and his own programs are printed in BASIC in a book rather than published as software.

Run

swift run --package-path Examples Example-Recreations-JohnWhitney-Permutations

Source

Examples/Recreations/JohnWhitney/Permutations/Sketch.swift

// A homage after John Whitney (1917-1995), made for learning. Not a reproduction
// of a specific film, and not affiliated with or endorsed by the artist or his
// estate. Written from his own statement of the rule in *Digital Harmony: On the
// Complementarity of Music and Visual Art* (Byte Books, 1980), where he sets a
// first element moving at a given rate, the next at two times that rate, the
// third at three, and so on, and where the example worked in the book puts those
// elements on a series of increasingly wider concentric circles. The reading of
// that construction was cross-checked against Jim Bumgardner's account of it in
// "The Whitney Music Box" (Bridges 2007, jbum.com/papers/whitney_paper.pdf).
// No code of anyone's was ported; the film that demonstrates the principle is
// *Permutations* (1968).
import Ollin

/// Whitney's rule: every point turns at a whole-number multiple of one rate, so
/// nothing wanders and the figure keeps coming back. Point 1 makes one turn in a
/// cycle, point 2 makes two, point 60 makes sixty. They leave together, fan into
/// a spiral, and gather again into two arms at half a cycle, three at a third, and
/// so on down the fractions. He called those gatherings harmonic resonance, and
/// held that they are to seeing what a chord is to hearing.
@main
final class Permutations: Sketch {

    /// How many points turn. Each one is a whole-number multiple of the first.
    @Param(6...600) var points = 150
    /// Seconds for the slowest point to come all the way around, which is also
    /// the period of the whole figure.
    @Param(6...120) var cycle = 30.0
    /// The whole number between one point's rate and the next.
    @Param(1...6) var step = 1
    /// How big the marks are.
    @Param(1...14) var dotSize = 5.0

    let palette = CosinePalette.neon

    override func draw() {
        background(Color(white: 0.04))
        noStroke()

        let reach = min(width, height) * 0.46
        let turn = time / cycle

        for index in 1...points {
            let fraction = Double(index) / Double(points)
            // The rule: rate is the index, times the whole-number step.
            let angle = .tau * Double(index * step) * turn - .pi / 2
            let at = center + Vector2(cos(angle), sin(angle)) * (reach * fraction)

            fill(palette.color(at: fraction))
            drawCircle(center: at, radius: dotSize * scale)
        }
    }
}