Circles that cannot let go of each other.
The Hopf fibration matches every point on a sphere with a circle in space, and any two of those circles are linked once, like two links of a chain. Here a few circles on the sphere are picked at random, and the circles over their points are drawn as families: hairlines that lace through one another, wound around tori that never touch. One to three circles in each family are heavy ribbons twisting along part of their length, one of them signal orange. The circle over the far pole runs off to infinity in both directions, so it is drawn as a straight line through everything.
Play it and it turns. Every so often the bundle unwinds and another is wound. Switch the ground to paper to draw it in black on white.
The fibre
javascript·desmi.js
// over the point (θ, φ) on the sphere: a circle on the 3-sphere,
// (cos(θ/2)·e^{it}, sin(θ/2)·e^{i(t+φ)}), projected stereographically into space
function fibre(theta, phi, t) {
const c = cos(theta / 2), s = sin(theta / 2);
const k = 1 / (1 - s * sin(t + phi));
return [c * cos(t) * k, c * sin(t) * k, s * cos(t + phi) * k];
}
The families
javascript·desmi.js
// the fibres over a circle on the sphere, centred at (θc, φc), radius ρ, cover a torus;
// families are kept short of the pole, where the circles swell toward infinity
const rho = random(0.15, min(0.9, LIMIT - thetaCentre));
for (let j = 0; j < count; j++) {
const [theta, phi] = around(thetaCentre, phiCentre, rho, phase + (j / count) * TAU * arc);
hairline((t) => fibre(theta, phi, t));
}
References
