Infinity Hair

Mathematica

Infinity Hair (2024): each frame is a new set of 1000 random-walk strands, and e moves them on. Click a picture for the full-resolution image.

a = Table[
  Graphics[Table[
    {White, Opacity[0.5],
     Line[
      Table[{Cos[θ],
        0.3 Sin[2 θ]}, {6}] +
      Accumulate[Table[
        {RandomReal[{0, 0.1 Sin[θ]}],
         RandomReal[{0, 0.1 Cos[θ]}]},
        {6}]]]},
    {θ, e, Pi + e, Pi/1000}],
   Background -> Black,
   PlotRange -> {{-1.5, 1.5},
     {-0.75, 0.75}}],
  {e, Pi, 5 Pi, 0.1}];

Export["infinity_hair_2024_02_18.gif",
  a, ImageSize -> 700]

Every strand of hair here is a random walk: it starts on a curve and takes a few random steps, each leaning the way the curve is heading. Redrawn every frame, the hair never sits still.

The portraits of Albert Einstein and Professor Calculus are drawn the same way: hair, moustache and eyebrows are groups of random walks rooted on arcs.

How it's made

A strand starts at a point \(b(\theta)\) on a curve and adds random steps: \(p_i = b(\theta) + \sum_{j \le i} (u_j, v_j)\), where \(u_j\) is uniform between \(0\) and \(0.1\sin\theta\) and \(v_j\) between \(0\) and \(0.1\cos\theta\). So every strand leans with \(\theta\), and its neighbours lean almost the same way.

For Infinity Hair the curve is the figure-eight \(b(\theta) = (\cos\theta,\; 0.3\sin 2\theta)\): a thousand strands of six steps, white at 50% opacity, rooted on half of it, \(\theta\) from \(e\) to \(e + \pi\). As \(e\) grows, the hair sweeps round the curve. Spiral Hair does the same along \((0.3\theta + \sin 2\theta,\; 0.3\theta + \cos 2\theta)\); Sunrise roots ten thousand strands at one point; Nebula lets sixteen walks run for 100,000 steps each.

Pick a drawing, set the sliders and press Generate; New variation grows new hair. First posted on January 10, 2016.