Koch Snowflake

This example demonstrates how the lq.path function can be used to generate a fractal − namely the Koch curve − in a diagram.

#let koch-snowflake(n) = {
  let complex-add(c1, c2) = { c1.zip(c2).map(array.sum) }
  let complex-multiply(c1, c2) = (c1.at(0) * c2.at(0) - c1.at(1) * c2.at(1), c1.at(0) * c2.at(1) + c1.at(1) * c2.at(0))
  let complex-inverse-array(a) = { a.map(c => c.map(x => -x)) }
  let complex-multiply-array(a, c) = { a.map(c1 => complex-multiply(c1, c)) }
  let complex-add-array(a1, a2) = { a1.zip(a2).map(cs => complex-add(..cs)) }
  
  let koch-snowflake-impl(n) = {
    if n == 0 {
      return (90deg, 210deg, 330deg).map(phi => (calc.cos(phi), calc.sin(phi)))
    }
    let pts = koch-snowflake-impl(n - 1)
    let diff = complex-add-array(pts.slice(1) + (pts.first(),), complex-inverse-array(pts))
    let points = (
      pts,
      complex-add-array(pts, complex-multiply-array(diff, (1/3, 0))),
      complex-add-array(pts, complex-multiply-array(diff, (0.5, -0.5 * calc.sqrt(3) / 3))),
      complex-add-array(pts, complex-multiply-array(diff, (2/3, 0))),
    )
    return array.zip(..points).join()
  }
  return koch-snowflake-impl(n)
}

#lq.diagram(
  width: 6cm, height: 7cm,
  xaxis: (ticks: none),
  yaxis: (ticks: none),

  lq.path(
    ..koch-snowflake(4),
    fill: blue, closed: true
  )
)