// Spirolateral engine:
// segments 1,2,...,n (x unit), turn a fixed angle after each, repeat to close.
// reverse the turn at chosen segment indices for the classic "reverse" figures.

// greatest common divisor (Euclid) -- used to find the closing period
def gcd(a, b) [
  a = abs(round(a))
  b = abs(round(b))
  let r = 0
  while b > 0 [
    r = a % b
    a = b
    b = r
  ]
  return a
]

// cycles until closure: net turn per cycle is angle*(n - 2*reversed),
// and the figure closes after 360/gcd(360, netturn) cycles.
def spiro_reps(n, angle, reversals) [
  let net = round(abs(angle * (n - 2 * len(reversals)))) % 360
  if net == 0 [ return 1 ]          // net rotation 0 -> open / drifting figure
  return 360 / gcd(360, net)
]

// build the polyline in maths-space (unit = 1) without sewing
def spiro_points(n, angle, reps, reversals) [
  let pts = [[0, 0]]
  let p = [0, 0]
  let h = 0
  repeat reps [
    for i = 1 to n [
      p = vadd(p, vfromheading(h, i))
      append(pts, p)
      if contains(reversals, i) [ h = h - angle ] else [ h = h + angle ]
    ]
  ]
  return pts
]

// centre on the origin and scale so the farthest point sits at radius maxr
def fit_points(pts, maxr) [
  let b = bbox(pts)
  let c = [(b[0] + b[2]) / 2, (b[1] + b[3]) / 2]
  let rmax = 0
  for q in pts [
    let d = vdist(q, c)
    if d > rmax [ rmax = d ]
  ]
  let s = 1
  if rmax > 0 [ s = maxr / rmax ]
  let out = []
  for q in pts [ append(out, vscale(vsub(q, c), s)) ]
  return out
]

// jump to the start, then sew the path in whatever stitch mode is active
def draw_spiro(pts) [
  up setpos(pts[0]) down
  sewpath(pts)
]

// fitted core: generate -> fit-to-hoop -> sew
def spirolateral(n, angle, reps, reversals, maxr) [
  draw_spiro(fit_points(spiro_points(n, angle, reps, reversals), maxr))
]

// auto-closing, auto-fitting (44 mm radius) -- the everyday entry points
def spiro(n, angle) [
  spirolateral(n, angle, spiro_reps(n, angle, []), [], 44)
]
def spirorev(n, angle, reversals) [
  spirolateral(n, angle, spiro_reps(n, angle, reversals), reversals, 44)
]

// raw: draw from the turtle's current spot, your own unit/reps, no fitting
def spiroraw(n, angle, unit, reps, reversals) [
  repeat reps [
    for i = 1 to n [
      fd i * unit
      if contains(reversals, i) [ lt angle ] else [ rt angle ]
    ]
  ]
]

// scene:
fabric "woven        // free underlay + pull compensation
seed 7
stitchlen 2
bean 3               // bold line; try satin 2 for a ribbon, or drop for min density

color 4
// --- figure ---
let sides = 6       // [3:12] segment count per cycle
let turn = 144      // [30:6:180] turn angle, degrees
spiro(sides, turn)
trim

// gallery: swap the call above for any of these (all vetted to close):
// spiro(3, 90)              4-fold, 12 segments
// spiro(5, 90)              4-fold, 20 segments
// spiro(7, 60)              6-fold, 42 segments
// spiro(8, 108)             5-fold rosette, 40 segments
// spiro(6, 100)             3-fold, 18 segments
// spiro(4, 72)              pentagon family
// spirorev(6, 90, [2, 4])   reversed turns
// spiroraw(6, 90, 3, 4, [])  exact 3 mm unit, no auto-fit