Turk's heads, by arithmetic.
Bights × leads decide the cords, the crossings and how the knot weaves — the app runs it, and so can you.
The two numbers that make the knot
A Turk's head is a bights × leads tube: bights run round the mandrel's rim; leads ("parts" in Grid Maker's language) travel the knot's length. Everything falls out of the pair:
| wants | answer |
|---|---|
| the cords to cut | gcd(bights, leads) — one cord only when they share no factor (the source: the knot article's own math). 9×5, 7×4... = the one-cord knots. 12×8 = two. |
| the crossings | (leads − 1) × bights — each lead crosses leads−1 others on its travels. 3×4 = 8 crossings. |
| the coding | an ordinary Turk's head = "over 1 under 1" row coding along the whole cord; a Gaucho = over 2 under 2 row coding; the herringbone = the column coding. (The app's own history: it labelled the column coding "Gaucho" until v1.0.9 — the sources settle it: the ROW coding is the Gaucho.) |
The run list
Tie from the run list, not the picture: each pass prints over/under only for the cord already laid — every crossing coded exactly once. Follow it and the knot comes even.
How much cord: the coverage
Each family of parallel cords has to fit round the mandrel: coverage ≈ passes × bights × cord ÷ (circumference × cos angle). "Fill the mandrel" aims at 95% — the app's convention. The calculator runs the crossings'; the coverage's estimate is the app's too.
The Gaucho/herringbone note: a column pattern needs 2 × bights a multiple of the pattern length, or the pattern breaks at the wrap's seam. The app's "locked edges" keeps the rows near each rim over-1/under-1 (Grid Maker's locked-coding behaviour).