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).
A worked knot
Tie a 5-lead × 4-bight Turk's head on a 25 mm handle:
- Cords: gcd(5, 4) = 1 — one cord, tied as one. Share a factor instead (9 × 6) and the count is 3.
- Crossings: (5 − 1) × 4 = 16, each half-pass crossing the 4 laid before it. The 3 × 4 knot's 8 crossings are the knot 8_18.
- Coding: row coding over-1-under-1 = the ordinary Turk's head; over-2-under-2 along the cord = the Gaucho; column coding = the herringbone. Column patterns need 2 × bights to be a multiple of the pattern length, or the pattern breaks at the seam.
- The cut: passes × 2 × bights × half-pass length, plus working tails — an estimate, labelled as one, with the extra adjustable.
The knot calculator answers the cord count, the crossings and the coding names for any bights × leads pair — the same arithmetic the grid checks in the app's tests.
Mistakes worth avoiding
| the mistake | what it costs |
|---|---|
| assuming one cord always suffices | gcd decides; a 9 × 6 knot needs 3 cords and the tool says so before the cut |
| column coding on a bight count that doesn't divide | the pattern tears at the wrap seam; 2 × bights must be a multiple of the pattern |
| calling every diamond weave a Gaucho | Gaucho = row coding over 2 under 2; column coding is herringbone — the app fixed its own labels in 1.0.9 |
| cutting cord from a length rule without the tails | each cord wants two working tails, and a shared factor of 3 means six; the estimate is adjustable for a reason |
The app helps
The Grid draws the knot from bights × leads, codes it row or column, locks the rim rows if asked, and lists the run so every crossing is coded exactly once — unit-tested from 2 to 12 leads and bights. The bench's calculator carries the cord count and crossings.
Sources: the gcd rule and the cord count from the knot references (Wikipedia's Turk's head entry); the crossings derived and pinned by the app's unit tests; the coding names fixed against the knot-grid sources in the 1.0.9 label pass — the app's craft-facts document carries every line.