Textile Structure
how cloth is put together
Theme

The thread: Exactly this many

There is no regular satin on six ends, and the reason is a one-line argument about common factors rather than a rule of thumb. Counts in this subject are more often theorems than tallies.
weaveleave the faces aloneexchange themplain1616basket16162/2 twill16163/1 twill16none5-end satin20noneevery candidate operation tried against the repeat Pattern and colour

Weaves as plane patterns

A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.

 ordermoves that give a satinhow many3 endsnone — every step shares a factor with the order04 endsnone — every step shares a factor with the order05 ends2 326 endsnone — every step shares a factor with the order07 ends2 3 4 548 ends3 529 ends2 4 5 7410 ends3 7211 ends2 3 4 5 6 7 8 9812 ends5 7213 ends2 3 4 5 6 7 8 9 10 111014 ends3 5 9 114enumerated from the coprimality condition, not recalled3–14 Weaves

There is no satin on six ends

Weavers have known it as a rule for centuries. It is a theorem with a one-line proof about common factors, and it rules out four ends as well.

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