Weaves
Plain, twill and satin
Three rules, and everything else in weaving is a variation on them. What separates the three is not appearance but one trade — how often a thread changes face against how far it runs when it does not.
The float decides
How far a thread runs on the face before it goes under is one number, and it sits behind lustre, drape, snagging, abrasion, tear strength and how densely the cloth can be set. Almost nothing else in the subject has that reach.
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.
Interlacings and firmness
Every time a thread changes face it has to bend, and a bend takes room. That one sentence decides how densely a cloth can be set, how firm it feels, and why the two run in opposite directions.