Themes
A weave is a matrix
Warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not an encoding of a weave — it is the weave, and it makes almost every question about it decidable.
Does it hang together
A draft can look entirely reasonable on point paper and describe cloth that falls into two layers, or has threads lying loose on the surface. The failure is invisible and the test is exact.
The float decides
How far a thread travels on the face before it goes under is the quantity behind lustre, drape, snagging, abrasion and how close the cloth can be set. Almost nothing else has that reach.
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.
A mechanism, not a material
Cloth on the bias stretches half as long again while no thread in it stretches at all. The deformation is kinematic — a trellis closing — and almost every explanation given for it is wrong.
Measured, not claimed
Thread count, breathability, the strength of satin: the trade is full of numbers that do not mean what they are taken to mean. Where a claim is refuted here, it is refuted by a computation.
Structure before fibre
Most of what a fabric does follows from how it is put together rather than from what it is made of. The same yarn gives a shirting or a satin, and they behave nothing alike.