Textile Structure
how cloth is put together
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The thread: Does it hang together — page 2

Page 2 of 2 of the essays on this thread.
The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways. Knits and other structures

What holds a crest apart

Two half periods meet at every crest of every course and, in this collection's model, run within a fiftieth of a yarn diameter of one another for more than a millimetre. In a fabric what holds them apart is the loop of the next course drawn between them — which is the loop this model does not have.

 A knitted interlacing, and the number that makes it a fabric. Two loops idealised to rings, one drawn through the other, which is what a needle does. The Gauss linking integral returns 1. A knitted fabric of n wales has that between every pair of adjacent courses n times over, and it is the whole of why a knitted fabric can be made from one thread and taken apart by pulling it. Knits and other structures

A tuck is the one stitch that links twice

Knitting has three stitches and only one of them makes a new link. A knit stitch links a loop to the loop below; a miss links nothing; and a tuck holds two loops in one head — which is why a tuck stops a run and why the three cannot be described by one number.

 8×2 and 2×8 over every origin. Two grids of the 16 relative origins of 2/2 twill under 8-end satin, the row being how many picks the ground is started along and the column how many ends. The left grid is the census at a block 8×2, the right at 2×8; a square is filled where some of the 65,536 profiles separate. 8×2 fails at 8 origins and 2×8 at 8; both fail at 0 and neither at 0. Turning the cloth over and through a right angle sends each origin to another, and the letters mark where: every letter lands on a square with the same answer, so the two shapes are one census read at relabelled origins. Pattern and colour

A turned block is a moved origin

An eight-end satin figured on a 2/2 twill fails on 55,536 profiles at a block eight picks by two ends and on none at two by eight, and that was read as a property of the block's shape. Sweep the relative origin as well and the two shapes trade places: at every one of the sixteen origins exactly one of them fails, and over the sixteen they fail equally often. A shape asymmetry that no origin removes exists, and it needs a satin whose move squared is not one.

 The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways. Mechanics and drape

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

 Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

Where this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

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