Munden's Stitch Density for Plain Knit

Munden's result is one of the cleanest in textile science, and it is genuinely surprising the first time you meet it. Knit a plain jersey, let it relax properly, and the dimensions it settles at depend on the stitch length alone — the length of yarn in one loop. Not on the yarn count. Not on the fibre. Not on the machine gauge. Change any of those while holding the stitch length constant and the relaxed fabric comes out with the same number of courses and wales per unit length. The loop, left to itself, adopts a shape that is a property of the knitted structure rather than of what it is made from.

\[ c = \frac{K_c}{\ell}, \qquad w = \frac{K_w}{\ell}, \qquad S = c\,w = \frac{K_s}{\ell^{2}}, \qquad K_s = K_c K_w \]

The source is D. A. L. Munden, "The Geometry and Dimensional Properties of Plain-Knit Fabrics", Journal of the Textile Institute Transactions, 50(7), 1959, T448–T471. His values for cotton plain knit run near $K_c = 5.0$, $K_w = 3.8$ and $K_s \approx 19.2$ in the dry-relaxed state, rising towards 21 fully relaxed. The ratio $K_c/K_w \approx 1.3$ is the loop shape factor, and it is why knitted loops are wider than they are tall and why jersey has the drape it does.

The Munden constants are DIMENSIONLESS, and stating that clearly is worth a paragraph because it is the single most common error in the online literature on this subject. Look at the arithmetic: $K_c = c\,\ell$ is a reciprocal length multiplied by a length, and $K_s = S\,\ell^{2}$ is a reciprocal area multiplied by an area. The units cancel completely. So the stitch length may be measured in any unit you like, provided you read the answer in the same one. Put $\ell$ in millimetres and $S$ comes out in loops per square millimetre. Put $\ell$ in inches and $S$ comes out in loops per square inch — the same 19.2, no conversion, no factor.

Munden himself worked in inches, and that is where the trouble starts. Sources quietly attach inches to his constants, notice that their own stitch lengths are in millimetres, and insert a 25.4 to "fix" the units — arriving at a stitch density wrong by $25.4^{2} = 645$. Others quote a separate set of constants "for metric units", which is a symptom of the same misunderstanding, since a dimensionless number has no metric version. If a source gives you a Munden constant with a unit attached, or two versions of it for two unit systems, distrust the rest of that source too.

Relaxation state is the real variable, and it is where the honest complexity lives. A fabric straight off the machine is under tension and its constants read low. Dry relaxation — hanging it up and leaving it — recovers some. Wet relaxation, and better still wet relaxation followed by tumble drying, recovers most of the rest and gives the "fully relaxed" state with $K_s$ nearer 21. The difference between the machine state and the fully relaxed state is essentially all of the shrinkage a garment will ever show, which is why Munden's work matters commercially rather than just theoretically: control the stitch length and you control the finished dimensions, whatever the yarn does.

Which is exactly how a modern knitting machine is run. Stitch length is set by the cam, but it is held by positive yarn feed — a device that delivers yarn at a fixed rate regardless of tension. Get the delivery rate right and the stitch length is right, and the fabric comes off the machine the same width every time. Measure it by unroving a known number of loops from one course, a hundred is convenient, straightening under a light standard tension, and dividing.