Textile Structure
how cloth is put together
After the loom

The constants say nothing about thickness

Munden's two constants give a knitted fabric's wale and course spacings from its loop length alone, and the tightness factor collapses every fabric's shape onto one curve. Neither reaches the third dimension: two knits that are one knit in plan are two different thicknesses.
13 min read 6 figures Measured, not claimedExactly this manyStructure before fibre

Worth reading first: Two knits with one tightness factor are one knit · A knit's dimensions come from its loop · How thick a knit is.

The most useful thing anybody has ever found out about knitted fabric is that its dimensions come from its loop length and nothing else. Wales per unit width are a constant over the loop length; courses per unit height are another constant over the same loop length; and the constants are the same for cotton, wool, nylon and everything else, in every count, on every machine.

That result is why a knitted fabric can be specified at all. It is also, this rung argues, silent about the fabric’s third dimension in a way that has never been stated because there was no third dimension to be silent about.

What the third dimension changes, and by how much. Every number the planar loop model produced, beside the same number with the climb in it, for a 20 tex cotton jersey at a 3.5 mm loop. Four of the five fall and none moves by as much as four per cent, which is the useful part of the answer: the planar model was not wrong about a jersey, it was a projection of the right curve. What it could not have at all is the quantity that is not on this list — the force through the fabric's thickness, 7.81 mN a stitch, which a model with no thickness has nowhere to put.
Fig. 1 The quantities the loop model produces, with and without the third dimension. Four of the five are shape quantities that the constants and the tightness factor cover between them. The one that is not on the list at all — the force through the fabric’s thickness — is the one that had nowhere to be.

What the constants cover

Two numbers, and between them a whole fabric’s plan.

The wale constant times the loop length gives the wale spacing; the course constant times the loop length gives the course spacing. From those two follow the stitch density, the areal weight, the cover, the openness, the extensibility and every force this collection computes.

They are measurements. Nothing predicts them, and this collection is explicit that the model takes them as input: the geometry is imposed and the yarn’s path inside it is solved, which is exactly the division of labour that makes the forces meaningful.

What the tightness factor covers

One number, and it is the group that makes the whole family one family.

A knitted loop’s shape depends on the yarn’s diameter over its loop length and on nothing else. Two knits matched on that ratio have the same solved loop, in units of the loop length, to fifteen figures — whatever they are made of and whatever count they are spun to. That is the geometric content of the collapse, and it is why a single dimensionless index does the work of a table.

The tightness factor is the trade’s version of the same group, with a constant made of the fibre’s density and its packing folded in.

What neither of them reaches

The thickness is two yarn diameters. It is a length in millimetres, not a ratio, and the group has no such length in it.

That is the whole of this rung and it deserves to be stated flatly. Take the pair this collection already uses to demonstrate the collapse: a 12 tex cotton at a 3.5 mm loop, and a 30 tex wool at the 5.96 mm loop that matches it on the ratio. Their solved loops agree to fifteen figures. Their wale and course spacings, in units of their own loop lengths, are identical. Every dimensionless quantity about them agrees.

Their thicknesses are 0.259 millimetres and 0.441 — a factor of 1.70, between two fabrics the collapse calls one fabric.

Why that is not a failure of the collapse

The collapse never claimed to cover a length, and this is a good place to say what a dimensionless group does and does not do.

A group collapses shapes. Two systems matched on it are geometrically similar — the same picture at a different scale — and every ratio computed from them agrees. What it cannot do is supply the scale, because that is precisely what has been divided out.

For a knitted fabric the scale is the loop length. So the collapse says two matched fabrics have the same shape in units of their loop lengths, and it is silent about their absolute sizes in every respect, including this one.

What makes the thickness worth calling out is that it is the only absolute length in the fabric’s own description. The plan dimensions are the loop length times a constant, so they scale with the loop. The thickness is the yarn’s diameter times two, and the diameter is not the loop length times anything — it is a separate input.

A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted.
Fig. 2 The two lengths in one picture. The drop across the courses is the loop length times a constant; the climb through the fabric is the yarn’s own diameter. The collapse is about the first and silent about the second.

Which is the same thing the forces do

There is a precedent and it is the reason to trust this reading.

The collapse is equally silent about force. Two knits matched on the tightness factor have identical geometry and their forces differ by the ratio of their bending stiffnesses over their loop lengths squared — which can be any factor at all. Munden’s constants are a statement about a knit’s dimensions and were never a statement about what it feels like.

The thickness is the same kind of exception with a different reason. The forces escape because the group has no stiffness in it. The thickness escapes because it has no length in it.

A dimensionless group is silent about exactly what was divided out to make it, and here that is a length and a stiffness.

What inherits the silence

Anything computed from a thickness, and there are three worth naming.

Warmth. A thermal resistance is a thickness over an effective conductivity, so two fabrics the collapse calls identical differ in warmth by something close to the ratio of their yarn diameters — 1.70 for the pair above, before the difference in their fibre fractions is counted.

Compression. The relaxed thickness is where a compression curve starts. Two matched fabrics start from different places.

And bulk density. Weight per unit area over thickness is a length divided by a length times a density, and the two lengths do not scale together.

Each of those is a quantity somebody might reasonably expect the collapse to cover, because they are properties of the fabric rather than of the yarn. None of them is covered.

The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.
Fig. 3 Where the missing length comes from. The two centre lines are a yarn diameter apart because that is what two threads in contact are — a distance in millimetres, unrelated to the loop length that sets everything else in the fabric’s plan.

One place the two lengths do meet

The tilt of the loop’s own plane is the one quantity on this ladder that is a ratio of the two lengths, and it is therefore the one the collapse does cover.

It is the climb over the drop — a yarn diameter over a course spacing and a diameter — which is a diameter over a loop length times a constant, and that is the tightness factor’s own group with the constant folded in. So two fabrics matched on the tightness factor have the same tilt, exactly, and the same share of their contact force acting through the fabric.

Which is a clean way to see what the collapse is: it covers every angle and every ratio, and it covers no length and no force. The tilt is an angle, so it is covered. The thickness is a length, so it is not.

Where a course of yarn sits through the thickness. The through-thickness position of one course of yarn, half period by half period, for 3 structures at a bed gap of 3 yarn diameters — 0.501 mm — on a 20 tex yarn whose diameter is 0.167 mm. Single jersey never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. A one-by-one rib crosses between the beds 4 times a course, travelling 0.501 mm through the thickness. A two-by-two rib crosses between the beds twice a course, travelling 0.668 mm through the thickness. The horizontal is a count of half periods rather than a length, because the repeats are not the same width and the comparison is not about their widths.
Fig. 4 The third length, on a fabric that has one. A two-bed structure’s thickness is a bed gap plus a diameter, and the gap is neither the loop length nor the yarn’s — it is a machine setting no collapse could reach.

The two lengths a knitted fabric has

It is worth being explicit that a knitted fabric has two independent lengths, and that most of its description uses only one of them.

The loop length sets the plan: the wale spacing, the course spacing, the stitch density, and how much yarn is in a unit area.

The yarn’s diameter sets the thickness, and enters the plan only through the tightness factor’s ratio.

Everything the trade specifies — the gauge, the stitch length, the areal weight — is about the first. The second is fixed by the count and the fibre and is not usually thought of as a fabric dimension at all.

So a knitted fabric is a two-parameter family and its usual description is a one-parameter one plus a count. That is fine for what the description is for; it is a trap for anybody who reads the collapse as saying more than it does.

And a third, once there are two beds

On a two-bed fabric there is a further independent length: the bed gap, which is a machine setting and is not derived from the yarn or the loop at all.

That takes a rib to three independent lengths, and none of the collapses covers the third. A rib’s thickness, its warmth and its through-thickness force are all set by a number that is not in Munden’s constants, not in the tightness factor and not in any published relaxation table.

Which is the reason this collection’s two-bed arithmetic takes the gap as an argument and refuses to give it a default. There is nothing to default it to.

A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.
Fig. 5 A rib in section, where the constants say least of all. Its thickness is the bed gap plus a diameter and the gap is a machine setting — so a fabric whose dimensions Munden’s constants describe exactly has a thickness they cannot reach at all.

Where the constants came from, and why they stop here

Munden’s constants are a measurement of a relaxed fabric’s plan. They were obtained by knitting fabrics, relaxing them and counting wales and courses under a glass, and they are excellent measurements — the collapse they demonstrate is one of the most robust results in the subject.

Nothing in the method could have reached the thickness. Counting wales and courses is a plan measurement, and a plan measurement of a fabric contains no information about its third dimension whatever.

So the silence is not an oversight in the original work. It is a property of what was measured, and it went unnoticed here for the same reason: a model with no thickness cannot notice that its inputs have none either.

What would extend it

A relaxation constant for the thickness — a number that, times something, gives the fabric’s depth — would be the natural completion, and this rung says what it would have to be.

For a single-bed fabric it is not a constant times the loop length; it is two, times the yarn’s diameter. So the “constant” is 2 and the length it multiplies is a different length from the one the other two constants use.

For a two-bed fabric it would be a constant times the bed gap, plus a diameter — and the constant would be 1, with the gap being the measurement nobody has published.

Neither of those is a relaxation constant in Munden’s sense, because neither of them relaxes. That is the deeper reason the constants are silent: a relaxation constant is a statement about a dimension that relaxes, and the thickness is not one.

What this does not settle

Whether the thickness really is two diameters. That is a prediction from the interlacing and it has not been measured here. If it is wrong, everything above is about a different number and the argument’s shape survives.

What packing factor to use. The yarn’s diameter comes from its count, its fibre’s density and a packing factor, and the packing factor is a modelling choice with a range. The thickness inherits its uncertainty directly.

And whether the collapse covers anything else it should not. This rung found one silence by asking what a group divides out. The same question could be asked of every dimensionless quantity in the collection, and has not been.

Why the silence went unnoticed for so long

It is worth asking why nobody has said this before, because the answer is not that it is subtle.

Munden’s collapse is a plan result and every use anybody has made of it is a plan use: predicting a fabric’s stitch density, its shrinkage, its areal weight, its dimensions after finishing. Nobody has asked it about a thickness, because a thickness has never been derived from a knitted loop’s geometry — it has been measured when it was wanted at all.

So the silence was invisible for the reason most silences are: nothing was being asked. This collection could not ask either, until the loop had a third coordinate, and the first thing that coordinate produced was a thickness with no constant in it.

A model that produces a quantity for the first time also produces the first opportunity to notice what the old inputs did not cover. That is the general shape and it is worth carrying, because it says a new output is a reason to re-read the inputs rather than only to check the output.

What else a length would escape

The argument generalises within this collection and it is short.

Any quantity that is a length, or that carries a length that is not the loop’s, escapes the collapse. The thickness is one. So is the canopy depth of the hair standing off the yarn, which is set by the fibre’s own length and stiffness and has no loop in it at all. So is the staple length that decides how a fibre is anchored.

Three quantities that belong to the yarn rather than to the fabric, all of them lengths, none of them collapsed. And each of them decides something a wearer cares about — warmth, softness, shedding — which is a reasonable summary of why a fabric cannot be specified by its structure alone.

What is genuinely new here

Two statements, and the second is the one to carry.

Two knits with one tightness factor are one knit in plan and two thicknesses. For a realistic matched pair that is nearly a factor of two, and it carries straight into their warmth and their compression.

And a dimensionless group is silent about exactly what was divided out. Here that is a length and a stiffness — so the collapse covers shape, is silent about force, and is silent about thickness, for two different reasons that are the same reason.

What the pictures cannot show

The bar chart on this page is about one fabric, and the argument is about two. There is no figure of the matched pair, because a figure of two fabrics with identical shapes at different scales is two identical pictures — which is the point and makes a poor picture.

The section drawing shows where the missing length is, and it shows it at one scale. Nothing in it says that scale is independent of the loop length; that is a fact about the algebra rather than about the drawing.

What is worth taking away

Three sentences.

The collapse covers every ratio and every angle a knitted fabric has, exactly, and it covers no length and no force.

The fabric’s thickness is a length that does not scale with the loop, so it is outside — and so is everything computed from it, which is the fabric’s warmth, its compression and its bulk.

And a dimensionless group is silent about exactly what was divided out to make it, which is a sentence about groups rather than about knitting and is the reason to check the inputs whenever a model produces a new kind of output.

Where a course of yarn sits through the thickness. The through-thickness position of one course of yarn, half period by half period, for 3 structures at a bed gap of 3 yarn diameters — 0.501 mm — on a 20 tex yarn whose diameter is 0.167 mm. Single jersey never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. A one-by-one rib crosses between the beds 4 times a course, travelling 0.501 mm through the thickness. A two-by-two rib crosses between the beds twice a course, travelling 0.668 mm through the thickness. The horizontal is a count of half periods rather than a length, because the repeats are not the same width and the comparison is not about their widths.
Fig. 6 Three structures’ traverses through the thickness, which is the quantity the constants are silent about. A jersey oscillates by one diameter, the ribs cross the whole gap — and the loop length is identical in all three. Everything Munden’s arithmetic knows is in the plane.

What a completed set of constants would look like

If somebody wanted to extend Munden’s table to cover a knitted fabric’s third dimension, this rung says what they would have to add and it is not a constant.

For a single-bed fabric: twice the yarn’s diameter, which needs a measured diameter rather than a relaxation experiment, and which does not change between states.

For a two-bed fabric: the bed gap plus a diameter, where the gap is a relaxed measurement nobody has published and which would need the same experiment the plan constants came from — knit, relax, section, measure.

So half of the completion is a microscope and half of it is a research programme, and the two halves are unequal in a way that says where the cheap improvement is.

Which rungs this stands on

The collapse itself, at two knits with one tightness factor are one knit, which asserts the agreement to fifteen figures and states in its own last paragraph that the group has no stiffness in it.

The thickness, at how thick a knit is, which supplies the length the group has none of.

And the constants themselves, at a knit’s dimensions come from its loop, which are measurements of a plan and were never anything else.

The argument is a subtraction rather than an addition: everything here was already in those three, and what is new is noticing what the intersection leaves out.

Where the ladder goes next

The one dimension the constants do not reach is also the one that does not need a relaxation state, and the two facts have the same cause: a state is a thickness too.

And a fabric with a thickness and an areal weight has a bulk density, which is a quantity the trade does not specify and which separates fabrics the collapse cannot: how dense a knitted fabric is.

Where this goes next

A loop is bent about as hard as its yarn allows. The tightest curvature anywhere on a relaxed loop, against the knitter's own tightness factor, in units of one over the yarn diameter — which is the curvature of a yarn wrapped hard round another of the same size, and the tightest bend any fabric asks for. Across the whole range a knitter can reach it stays between 0.73 and 1.27, crossing one at a tightness factor of about thirteen — which is where the trade's own usable band begins. Nothing arranged that. The only things imposed are the loop length, the yarn diameter and the two measured spacings, and the curvature is whatever the minimisation returns. Two knits with one tightness factor are one knit The loop model has exactly one dimensionless group in it — the yarn's diameter over the loop length — so two fabrics that share it have the same loop, to fifteen figures, whatever they are made of. That group is the knitter's own tightness factor, and it explains why an index quoted as empirical works as well as it does. A knit's dimensions come from its loop length. Courses and wales per centimetre against loop length, for a plain weft-knitted fabric in one relaxation state. Neither axis carries a yarn count, a fibre or a machine gauge, and that is the finding: every plain knit measured sits on these two curves whatever it is made of. A knit's dimensions come from its loop A relaxed plain knit's courses, wales and stitch density depend on the loop length and on nothing else — not the yarn count, not the fibre, not the machine gauge. The constants are measured rather than derived, and the interesting thing about the published set is that it does not quite satisfy its own arithmetic. The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports. How thick a knit is Two centre lines pass at a diameter and each has a radius on either side, so a plain jersey is two yarn diameters thick with nothing fitted. It does not depend on the gauge, it lands inside the band of this collection's woven cloths, and it is lower than any gauge will read.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AnisotropyAreal densityCloth thicknessKnit geometryLoopLoop lengthMunden constantsSpecificationTightness factor