Concept

Munden constants — where it appears

Four fitted numbers relating a relaxed plain knit's courses, wales and stitch density to its loop length and to nothing else. They very nearly satisfy the identities that relate them and do not exactly, which is what separately fitted regression constants look like.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

A fashioned edge at 1 wales in 2 courses. A knitted panel narrowing by 1 wale every 2 courses, drawn at the fabric's own aspect: a wale is 1.2791 times as wide as a course is tall, which is Munden's ratio of the two published constants. The edge therefore stands at 32.60 degrees from the wale, and that angle is the same in every yarn, at every gauge and at every loop length. Marks show where the 8 transfers fall.

A fashioned edge has a quantised angle

A knitted panel is shaped by transferring loops, so its edge steps by whole wales at whole courses and its angle is the arctangent of a fraction. The available angles turn out to be the same for every plain knit there has ever been — in any yarn, at any gauge, at any loop length — because the constant they scale by cancels the loop out. There are eighteen of them, and 16.67° between the last two.

knits · Shaping

How far a tube can be tapered by its loop. The same 10 wales by 10 courses of plain knit at the two ends of the usable loop range for a 20 tex yarn — 3.44 mm at the loose end and 2.80 mm at the tight one — drawn at a common scale in centimetres. On 240 needles the circumference falls from 192.0 cm to 156.0 cm, a taper of 18.8 per cent, and the fabric becomes 1.51 times as dense.

A tube can only be shaped by its loop

On a circular machine the needle count is the cylinder, so a seamless tube's circumference is its wale count times its wale spacing — and the wale spacing is the loop length over one constant. The loop is the only free quantity, the yarn bounds it at both ends, and what is left is a taper of 18.8 per cent bought at the price of a fabric half again as dense.

knits · Shaping

A plain knit's two relaxation steps. Munden's three relaxation states are usually given as three sets of constants. Read as a path they are two steps, and the two compose to the whole exactly — which is a real check, because the three sets were measured independently. The first step is the larger in the course direction and the smaller across the wales, and the second is 0.64 of the first lengthwise. That is the shape of a laundering series and it is the same mechanism: a fully relaxed state is reached by tumbling rather than by waiting, so what the standard specifies is a quantity of agitation and not a duration. What the bars cannot show is the loop length, which cancels out of all four numbers because every dimension of a knit is a loop length times a dimensionless constant.

A knit relaxes for as long as it is allowed to

Munden's three states are usually given as three sets of constants. Read as a path they are two steps, they compose exactly, and the second is not a smaller version of the first — the fabric shrinks twice as much along its courses as across its wales on the first step and rather less than half as much on the second.

knits · Knit

When a knit has a hole between its loops, and when it has none. A loop's occupancy is its length times its diameter over the cell it sits in, and Munden's constants make that cell ℓ²/(k_c·k_w). The loop length cancels once and what is left is d·k_s/ℓ, where k_s is Munden's own stitch-density constant — so whether a knit has a hole between its loops depends on d/ℓ and on nothing else: no gauge, no count, no fabric dimension. The three curves are the three relaxed states of 20 tex cotton. Each crosses one at a loop length of 3.34, 3.63, 3.95 mm respectively, and a jersey is knitted at 2.63 to 3.44 mm — the shaded band. Every commercial jersey is therefore on the wrong side of the threshold in at least two of its three states: it closes its own holes as it relaxes, and its air goes through its threads rather than between them.

A knit has no hole to lose

Every argument in this ladder is planar: threads at a spacing, a rectangle between four of them, a channel down it. Applied to a jersey it returns nothing at all, and the nothing is the finding. A loop's occupancy is its diameter times Munden's own stitch-density constant over its loop length, with no gauge and no fabric dimension in it — and the whole commercial range of tightness is on the wrong side of the threshold.

knits · Knit geometry

single jersey, as loops. Three courses of the same structure drawn as yarn. A knitted needle takes a new loop through the one below; a tuck takes the yarn into the loop below and leaves it on the needle, so that loop is held for another course; a miss floats straight past. Six needles is as many as a loop diagram can carry, which is why the array beside it exists.

A jersey has two surfaces

The face of a plain knit shows the legs of its loops, which run along the wale; the back shows the heads and feet, which run across it. So the two faces carry their crowns at right angles — the same situation as a damask's figure and its ground, in a fabric with no warp, no weft and no float.

knits · Knit

A knitted loop, solved rather than drawn. 3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings.

A loop is nine tenths free run

Half the yarn in a knitted stitch is slack — the straight line between two interlacings is a little over half the thread available to span it. That is two orders of magnitude more room than a woven thread has, and it is why a knitted loop is a shape that can be solved rather than a shape that has to be constructed.

knits · Elastica

Everything a plain knit can be, at one loop length. Bending energy over the two spacings a plain knit has to choose, for a 20 tex cotton yarn at a 3.5 mm loop, as a multiple of the energy the fully relaxed fabric holds. Darker is more. The solid edge is where the straight line between two interlacings reaches the yarn between them — the geometry's own limit, with nothing elastic in it — and there is no state beyond it at any force. Munden's three relaxation states are marked, and the thing to see is that they are not in a hollow: they lie along a slope, in order, with the most completely relaxed of them the highest. An unset yarn would slide down and to the right until it met the edge. Real fabrics sit where they were left.

The relaxed knit is not at a minimum

Differentiate a loop's bending energy along the fabric instead of across it and the answer should be zero, because a relaxed fabric is one nothing is pulling. It is not zero. It is tens of newtons a metre, downhill in both directions at once — and the three relaxation states everybody measures run the wrong way up the slope.

knits · Elastica

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.

finishing · Knit geometry

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.

A state is a thickness too

This collection's rule is that a fabric dimension quoted without its relaxation state is not a measurement. A knitted fabric has three dimensions and only two of them obey the rule: its thickness is the same in every state, because the interlacing that sets it does not relax.

finishing · State

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.

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.

finishing · Knit geometry $A knit's warmth is its thickness, and a rib's thickness is a machine setting. Thermal resistance in tog for a 20 tex cotton at a 3.5 mm loop, for a plain jersey and for a one-by-one rib at four bed gaps. The bar is the lower bound and the mark beyond it is the upper — Wiener's two bounds on a mixture of fibre and air at the fabric's own fibre fraction, which is 26.6% for the jersey and falls to 8.9% for the widest rib. The jersey holds 0.088 tog and the rib at five diameters 0.334 — a factor of 3.8, bought entirely by opening the beds. Every one of these is a small fraction of the still-air layer that clings to a garment's outside, which is worth about 1.2 tog on its own.$

How dense a knitted fabric is

A fabric's areal weight is what the trade specifies and it says nothing about bulk. Divide it by a thickness and the answer is a density — 0.40 grams a cubic centimetre for a jersey, a quarter of the fibre it is made of — and that quarter, the share of the volume that is not air, is the number every other property follows.

setting · Weight

A loop is a plane curve in another plane

A knitted loop was solved as a flat curve because two curves in one plane cannot pass through one another and a loop must. Letting it out of the plane turns out to change nothing about its shape: the loop is still planar, and its plane is the fabric's turned through twelve degrees.

mechanics · Elastica

A short-row heel on 64 needles, laid flat. The 32 heel stitches of a 64-needle sock, each column drawn as the courses it was worked in during the short rows: nought at the heel's two edges, rising by four courses a stitch to 42 across the 11 stitches left at the turn. The cells are drawn at the knit's own aspect, a wale 1.2791 times as wide as a course is tall. The instep's 32 columns beside it are worked in none of the short rows. Down the back line the extra length turns the tube through 92.4 degrees, and the whole trapezoid is 65.6 per cent of the fabric a true bend of that angle would need.

A heel turns a right angle because of the loop

A sock's heel is knitted in short rows on half the needles until a third are left, then back out again. Down the back line that adds a length of courses, and a tube's back line longer than its front by that much has turned through π times two thirds over the loop's own aspect — 93.8 degrees, on any needle count, in any yarn. The trade's third is the loop's shape in disguise. What the rule does not do is make a bend: the heel supplies two thirds of the fabric a true right angle needs, and every missing stitch is on its sides.

knits · Shaping

A knitted disc is flat at one shape of loop

A disc lies flat only if its circumference grows by exactly 2π per unit of radius, and a knitted disc's growth is a count of loops in one direction over a count in the other. So it is flat at one value of the loop's aspect and no other. Knitted sideways in wedges of short rows the growth falls as the loop gets wider; knitted outward in rounds it rises. Four wedges and ten increases every two rounds are each flat at an aspect inside the range plain knit moves through when it is washed — and they cross it in opposite directions.

knits · Shaping

Every count that can make 150 g/m² as plain jersey. Areal weight against count for plain jersey in the fully relaxed state: the shaded wedge is every knit between a tightness factor of 1.3 and 1.6, whose weights are kₛ times the tightness times the root of the count. The rule at 150 g/m² crosses it between 15.8 and 23.9 tex, a ratio of 1.51. A plain woven cloth of the same weight can be made from anything between 20 and at least 200 tex, a range of at least 10 to one against the knit's 1.51.

A knit's weight nearly names its yarn

A woven cloth's weight is one equation in four unknowns, and a hundred and fifty grams can be woven from anything between twenty tex and two hundred. A plain jersey's weight has the loop in it and nothing else to spare, and the loop is bounded by the yarn it is knitted from. Put the two together and the loop cancels: the weight is a constant times the tightness times the root of the count, so at one weight the count is fixed to within half again — and in each relaxed state it is fixed to a different half.

setting · Weight

One gore of a 4-gore knitted ball, laid flat. A gore of 30 stitches from pole to pole, knitted sideways in short rows, drawn flat with its rows stacked at the plain-knit loop's fully relaxed aspect of 1.2791. Turned on the sine, its 10 row pairs turn at stitches 0, 1, 2, 3, 4, 5, 6, 8, 9, 11 from each pole — one stitch apart near the pole and further apart towards the equator — and its edge follows the width a sphere asks for. Turned one stitch every pair, the same gore is a diamond of 15 pairs, 1.50 times as wide at the equator. What the drawing cannot show is the fabric smoothing its own staircase edge.

A knitted ball's short rows have to slow down

A knitted disc needs one count to meet 2π; a sphere needs a count that follows a sine. Knitted sideways in gores, a ball's short rows must turn one stitch apart at the pole and ever further apart towards the equator — evenly spaced turns knit two flat discs joined at the rim. The pole is the disc again, so gores come in fours; the equator's row pairs have to come out whole, so only some sizes knit round; and a wash moves a four-gore ball from a ruffled pole to a round one.

knits · Shaping

A jersey's bending length against its tightness. The wale-wise bending length of plain jersey in the fully relaxed state at 10, 20, 40 tex, across the whole tightness band from 1.3 to 1.6, at the two ends of the yarn's stiffness bracket. At the free bound every count and every tightness gives 10.13 mm. At the locked bound the lines separate by count — 55 mm at 10 tex, 70 mm at 20 tex, 88 mm at 40 tex — and are still flat. The weights along the lines run from 97 to 239 g/m². What the plot cannot show is friction between the loops, which is where a tight jersey's firmness has to come from.

A jersey's drape does not know its loop

A woven cloth's bending length at the free bound turned out to hold neither its count nor its weight — only the fibre. A plain jersey goes further. Its stiffness per width and its weight per area are both a number of loops per millimetre times something about one loop, so the loop cancels at every stiffness the yarn could have; at the free bound the count cancels too, and what is left is the fibre and which relaxed state the fabric is in.

setting · Weight

A ball knitted in rounds, count by count. The stitch count of each round of a ball knitted outward from one pole in 40 rounds, from the pole to the equator, in the fully relaxed state, against the count a sphere asks for. Increasing 5 at a time it reaches 65 at the equator, increasing on rounds 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 14, 16; Increasing 10 at a time it reaches 60 at the equator, increasing on rounds 3, 5, 8, 11, 15. What the chart cannot show is where round the ball the increases fall, which spirals or stacks them and is a choice of the pattern.

A ball knitted in rounds washes the other way

A ball can be knitted sideways in gores or outward from one pole in rounds, and both have to follow the same sine. But a gore counts its meridian in stitches and its circumference in rows, and a round counts them the other way, so the loop's aspect sits underneath one construction and on top of the other. A wash raises the aspect — and closes a gored ball's ruffled poles while it opens a round-knitted ball's.

knits · Shaping

Stacked increases knit a ball with flat sides

A ball knitted in rounds can put its increases anywhere in each increase round. Stack them in lines from pole to pole and every stitch between two lines has a flat knit's neighbours, so the ball is made of flat panels and all its curvature sits on the lines. Flat panels with straight rows close up without stretching in exactly one way: every round a regular polygon, each pole a point. At five lines that ball is 13 per cent taller than it is wide across its ridges and 40 per cent across its flats, and to make it round the stuffing has to stretch the middle of every panel by π²/4m² — nine per cent at five lines, less than a wash moves a course at ten.

knits · Shaping

Named alongside it

The objects these essays reach for when they reach for this one.

Loop length Tightness factor Relaxation Wale Course Shaping Loop aspect Loop Cloth thickness Elastica Gaussian curvature Short-row

All concepts