Textile Structure
how cloth is put together
Concept

Course — where it appears

A row of loops running across a knitted fabric, which is the knit's counterpart of a pick. Its spacing is a constant times the loop length, and the ratio of it to the wale spacing is what makes a loop taller than it is wide.

Named by 24 essays across 2 fields — each of them below, with the objects they name alongside it.

The knitted loop. One thread, bent into a course of loops, each of them drawn through the loop below. Nothing here is straight, which is why a knit extends in every direction while a woven cloth extends only on the bias.

The loop

A knit is one thread bent into loops, each drawn through the loop below. Nothing in it is straight, which is why it stretches in every direction while a woven cloth stretches only at an angle.

knits · Knit half-cardigan as an array. One cell per needle per course, over 2 repeats each way. K knits a new loop and casts the old one off, T tucks the yarn into the loop below without casting off, M misses the needle and floats past it. The margins carry the two conditions: a wale with no K in it never casts off, and a course with no K in it is never caught by anything. Both hold here.

Knit, tuck and miss

A weave is a matrix over two symbols and a weft knit is a matrix over three. The site's central question survives the translation intact: a weave falls apart when its above-and-below relation is disconnected, and a knit falls apart when a needle never knits.

knits · Stitch notation Yarn per needle, by structure. Wale spacings of yarn per needle position, at one loop length. A knitted loop is 4.30 of them — Munden's constant, measured rather than derived — a float across one needle is exactly one, and a tuck is taken as a stated multiple of a loop. Everything else follows by counting, so the percentages beside the bars are not estimates.

What a tuck costs

A knitted loop is about four wale spacings of yarn and a float across one needle is exactly one, so replacing a knit with a miss removes three quarters of a loop. That much is a count. What it does to the fabric's size is not, and this essay is careful about which is which.

knits · Stitch notation tubular, as a graph of its wales. Every wale of tubular as a node — 2 on the front bed and 2 on the back — with one chain per course joining everything that course takes yarn on, because a course is one traverse of one yarn. The nodes are filled by which component they fall into. This structure comes out as 2 fabrics: F0+F1 and B0+B1. The same answer is obtained a second way, by walking every partition of the wales and asking whether any course straddles it, and the two are required to agree.

Does a double jersey hang together

Two beds knitting with nothing passing between them are two fabrics that happen to have been made at once. Of 6,561 two-bed arrays, 1,135 are fabrics and 50 of those are two fabrics — and 28 of the 50 split across the beds rather than along them, so neither half is a layer.

knits · Two-bed 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 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.

finishing · Knit geometry The loop that costs nothing to extend. A plain knitted loop at rest and extended by 35 per cent, with the arcs marked. The arcs' radius is the diameter of the yarn the loop wraps, 0.167 mm, and it is set by contact rather than by the fabric's dimensions — so extending the fabric lengthens the legs and bends nothing further. The bending energy is 0.0176 N·mm at both, and the model therefore asks no force at all for an extension a woven cloth would refuse. What the drawing cannot show is what a real knit's first few per cent do cost, which is friction and yarn flattening and is not a bending property.

A knit is soft because it bends

Ask the same energy question of a woven cloth and a knitted one and the answers are not different by a factor — they are different in kind. A woven cloth's bending energy changes the moment it is extended. A knitted loop's does not change at all, exactly, over the whole of its extension, because its arcs are held to a radius by contact rather than by the fabric's dimensions.

knits · Knit The weight against the constant nobody has measured. Areal weight against the stitch-density constant, for four two-bed structures at 20 tex on a 0.35 cm loop with a tuck taken as 1.15 of one. Every line is exactly straight through the origin, because the weight is tex times yarn per repeat times k_s divided by the area of the repeat and there is no fitted constant anywhere in that division. The three vertical rules are the only measured values this site has — Munden's published k_s for plain single jersey in three relaxation states — and none of them is the right value for any structure drawn here. Where each line should be read is the whole of what is missing.

The constants do not compose

The yarn in a knitted structure is a sum of what each element takes, exactly, over structures nobody has measured. The size the structure relaxes to is not a sum of anything — and the whole gap between the two is one number per structure, which would cost forty-five fabrics each to obtain.

finishing · Knit geometry Which knitted structures spiral, at a twist factor of 4.0. A yarn leaves the spinning frame with a torque it has not been allowed to release, and a loop knitted from it leans. The lean per unit of twist factor above balance is measured; what is counted here is the structure. A loop on the front bed and one on the back are mirror images, so their torques have opposite signs, and a fabric that knits equally on both beds nets to zero whatever the yarn is doing — which is why 1x1-rib, 2x2-rib, interlock do not spiral and plain, half-cardigan, tubular do. The count has to be made per fabric and not per structure: an interlock and a tube both knit equally on the two beds, and they are opposite cases, because an interlock's two components each straddle the beds while a tube's are each wholly on one. The integrity criterion, which asks what nothing holds together, is what tells them apart. What the bars cannot show is the tube's second face, which leans the other way.

A jersey leans because its yarn still turns

A single-jersey T-shirt comes back from the wash with its side seam spiralling round the body, and a rib does not. The difference is not the yarn: it is a count. Loops on opposite beds are mirror images, so their torques oppose, and a fabric that knits equally on both nets to zero whatever the yarn is doing.

knits · Spirality The same error twice: scattered, and in a period. Two bands of 96 ends, drawn at spacings that differ from the reed's by the same root-mean-square amount — 5% of a dent, which is 21 µm. The upper band's errors are independent; the lower band's repeat every 4 ends, which is what one shaft set forward or one dent of the reed too wide produces. The displacement is drawn 8 times over scale, because at true scale it is half a pixel and both bands are picket fences; the arithmetic below it is at the true amplitude. The upper band reads as an even cloth with a little texture in it and the lower one has stripes, and nothing about the eye is needed to say why: at 96 ends the periodic arrangement is 7.8 times stronger at its own frequency than the scattered one is at any frequency, and over the 256 ends a buyer takes in at once it is 12.8. The ratio grows as the square root of how much cloth is looked at, which is why a fault-finding sweep is done at a distance rather than close up.

A course is one thread and a warp is many

A woven fabric draws its warp from two thousand packages side by side, so a yarn's drift averages out across the width. A weft knit takes whole courses from one package, so the same drift becomes a band — and the standard remedy for that turns an invisible error into a visible one.

knits · Knit The stripe a knitting machine chooses. A circular machine takes its yarn from a fixed number of feeders arranged round the cylinder, and feeder k lays every F-th course. So any difference between packages — a shade, a count, an evenness — is reproduced in the fabric with a period of exactly F courses, and at 20 courses per centimetre that is a band every 48.0 mm on a 96-feeder machine. The machine chooses the period, not the yarn. The spacing is proportional to the feeder count and inversely proportional to the course density, as the turn of the cylinder requires whatever the bars show, and a one-feeder machine produces no band at all from the same packages.

Why a knit shows a thick place

A woven cloth has hundreds of separate warp ends and averages a yarn's faults among them. A knit has one thread and a machine that repeats — so a difference between two packages becomes a stripe, and the machine chooses its period.

knits · Variation 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 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.

A rib climbs a gap

A jersey's yarn crosses one diameter between interlacings because that is what a crossing of two threads is. A rib's crosses the whole distance between the two beds. Nothing else in the model changes, and that one length is the whole mechanical difference between the fabrics.

knits · Two-bed How much of a fabric's yarn crosses between the beds. The share of half periods that cross from one bed to the other, read off each structure's own traverse rather than quoted. It is the mechanical difference between these fabrics in this account: a half period that crosses climbs the whole bed gap and one that does not climbs a yarn diameter. Single jersey and a tubular fabric come out at zero — the tubular one because its two faces are made on separate courses and never meet — and a one-by-one rib comes out at one, with every sinker loop crossing. A two-by-two rib is at a half, which is the number a reader would guess and is here counted.

Where a two-bed fabric's yarn is

Thirteen named structures, and for each of them the share of its yarn that crosses between the beds — read off its own traverse rather than quoted. It separates the fabrics into three groups, and one of the groups turns out not to be a fabric at all.

knits · Two-bed How much of a fabric's yarn crosses between the beds. The share of half periods that cross from one bed to the other, read off each structure's own traverse rather than quoted. It is the mechanical difference between these fabrics in this account: a half period that crosses climbs the whole bed gap and one that does not climbs a yarn diameter. Single jersey and a tubular fabric come out at zero — the tubular one because its two faces are made on separate courses and never meet — and a one-by-one rib comes out at one, with every sinker loop crossing. A two-by-two rib is at a half, which is the number a reader would guess and is here counted.

A tube and an interlock balance for different reasons

Two structures come out identically flat on a signed count and are as unalike as two knitted fabrics get. One is two jerseys that curl in opposite directions and are joined at the edges; the other is a fabric whose every course crosses. Telling them apart needs a different question asked of the same grid.

knits · Two-bed Two courses of a knitted tube, and the number between them. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, wrapped onto a tube 12 wales round, drawn with the course below it. Each is a closed curve, which is what a linking number needs. The Gauss integral over the pair returns 0.0000 — zero, to four places. In a knitted fabric the answer is 12: every needle loop of one course is drawn through the loop below it. The model puts the interlacing at a point where two centre lines pass one yarn diameter apart, and two curves that pass beside one another are not linked however close they come.

A jersey's course has no writhe

The mechanism everybody quotes for why a hard-twisted jersey leans is that the fabric relieves the yarn's twist by writhing. This collection's own solved course has a writhe of minus six parts in a million, and it is not small — it is exactly zero, by a symmetry, and the symmetry is a statement about what the model left out.

knits · Topology What links what, in the two ways of making cloth. The linking number between two adjacent courses, for a knitted tube of 12 wales, for the same tube as this collection's model draws it, and for a woven cloth's two thread systems. The fabric's is 12 — one for every needle loop drawn through the loop below. The model's is -0.0000, because it places the interlacing at a point where two centre lines pass a diameter apart and two curves passing beside one another are not linked. The woven cloth's is -0.0000 and always will be, at any crimp and for every weave. That last row is not a defect of any model: a woven cloth really is unlinked, and it is the reason it frays where a knitted fabric runs.

A point cannot link

A knitted fabric of n wales has a linking number of n between every pair of adjacent courses. This collection's model of the same fabric has zero, and it has zero because the interlacing was declared to be a point where two centre lines pass a diameter apart — which is a near miss, and a near miss is not a knot.

knits · Topology 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.

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.

knits · Contact The course helix of a 30-inch machine with 96 feeders, unrolled. A knitted tube from a 30-inch machine with 96 feeders unrolled flat, one round wide, with its wales drawn straight along it and its courses climbing across it. The machine lays 96 courses in each turn, so each course climbs 96 course spacings — 48 mm of fabric — in one round of 2,262 wales, 162 cm: an angle of 1.70° from the tube's cross-direction. The course drawn heavy is followed round one turn. The vertical scale is exaggerated 6 times. What the drawing cannot show is the hand of the helix, which is set by the direction the cylinder turns.

A circular machine leans its courses whatever the yarn

Spirality is blamed on the yarn, and the yarn is most of it. The rest is the machine's. A circular knitting machine's needles make wales that run straight along the tube, and its feeders lay one course each per turn, so every course climbs its feeder count in every round: on a 96-feeder machine, 48 millimetres round 162 centimetres of tube, an angle of 1.7 degrees. The helix has no machine size in it, its hand is set by the way the cylinder turns, and it survives every remedy aimed at the yarn — a steamed yarn, a plied one, S and Z on alternate feeders, and a rib.

knits · Spirality 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 Where a knitted disc is flat. The circumference a knitted disc grows per unit of radius, as a share of the 2π a flat disc needs, against the loop's aspect, for 4 wedges (flat at 1.2732), 10 every 2 rounds (flat at 1.2566). Wedges of short rows fall as the aspect rises and rounds with increases rise. The shaded band is the aspect of plain knit between its dry-relaxed value, 1.2500, and its wet-relaxed value, 1.2927.

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 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 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 A ball knitted in rounds with its increases stacked in 5 lines. The shape a ball of plain knit takes unstuffed when its increases, 5 at a time, are stacked in 5 lines from pole to pole, with the same stitches in every round as a sphere. Every stitch between two lines is flat knit, so the 5 panels are flat and the one shape they close up into without stretching has 5 flat sides in every round and 5-sided points at its poles: 2.42 sphere radii tall, 2.14 across its ridges and 1.73 across its flats, so 1.13 to 1.40 times as tall as it is wide. The knitted counts of a 40-round ball give 1.13 to 1.40, against 1.02 for the same counts with the increases spread round, whose rounding is its own. What the drawing cannot show is stuffing, which would stretch the panels towards round.

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.

WaleLoop lengthLoopMunden constantsLoop aspectRelaxationShapingInterlockTwo-bedCloth integrityConnectivityElastica

All concepts