Textile Structure
how cloth is put together
Concept

Gaussian curvature — where it appears

The product of a surface's two principal curvatures, which is zero exactly where a flat sheet can take the shape without stretching. Its sign decides whether draping a cloth closes the cells or opens them, and getting the sign backwards would draw a fabric that gathers where it should stretch.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

A cloth laid over a sphere. Every thread segment is exactly one pitch and none has stretched. What has changed is the angle at each crossing, and the amount is decided by the surface — a developable one costs nothing and a curved one costs more the further the cloth goes.

Why clothes need darts

A flat cloth goes round a cylinder for nothing and cannot go round a sphere at all without shearing. The amount of shear is decided by the curvature, and when it exceeds what the threads allow, something has to be cut out.

mechanics · Drape A cloth laid over a sphere. Every thread segment is exactly one pitch and none has stretched. What has changed is the angle at each crossing, and the amount is decided by the surface — a developable one costs nothing and a curved one costs more the further the cloth goes.

Shear locking in a composite preform

Laying a woven reinforcement over a mould is the bias mechanism doing useful work, and it stops dead at the angle where the threads jam. Where the cloth wrinkles is a geometric prediction with a radius attached.

mechanics · Bias How much of the curvature a cloth can take without being cut. The dart angle a spherical cap still demands after the fabric's own shear has absorbed what it can, against how closely the cloth is set. The total the cap demands is fixed by Gauss–Bonnet and is the same for all of them; what changes is how much of it the trellis can supply before its threads jam. An open cloth drapes a hemisphere with no dart at all; a closely set one has to be cut from the start.

A hemisphere costs one full turn

The total angle a pattern must remove to cover a hemisphere is exactly 360 degrees, and it is the same for a hat and for a stadium dome. What changes with the fabric is how much of that the cloth can supply by shearing instead of by being cut — and that is a property of the sett, computed from the angle at which its threads jam.

applied · Cutting 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.

CourseLoop aspectMunden constantsShapingWaleRelaxationShort-rowDartDevelopableDrapeFishnetInextensible

All concepts