Textile Structure
how cloth is put together
Concept

Euler's totient — where it appears

The count of whole numbers below one that share no factor with it, which decides which satins exist and which outlines can be drawn. It is an arithmetic function evaluated on the repeat, so which satins exist is settled before any yarn is mentioned.

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

Which satins exist. For each order, the moves that give a satin — a step coprime with the order, and not the steps of one and one less which give a twill instead. Four and six admit none, so no regular satin exists on four ends or on six.

There is no satin on six ends

Weavers have known it as a rule for centuries. It is a theorem with a one-line proof about common factors, and it rules out four ends as well.

weaves · Satin The staircase a woven outline is. Four straight edges on a block grid, stepping 1 across in 1, 1 across in 2, 2 across in 1, 1 across in 4. Each tread is one repeat of 8-end satin, which at 24 by 22 threads per centimetre is 3.33 mm across and 3.64 mm up. A jacquard hook at 140 cm width is 1.17 mm, so the machine resolves 2.9 times finer than the cloth can use.

A woven outline is a staircase

A jacquard's resolution is quoted as its hook count, and on a 1,200-hook machine 140 cm wide that is a step of 1.17 mm. The cloth cannot use it. A figure's smallest feature is one repeat of the ground weave, which on an eight-end satin is 3.33 mm — so the machine resolves nearly three times finer than the fabric can hold, and a finer machine buys nothing at all.

pattern · Blocks

Named alongside it

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

Block figureCoprimeDamaskFareyFigure and groundJacquardMove numberProfile draftRegular satinRepeatResolutionSatin

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