Textile Structure
how cloth is put together
Concept

Two-colour symmetry — where it appears

A cloth whose threads carry two colours, so every intersection shows one of the two. The pattern a reader sees is a function of the weave and the two colour orders together, and how much of the weave that surface reveals depends only on how many threads of each colour the two systems carry.

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

A draft is a two-coloured pattern. For each weave, the symmetries that leave warp-up as warp-up and those that exchange the two. Only a balanced weave has any of the second kind, because exchanging warp and weft is only a symmetry when there is as much of one on the face as the other.

Weaves as plane patterns

A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.

pattern · Pattern What a pair of colour orders does to the catalogue. Dark ends in the warp across against dark ends in the weft down, for the sixteen two-colour orders at four ends. Each cell gives the blind intersections of sixteen and the number of distinct surfaces the 22874 drafts collapse onto. Every cell stands for between one and thirty-six colour orders and they all behave identically, so the arrangement of the colours does not matter and only their counts do. The corner at four against nought is blind nowhere and separates every draft.

The colour order that hides least

A caption on this account's top essay left a question: whether a warp of long runs against a weft of short ones hides less than either would against itself, and called it a lever nobody uses. Sweeping all sixteen orders against all sixteen says the lever is real and the caption named the wrong variable. Run length has nothing to do with it — the blind count, the separation and the largest confused class all depend on the two orders' colour counts and on nothing else, and what a mixed pair saves is exactly the square of the difference between them.

pattern · Pattern

Named alongside it

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

AppearanceBalanceBlind intersectionCensusColour and weaveColour orderEnumerationPlane patternRepeatSymmetry

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