Textile Structure
how cloth is put together
Concept

Burnside — where it appears

The counting argument that turns a group acting on a set into a count of orbits, used here to count cloths where a sweep has counted drafts. Applying it needs the group and the fixed points, both of which come out of the drafts themselves rather than out of a table.

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

How many four-by-four weaves there are. The same census counted four ways. A draft is a notation; shifting the repeat's origin, turning the cloth over and turning it end for end all change the matrix and not the fabric. Each bar is the number of distinct objects left once those identifications are made, counted by canonical form and checked against Burnside's lemma.

How many cloths are there

Twenty-two thousand eight hundred and seventy-four four-by-four drafts. Four hundred and twenty-six four-by-four cloths. And the site's own separation rate — the fraction of drafts that look like fabric and are not — more than doubles when fabrics are counted instead of notations, which is the opposite of what was expected.

pattern · Pattern 2/2 twill: written at 4×4, repeating on 4. On the left the draft as written, with the smallest rectangle that tiles it outlined. On the right the same draft with every translation that leaves it unchanged marked from the top left corner: there are 4 of them, so a fundamental domain holds 4 intersections rather than the 16 the rectangle claims and the 16 the point paper carries. Nothing in the drawing on the left says so.

What a repeat repeats

Point paper is ruled in squares, and the rectangle a draft is written on is a decision by whoever drew it. The unit a cloth actually has is smaller — two intersections for a plain weave, four for a 2/2 twill — and it is not a rectangle at all.

cloth · Repeat

Named alongside it

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

OrbitPlane groupRepeatSymmetryCensusCloth integrityFundamental domainPlain weavePoint paperSatinTranslation lattice

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