Textile Structure
how cloth is put together
Concept

Orbit — where it appears

All the ways one object can be written once a group of operations is allowed, whose size is what a census divides by to count objects rather than writings. Its size is the number of ways one object can be drawn, so counting arrangements and counting cloths are different questions with different answers.

Named by 6 essays across 3 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 The basic weaves at four by four. Plain weave and the three twills a repeat of four admits, each drawn with its longest float and its layer count computed from the matrix. The fifth frame is empty: a satin needs a move coprime with its order and neither one nor one less than it, and four ends admits 0 such moves. So the smallest interesting repeat contains two of the three weaves every manual begins with.

The three basic weaves do not generate the rest

Every weaving manual opens with the same sentence: there are three basic weaves, and everything else is derived from them. The complete catalogue of the smallest interesting repeat is in hand, so the claim can be checked instead of repeated. Starting from plain weave, every twill and every satin, and applying every derivation the manuals name, reaches nine of the 426 cloths that exist there.

weaves · Derivation How many cloths any one cloth derives into. The 426 four-by-four cloths sorted into the orbits the manuals' derivations cut them into. 12 orbits hold 1 cloth; 83 orbits hold 2 cloths; 62 orbits hold 4 cloths. The largest orbit in the whole catalogue holds 4, so no cloth derives into more than 3 others by any sequence of the named operations, however long. The derivations generate a group of 256 elements and it cuts the catalogue into 157 pieces. What the bars cannot show is which cloths are in which orbit, which is the next figure.

No cloth derives into more than three others

Every weaving manual opens by saying the three basic weaves generate the rest. This collection counted the reach and found nine of 426, and left the nine as a count. It is not a count: every derivation the manuals name is a relabelling of the grid or a complementation of it, both invertible, so they generate a group — and that group cuts the 426 cloths into 157 closed pieces of which the largest holds four. The claim is not merely wrong about how much derivation reaches; derivation cannot reach more than four cloths from anywhere, by any sequence of operations, however long.

weaves · Derivation How many derivation orbits each measure separates. For the 426 four-by-four cloths in 157 derivation orbits, the number of classes each invariant measure cuts the orbits into: marks, up to exchanging face and back, 5; interlacings, 8; layers, 2; plane group, 12; floats of both faces, both systems, 60; changes of face per end and per pick, 12; distinct ends and distinct picks, 4; all seven familiar measures, 120; census of two-by-two patches, 127; all seven, and the two-by-two patches, 153; census of three-by-three patches, 157. Only the census of three-by-three patches reaches 157. What the bars cannot show is which orbits a measure confuses, which the pair figure draws for the seven measures together.

A cloth's derivation class is its census of small patches

The manuals' derivations cut the 426 four-by-four cloths into 157 orbits, and an orbit is found by searching a group of 256 operations. The question left was whether a short list of numbers read off a draft could do the same job. The familiar ones cannot. Marks, interlacings, layers, plane group, float lengths, crossings per thread and distinct ends and picks are all invariant, and all seven together tell 120 of the orbits apart. A census of the two-by-two patches a draft contains tells 127 apart. A census of its three-by-three patches tells all 157 apart — a complete invariant of derivation, computed by counting windows rather than by searching operations.

weaves · Derivation Which window sizes name every derivation orbit, on every repeat small enough to walk. Every draft of eleven repeat shapes from three picks by three ends to five by five, every thread interlacing, cut into the orbits the derivations that keep a shape can reach — slides, reversals, the counterchange and, on a square repeat, the quarter turn — and tested with censuses of two-, three- and four-thread-square windows. 3 × 3: 3 orbits; window 2 names all, window 3 names all. 3 × 4: 19 orbits; window 2 confuses 2, window 3 names all. 3 × 5: 66 orbits; window 2 confuses 43, window 3 names all. 3 × 6: 365 orbits; window 2 confuses 328, window 3 confuses 12. 3 × 7: 1,680 orbits; window 2 confuses 1,649, window 3 confuses 211. 3 × 8: 8,988 orbits; window 2 confuses 8,949, window 3 confuses 2,324. 4 × 4: 157 orbits; window 2 confuses 49, window 3 names all, window 4 names all. 4 × 5: 2,875 orbits; window 2 confuses 2,153, window 3 names all, window 4 names all. 4 × 6: 36,380 orbits; window 2 confuses 34,439, window 3 confuses 204, window 4 names all. 4 × 7: 447,756 orbits; window 2 confuses 444,174, window 3 confuses 6,503, window 4 confuses 2. 5 × 5: 45,077 orbits; window 2 confuses 42,480, window 3 names all, window 4 names all. On every shape a window no wider than half the ends confuses some orbits, and a window wider than half names all of them but for one pair at four picks by seven ends. What the grid cannot show is a shape too large to walk, six by six among them.

The window that names a cloth is half its repeat

At four by four a census of three-by-three windows tells every derivation family from every other, and the question it left was whether three stays enough as the repeat grows. Walked whole over eleven repeat shapes, three is still enough at five ends — all 45,077 families of the five-by-five cloths apart — and fails as soon as the repeat reaches six. On every shape a window no wider than half the repeat confuses some cloths, and a window wider than half names them all, but for one pair of seven-end cloths that differ in a single end.

weaves · Derivation

Named alongside it

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

CensusDerivationRepeatSymmetryPlane groupBurnsideCatalogueCounterchangeSatinBasic weavesClosureCloth integrity

All concepts