Textile Structure
how cloth is put together
Concept

Derivation — where it appears

Making a new weave from an old one by an operation a manual names, which reaches far less of the catalogue than the manuals imply. How little of the catalogue it reaches is a count rather than an impression, and the shortfall is large at every repeat size tried.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 What combination adds to the manuals' reach. The three counts. The manuals' own operations on their own seeds reach 9 of the 426 four-by-four cloths. Admitting stripes, checks and figures of any two seeds — 17,787 constructions, of which 177 repeat inside four ends and four picks — takes it to 28, a gain of 19. That is a threefold rise and it leaves 398 cloths unreached, which is 93.4 per cent of the catalogue. What the chart cannot show is combinations of combinations, which are excluded on purpose: the closure's seeds have to be what a chapter actually teaches or the count measures something else.

What combining two weaves reaches

The account before it found that the manuals' own operations on their own basic weaves reach nine of the 426 four-by-four cloths, and recorded one exclusion honestly: combination — striping, checking and figuring — was left out, because a combination of two four-end weaves is eight ends wide and so is not a four-by-four cloth at all. Admitting it triples the reach and leaves ninety-three per cent of the catalogue outside.

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.

CensusOrbitRepeatCatalogueCounterchangeFloatSatinSymmetryTwillBasic weavesClosureCloth integrity

All concepts