Concept

Counterchange — where it appears

The operation that turns every warp mark of a draft into a weft mark and every weft mark into a warp one — the cloth seen from its other face. It preserves the shaft count, the float lengths and the interlacing rate, which is why a damask's figure and ground cost the same.

Named by 2 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

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

Named alongside it

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

Catalogue Derivation Orbit Basic weaves Census Closure Enumeration Hopsack Repeat Satin Twill Weave matrix

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