Draft — where it appears
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
The draft is a matrix
Point paper is not a diagram of cloth. It is a grid of yes-or-no decisions about which threads to lift, which makes a weave a binary matrix and almost every question about it arithmetic.
A stripe is a partition of the warp
A striped cloth is not a weave with decoration on it — it is one matrix in which different bands of ends obey different rules. The shaft count is a set union rather than a sum, and of forty-five pairs of standard weaves only three share a single column.
A proof plan has no slide in it
A threading error that a 2/2 twill hides is invisible in the cloth and still wrong on the loom, and it shows only when the loom is re-pegged. Re-weaving every silent error under eight other plans says which do the showing. The twill run the other way and plain weave never expose one. A 1/3 twill exposes 44 per cent, a broken twill 70, and a plan of eight unrelated rows all but 864 — and those 864 are not errors at all, because each is the right threading started at another end. So the plans nest: whatever a broken or a point twill hides, the 2/2 hides too, and a threading proved under a 2/2 twill has passed the weakest test there is.
A blind cell costs half the cloths at any real repeat
At four by four, spreading a blind set over more threads kept half as many cloths apart again as gathering it. At eight by eight the same rule holds, and it is almost nothing: three per cent for four blind cells, eight for eight. The reason is one factor. A blind set's surfaces are the count's bound, 2^(n² − b), times the chance that every thread it misses interlaces on its own, and at eight threads a thread lies all on one face once in 128 tries. So shapes that touch the same number of threads tie exactly, whatever their rectangles; every blind cell costs very nearly half the catalogue; and by twelve threads the arrangement is free.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
CensusInterlacingPoint paperRepeatSymmetryThreadingBlind intersectionCloth integrityColour-and-weaveColour orderColumn setDerivation