A weave holds its selvedge with two of its own columns, or not at all

A selvedge holds only where its edge end changes face found the one question a weft turn asks. Between two picks the weft turns round the outermost end, and the turn is caught only if that end is on the other face on the second pick from the first. Asked of every four-by-four draft at every placement of its repeat in the width, the answer was bleak: four drafts in ten cannot hold a selvedge at any width, and only one in seven holds at a width of whole repeats, the width a cloth usually has.

That count treated the edge end as whatever column of the repeat happened to fall at the edge. A weaver is not bound to that. An extra end at each edge can be entered on any shaft the loom has, and the question becomes which shafts would serve. The answer turns out to depend on the weave’s set of columns and on nothing else — not the width, not the arrangement, not the other edge — and it divides the drafts that can be rescued by two re-entered ends from the ones that need a shaft of their own.

The two edges ask different questions #

Throw the first pick from the left. The weft reaches the right edge at the end of pick one and turns; it reaches the left at the end of pick two; the right again at the end of pick three. The right edge turns at every odd boundary and the left at every even one — between picks one and two, three and four at the right; between two and three, four and five at the left.

So the two edges ask different questions of the end standing at them. An end at the right edge must change face across the odd boundaries and may do what it likes across the even ones, because no turn happens there. An end at the left edge must change face across the even boundaries. Plain weave’s ends change face across every boundary and answer both. Almost every other end answers one question or neither.

The hero figure is the sharpest case. A hopsack lifts its ends in pairs for two picks at a time: two up, two down, so each end changes face every second pick. Its changes all fall on the same boundaries, between picks two and three and between four and one — the even ones. Every end of a hopsack can hold the left edge and none can hold the right. Thrown from the left, the left selvedge is perfect and the right one slips at every turn, whatever width the cloth is cut to. Thrown from the right, the two exchange.

Sixteen columns, four kinds #

Over a repeat of four picks an end has sixteen possible columns — sixteen ways of being up or down on each pick — and each belongs to one of four kinds.

Two columns hold either edge: plain weave’s, up on alternate picks. Two hold only the right edge, changing face between picks one and two and between three and four; two hold only the left, changing between two and three and between four and one. Ten hold neither. The ten include the two that never change at all, and every column with a float of three, which the earlier essay proved must let a turn slip at both edges.

This is the whole of the selvedge question at four picks. A weave can hold both selvedges if and only if its columns include one that holds the right edge and one that holds the left — or a plain one, which counts as both. The width decides only which of its columns happen to stand at the edges. Given one of each kind somewhere in the repeat, a width can be chosen that puts them there, and if the width is fixed, the right ends can be put there by entering them.

Two re-entered ends rescue three drafts in five #

That turns into a census at once. For every four-by-four draft, ask whether its columns include one of each kind.

The criterion reproduces the earlier census exactly. Every one of the 13,238 drafts that hold at some placement of the width has a column of each kind, and none of the 9,636 that hold at no placement does. The count of placements was a count of which pairs of columns fell at the edges; the kinds say which pairs could.

So re-entering two ends turns the census round. A cloth woven at a width of whole repeats holds its selvedges if it happens to have the right kinds at its first and last columns; 3,262 drafts do. Add an end at each edge, threaded on a shaft whose column is of the right kind, and every one of the 13,238 holds at a width of whole repeats — the other 9,976 need only that. The cost is two heddles and the weaver’s attention to which shaft each goes on. No shaft is added and no selvedge weave is drafted.

A 2/2 twill is the everyday example. Its four columns are two of each kind, so at a width of whole repeats it holds from one starting side and slips everywhere from the other, as the earlier essay found. With an extra end at the left on the shaft of its second column and one at the right on its first, it holds from the left at any width.

The 9,636 lack a kind #

The drafts that cannot be rescued this way split three ways, and the split has a symmetry in it.

3,614 lack a column for the right edge, 3,614 lack one for the left, and 2,408 lack both. The two one-sided counts are equal because reversing the picks of a draft exchanges the odd boundaries with the even, and so exchanges the two kinds: every draft that holds only its left selvedge has a reverse that holds only its right. The hopsack is in the first group or the second depending on which way its picks are counted, and it is the named example of both.

The 2,408 that lack both are the drafts whose every column misses a turn at each edge. They include the 3/1 and 1/3 twills, every one of whose ends floats three picks, and every draft whose ends all change face on boundaries that suit neither edge. For these, and for the one-sided drafts at their bad edge, no end the draft can lift will hold, however it is entered.

An odd repeat has no edge column at all #

Every count so far is over four picks, an even number. With an odd number of picks the question changes character, and the answer is absolute.

A 2/1 twill has three picks to its repeat. Thrown from the left, the weft turns at the right edge after picks one, three and five and at the left after two, four and six. Over two repeats, every boundary of the repeat is a turn at both edges once: after pick three the weft turns right, and pick three’s boundary is the repeat’s last, which after the next repeat is a turn at the left. So an end that is to hold either edge must change face at every boundary of the repeat.

No column of odd length can do that. A column that changes face at every boundary alternates up, down, up, down; round a cycle of three it arrives back at its start after three changes, which is an odd number, and it cannot be both up and down there. Every column of every odd repeat misses at least one boundary, and the figure marks where each of the 2/1 twill’s three does. The same is true of the five-end satin, the seven-end, every twill with an odd repeat, and every draft of odd pick count whatever its ends: four hundred random odd repeats of three, five and seven picks were tried and none had a single column of either kind.

So a weave with an odd number of picks to its repeat can never hold its own selvedge, at any width, with any entry of any of its own ends. The earlier essay found that the 2/1 twill holds at none of its thirty-six placements and the five-end satin at none of its twenty-five. The theorem says those were not unlucky weaves. No odd repeat has a chance.

One shaft lifted on alternate picks holds any weave #

The drafts that lack a kind, and every odd repeat, need an end the weave does not have. The cheapest one is an end whose column is plain: up on one pick, down on the next.

A single extra shaft, lifted on every other pick, holds both edges of any weave, because a plain column changes face at every boundary and so holds whichever edge it stands at. Both edge ends can be on the same shaft; they lift together, and each changes face at every boundary regardless. The weave’s own plan is not touched.

For an odd repeat the price is a longer plan. The extra shaft’s column repeats every two picks and the weave’s every three or five, so the loom’s plan must run to their least common multiple: six picks for a 2/1 twill, ten for a five-end satin. On a dobby that is a few more lags or a longer card, and what a dobby stores counts exactly that — the pattern chain is as long as the plan, and a selvedge shaft can double it.

The other way out, which the earlier essay described, is a floating selvedge: an extra end at each edge threaded through no shaft, which the weaver passes the shuttle over and under by hand. It costs no shaft and no plan, and it costs a hand at every pick, which is why it belongs to hand looms and a selvedge shaft to power looms.

What the count means at the loom #

The practical content is a decision tree, and every branch is now decidable from the draft.

First, list the draft’s columns by kind. If there is a plain column, any end on it holds either edge, and the question is closed. If there is a column of each one-edge kind, enter an extra end at each edge on a shaft of the kind that edge needs, and the selvedges hold at any width; which edge is which is decided by the starting side, so the weaver must throw the first pick from the side the entry assumed.

If a kind is missing, the weave cannot hold that edge with its own shafts. On a loom with a spare shaft, one shaft lifted on alternate picks takes both edge ends. On a loom whose shafts are all used by the weave — a four-shaft twill on a four-shaft loom — only a floating selvedge is left.

And if the repeat is odd, both kinds are missing by necessity, and the plan will double when the selvedge shaft is added. A designer choosing between a 2/1 twill and a 2/2 twill for a cloth whose edges matter is choosing, among other things, between a selvedge that costs two heddles and one that costs a shaft and twice the pattern.

The float theorem, from the other side #

The earlier essay proved one necessary condition: an edge end that floats past two picks always lets a turn slip, because one of any two consecutive boundaries is a turn at that edge. The kinds give the full condition, and the float theorem falls out as a corollary.

A column holds the right edge if it changes at every odd boundary. Between two odd boundaries there is one even boundary, where it may or may not change. So its runs of one face can be one pick long or two, never three; and where a run is two picks long it must straddle an even boundary. That is the float theorem sharpened: floats of at most two, with every two-pick float placed across the boundaries of the other edge. Interlacings and firmness counted how often a weave’s threads change face, and a cloth slips at its least-interlaced thread found the same kind of statement about firmness — a property of the worst thread, decided by its float — and the selvedge is that argument applied to the one thread whose float the weft’s own turn tests.

It also says why the float decides so much about a weave’s edges. A weave chosen for long floats has, by construction, few columns that change face often, and the columns it does have change on a weave-wide rhythm. A satin’s rhythm is its move, and its columns are all shifts of one column with a float of all but one pick, so every one of them misses turns at both edges. That is why satins are woven with selvedges of their own and twills of the ordinary kind are not.

How the census was taken #

The columns are classified exactly. For a repeat of even length, a column holds the right edge (first pick from the left) if its value differs across every boundary from an odd pick to the next, and the left edge if it differs across every boundary from an even pick to the next, the boundary from the last pick to the first included. For an odd repeat both edges require a change at every boundary. The four-by-four census applies that to the 22,874 drafts in which every thread interlaces, and records for each whether it has a column of each kind.

What is required of it. The kinds must agree with the earlier placement census on every draft: a draft has one of each kind exactly when some placement of its width holds. The re-entry count and the shaft count must add to the earlier essay’s 13,238 and 9,636. The 2/1 and 1/2 twills, the five-end satin, the 3/2 twill and the 1/1/1/2 twill must have no edge column, and so must 400 seeded random drafts of three, five and seven picks. And one shaft lifted on alternate picks must hold both edges of a 3/1 twill, a hopsack, a 2/1 twill and the five- and eight-end satins.

What the count leaves out #

The edge end is not the only thing at the edge. A selvedge usually has several ends woven tighter than the body so the edge does not draw in, and the draw-in depends on the edge ends’ crimp and the weft’s tension on the turn, which how far a cut edge frays and the beat-up essays price and this count does not. An end that holds every turn can still be pulled in by a tight weft, and an edge that holds is not an edge that is straight.

And the width is not free. A re-entered end at each edge costs no shaft but it adds two ends to the warp, and a repeat has to fit the width found that the ends a loom can carry are counted to the one. At the edge that is rarely binding. It is worth saying that the count assumes it is not.

Who worked out which part #

Selvedge weaves and floating selvedges are ordinary weaving practice, and every manual advises a plain or special weave at the edge for twills and satins. The turn criterion and the placement census are the earlier essay’s. What is added here is the classification of columns by the edge they can hold, the finding that it decides the census exactly, the re-entry count, the one-sided symmetry, and the theorem that an odd repeat has no edge column at all — the reason every weaver of a 2/1 twill or a five-end satin has to add something at the edge, stated as arithmetic rather than as advice.

Still open: whether a weave can hold with an edge end that is not an end #

Everything here puts the edge on an end threaded like the others. Some looms catch the weft with a catch cord or a tuck-in, where the weft’s tail is laid back into the next shed rather than turned round an end, and on a rapier or air-jet loom there is no continuous weft to turn at all: each pick is cut, and the edge is held by a leno or a tucked-in end.

For those the question is different in kind, and the arithmetic above does not apply: a cut weft has no turn to catch. What does carry over is the column test, asked of whatever ends hold the tucked-in tails — and whether a weave whose columns lack a kind can be edged by a tuck-in on its own shafts, or needs the leno ends a leno easer was built for, is a count this account has not taken.