The four named weaves are corners of a family

The rung this ladder starts from established that warp rib, weft rib and hopsack are one construction applied three ways: take plain weave and double its threads — the picks, the ends, or both.

Three ways, and there are more. Double the picks by two and the ends by three and the result is a perfectly ordinary cloth, weavable on two shafts, that the trade calls an oblong matt and that no account of the derived plain weaves treats as anything.

Two knobs, and the trade turns them together #

The construction has two parameters: how many ends to a group and how many picks. Call them a and b.

The four named weaves are four points:

• (1, 1) is plain weave, ungrouped;
• (1, b) is a warp rib, the picks grouped and the ends single;
• (a, 1) is a weft rib;
• (a, a) is a hopsack.

Everything else in the grid is a cloth and none of it has a name that means anything. “Oblong matt” is the trade’s term and it covers the whole interior — a 2×3 and a 4×7 are both an oblong matt — which is a name for the absence of a specification rather than for a construction.

The three closed forms #

The useful thing about a two-parameter family is that its quantities can be given as formulas rather than as a table, and three of them come out exactly.

The shaft count is two, everywhere. A grouping adds no distinct column: every end in a group has the identical lifting pattern, so the whole grid weaves on the plainest harness there is. A shaft is a distinct column, and a 4×7 oblong matt on fifty-six squares has exactly two of them.

The fundamental domain is 2ab. The unit a cloth actually has is the repeat divided by its own translations, and the rung below found that each doubling multiplies it: plain weave’s is two, a k-fold rib’s is 2k and a k-fold hopsack’s is 2k². Those are the three corners of one expression, and the interior fills it in: 2ab, for every member.

So a 2×3 matt has a unit of twelve intersections and a 3×2 has a unit of twelve, and a 4×3 has twenty-four — a larger cloth than a 2/2 twill by a factor of six, on half its shafts.

And the longest float is the larger of the two groupings. An end floats over b picks and a pick floats over a ends, so the draft’s longest run is max(a, b) — which means the float limit binds the larger knob and leaves the smaller one free.

That last one is the practical form of the whole family. A designer working to a float limit of three can use any (a, b) with both at three or under, which is nine members of the grid rather than the three the named weaves offer.

The setts, where the two knobs act separately #

The one quantity that does not have a single formula is the pair of setts, and its behaviour is what the extra dimension is actually for.

Grouping a system lets it be set more densely, because the grouped threads share their interlacings: the rung below gives a k-fold rib’s two setts as (k + 1) to 2k. In the two-parameter family the two systems get that treatment independently, so the sett ratio is the ratio of two such expressions and depends on both knobs.

The ratio is one exactly on the diagonal and nowhere else, and off the diagonal it favours the more heavily grouped system. That is the family’s own version of the unbalanced cloth: the imbalance is not a choice of sett, it is forced by the construction, and its size is a ratio of two integers.

And it fills a gap the symmetric family leaves. A 2/2 warp rib is unbalanced by three to four and a plain weave by one to one, and there is nothing between them — but a 2×3 matt sits at 0.889 and a 3×4 at 0.929, so the interior of the grid offers intermediate imbalances the named weaves do not.

What it costs, which is the unit and nothing else #

Set the three closed forms side by side and the family’s economics are unusually clean.

The harness cost is flat. Two shafts, whatever a and b are, so a mill can weave any member of the grid on the loom it already has threaded — and can move between members by changing the lifting plan alone.

The notation’s cost is 2ab and it is the only thing that grows. A 4×4 hopsack is written on sixty-four squares and is a cloth of thirty-two; the dobby chain that stores its lifting plan is 2b rows long, which is the picks in the repeat and is the only place the doubling is paid for.

So the family is free in the warp and paid for in the weft. Changing a — the ends per group — costs nothing at all on a threaded loom, because the threading is two shafts either way and the lifting plan’s height does not depend on it. Changing b costs pattern chain.

That is an asymmetry between the two knobs that has nothing to do with the cloth, and it is a reason to prefer one over the other that no account of the derived weaves gives. A mill with a short pattern chain should buy its effect with the ends and not with the picks.

The transpose, which is the same cloth turned over #

There is one symmetry in the grid and it is worth naming because it halves what has to be considered.

Swapping the two groupings transposes the draft. An a×b matt with its ends grouped by a and picks by b is, read the other way up, a b×a matt — so the grid is symmetric about its diagonal and every member off it has a partner that is the same cloth with the warp and weft exchanged.

Every quantity respects that. The shaft count is two on both sides; the unit is 2ab either way; the float is max(a, b); the two setts swap and the ratio inverts. Nothing distinguishes a 2×3 from a 3×2 except which system is the warp, which is not a property of the cloth at all.

That matters practically because the two are not equally easy to make. The warp is on a beam and the weft is in a shuttle, so exchanging them exchanges a threading for a pattern chain and a warp tension for a pick density — and, from the section above, one direction is free and the other is paid for. So a mill choosing between a cloth and its transpose is choosing between two costs and not between two cloths.

That is the same asymmetry a warp rib and a weft rib have, generalised: the geometry is symmetric in the two systems and the machine is not, and every decision in this family is made on the second.

And the cord, which the diagonal does not have #

The rung below found that a rib’s cord exists because one system’s crimp amplitude exceeds the other’s, and that a hopsack has no cord because the two are equal.

The grid says what happens in between. Off the diagonal the two groupings differ, so the two crimps differ, so there is a cord — running in the direction of the more heavily grouped system, and at a height set by how far apart the two groupings are rather than by either alone.

So the interior of the grid is exactly the region with a weak cord: a 2×3 matt has a cord and it is a much slighter one than a 1×3 rib’s, because the ends are grouped too and the two crimps have come partly back together.

That is a design axis the named weaves cannot reach. A weaver wanting a faint cord has, in the symmetric family, a choice between a rib and a hopsack — a full cord and none — and in the full family has a continuum, indexed by how far off the diagonal the cloth sits.

How large the family is, against how much of it is used #

The grid has a size and it is worth counting, because the count says how much of the space the four names cover.

Bound the two groupings by a float limit L — which is what actually stops a cloth being made — and the family has L² members, of which four have names: plain at (1, 1), L − 1 warp ribs down one edge, L − 1 weft ribs along the other, and L − 1 hopsacks on the diagonal.

So the named members are 3(L − 1) + 1 of L², and everything else is an oblong matt. At a float limit of three that is seven of nine; at four, ten of sixteen; at six, sixteen of thirty-six. The named fraction falls as 3/L, so the larger the float a cloth can carry the smaller the share of the family anybody has a word for.

That is a small piece of arithmetic and it changes what the essay is claiming. The interior is not a curiosity at the edge of a well-covered space — beyond a float limit of about four it is most of the space, and the naming convention is covering a shrinking fraction of what the construction produces.

And the useful members are exactly the interior ones. The edges are the extreme cases: a rib is maximally unbalanced and a hopsack is maximally coarse for its float. A designer wanting a cloth between those has to reach into the region with no vocabulary, which is a real reason a construction goes unused.

What the interior would be called if anybody named it #

A construction with no vocabulary does not get specified, so the last thing worth doing is to say what a name would have to carry.

The family has two integers and every quantity in it is a function of the pair, so a name that carries the pair carries everything: a·b matt, written as two numbers in a fixed order, gives the shaft count, the unit, the float, the two setts and the direction of the cord without any further information at all.

That is more than any of the four existing names does. “Hopsack” carries one number and leaves the reader to know that both groupings are it; “warp rib” carries one number and a convention about which system; “plain” carries none. And “oblong matt” carries neither number and only the fact that the two differ.

So the notation the family wants is the one the derived-weave section already half uses — a 2/2 hopsack, a 3/3 warp rib — extended to let the two figures differ. A 2/3 matt and a 3/2 matt would then be as readable as a 2/2 hopsack, and would say which way the cord runs.

Whether that would have changed anything is a fair question and the answer is probably yes at the margin: a designer choosing between named things chooses a named thing, and the interior of this grid is exactly the region a designer would want for a faint cord, an intermediate sett ratio or a float budget spent unevenly. A construction with no name is a construction nobody specifies, and this one has three closed forms and no vocabulary.

What was counted, and how #

The drafts are generated from the rule and then measured, not typed out: an end is up when its two group indices have the same parity, which is the same rule the symmetric case uses with the two divisors separated.

Each of the three closed forms is asserted against the measurement over the whole grid — the shaft count against two, the unit area against 2ab, the longest float against max(a, b) — rather than derived and trusted. A formula that agrees with an enumeration is a formula and one that does not is a bug, and all three are the kind of expression that is right at the corners and wrong in the middle.

The sett claim is asserted as a sign rather than as a value. On the diagonal the ratio must be exactly one and off it the ratio must exceed one exactly when a exceeds b — which is a statement about the family, where a table of ratios would be a statement about the members tried.

And every member is required to be one cloth, which is not automatic: a grouping that made a draft fall into layers would be a real finding and the check is what would report it.

What a stripe of two members costs #

The family’s flat harness cost has a consequence at the next scale up, and it is the one a designer would actually use.

A stripe is a partition of the ends, and its harness cost is the number of distinct columns in the whole striped draft rather than the sum of the bands’ shaft counts. Two weaves that share their columns cost nothing extra.

Every member of this family with the same b has the same two columns, because a column records only which picks an end is up on — and that is decided by the pick grouping alone. So a 1×3 warp rib, a 2×3 matt and a 3×3 hopsack all have the identical column set, and a cloth striping all three of them weaves on two shafts in total.

That is a striking amount of design for nothing. A weaver can put a plain rib beside a matt beside a hopsack, in bands of any width, and the loom cannot tell the difference — the harness is two shafts and the lifting plan is the same 2b rows for all of them.

What changes across the boundary is the ends per group, which is a threading decision — and here the threading is not a shaft assignment but which shaft the next end goes on. So the bands differ in the order the two shafts are used and not in how many there are, which is exactly the freedom the harness census says a two-shaft harness has and which nothing else on this site has had a use for.

The rung below found the same thing for the symmetric family — a warp rib and a hopsack of one k stripe for two shafts total — and the two-parameter form says how far it goes: the whole column of the grid at fixed b stripes for free, which is L cloths rather than two.

Where the model stops #

The grid is plain weave’s derivatives only. The same doubling can be applied to a twill, a satin or anything else, and the resulting family is much larger and its shaft count is not two. What makes this family clean is that its base has one distinct column pair, and nothing here says what happens when the base does not.

The setts come from the counting model in cloth.js, where a thread takes one diameter and each interlacing takes one more. Peirce’s circular thread gives different numbers; what survives either model is the ratio, because it is a ratio of interlacing counts and the diameter cancels.

The cord claim is qualitative here. That a member off the diagonal has a weaker cord than a rib follows from the two crimps being closer together, and how much weaker needs the geometry the rung below runs — which is not swept over the grid.

And nothing here is about appearance. Two cloths with the same a and b and different yarns look nothing alike, and the family’s whole content is structural.

Who found it, and when #

The oblong matt is in the manuals — Watson gives it, Grosicki gives it, and both treat it as a variant of the matt rather than as a member of a family. The drawings are there and the parameters are not separated.

What seems genuinely absent is the two-dimensional framing, and the reason is probably that the manuals organise by name: plain, rib, matt, and the oblong matt as an aside under the matt. A family with two integer parameters organises by arithmetic instead, and then the three closed forms are the obvious thing to want.

The forms themselves are elementary once asked for. What is worth carrying is the asymmetry between the two knobs — free in the ends, paid for in the picks — because it is a fact about the loom rather than about the cloth and it points at a member of the family a mill would otherwise not reach for.

Where the ladder goes next #

Everything in this ladder has been about what a group is. What a group does in use is a different question and it has one clear answer: a group is a set of threads nothing separates, so at a cut edge nothing separates them there either — and a hopsack frays in a way a twill of identical firmness does not.