Textile Structure
how cloth is put together
Pattern and colour

What a figure costs the loom

A block design's shaft count does not depend on how wide the figure is, or how deep, or how many blocks it has. It depends on how many *different* block-columns it has, and each new one costs exactly one repeat of the ground weave — 8 shafts, then 16, then 24, in a straight line with no slope to fit.
15 min read 7 figures Exactly this manyA weave is a matrixStructure before fibre

Worth reading first: A figure is not a stripe · How many shafts a draft needs.

A damask tablecloth carries a figure that may be forty centimetres across. At thirty ends to the centimetre that is twelve hundred ends of pattern, and the obvious reading of a loom’s shaft count — one shaft per group of ends that do the same thing — makes twelve hundred ends of unrepeating pattern look like an enormous number of shafts.

It is not, and the reason is not the one the trade usually gives. A jacquard does abolish the shaft budget, and the essay about that is elsewhere. This one is about a design that is not on a jacquard, on an ordinary dobby with sixteen or twenty-four shafts, weaving huckaback or M’s and O’s or summer and winter — and about the exact arithmetic that says what such a design costs.

The answer is one sentence: the shafts a block design needs are the number of distinct block-columns it has, multiplied by the repeat of the weaves in them. Not the number of blocks. Not the width. Not the number of rows of blocks, which costs nothing at all.

What a figure costs in shafts. Two families of block designs in 8-end satin on 8-end sateen at blocks of 8. One repeats three block-columns however wide it gets and costs 24 shafts at every width from 24 to 120 ends. The other gives every block-column a different pattern and costs 16, 24, 32, 40, 48 shafts as it grows — exactly 8 more per new column. Shafts are counted as distinct columns of the composite matrix.
Fig. 1 Two families of block design in an eight-end satin figured on its own reverse, grown sideways. The flat line is a design that repeats three block-columns however wide it gets: twenty-four shafts at twenty-four ends and twenty-four shafts at a hundred and twenty. The rising line gives every block-column a different pattern and costs exactly eight shafts more for each — the repeat of the weaves, every time, with no slope to fit. What neither line shows is the depth: growing the design downwards moves neither of them.

What decides a shaft

A shaft is a set of warp ends that are always lifted together, so the shaft count of any draft is the number of distinct columns of its matrix. That is the whole definition and this site has an essay on nothing else.

In a block design an end’s column is decided by two things and no more. The first is which block-column of the profile the end lies in, because that decides which weave is in force at every pick — the ground weave where the profile is blank and the figure weave where it is not, all the way down the design. The second is the end’s phase within its block: an end three squares into a block sees the third column of whichever weave is acting.

So two ends have the same composite column exactly when their block-columns have the same profile pattern and their phases agree. There are at most (distinct block-columns) × (block width) of those, and for two weaves that share no columns the bound is attained.

2/2 twill figured on 3/1 twill. A 4 by 4 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 2/2 twill and the ground is 3/1 twill, both single cloths on their own; each block is 4 ends and 4 picks. The composite is 1 cloth, with a longest float of 3 and 0 threads lying loose, all counted from the matrix that drew the picture.
Fig. 2 A four-block check in a 2/2 twill on a 3/1 twill, blocks of four. The profile has two distinct columns and the block is four ends wide, so the ceiling is eight shafts; the reading beside the draft says what it actually costs, which is lower because the two twills share columns. A stripe’s harness cost is a union rather than a sum, and so is a figure’s — the same arithmetic, in two directions instead of one.

Why depth is free and width is not

This is the asymmetry, and it is exact rather than approximate.

Adding a row of blocks to a design lengthens every block-column’s pattern by one entry. If the new row repeats a pattern the design already had, no block-column’s pattern changes in a way that separates it from another, so no end acquires a new column and the shaft count does not move. Adding a column of blocks with a pattern the design has not used gives its ends columns nothing else has, and the shaft count rises by the block width.

The two budgets a loom has are therefore spent by the two directions of the design separately. Shafts are a property of the profile’s columns; the pattern chain, which stores the distinct lifting plans, is a property of the profile’s rows. A design can be arbitrarily deep on a small harness and arbitrarily wide on a short chain, and it is only expensive when it is genuinely new in both directions at once.

That is the harness does not grow restated for two dimensions. There the finding was that a reversed twill on ninety-six ends weaves on the four shafts its base twill needs, because a reversal reuses columns. Here the same fact is a design rule with a number attached: reuse a block-column and it is free; invent one and it costs a repeat.

The draft for 8-end satin, as a loom holds it. The 8-end satin written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.
Fig. 3 The factorisation the whole argument rests on, drawn for a single weave: the threading says which shaft each end is on, the lifting plan says which shafts rise on each pick, and multiplying them back out has to reproduce the draft exactly. A block design is the same factorisation with the threading built out of block-columns.

The block weaves, and what they were designed around

The traditional block weaves are the evidence that this arithmetic was known long before anybody wrote a matrix down, because every one of them is built to keep the block-column count small.

Huckaback is two blocks. M’s and O’s is two blocks. Summer and winter is two blocks and a tie, and its whole reputation is that any number of blocks can be added at four shafts apiece. Crackle is four. Each of them is a relief or a spot weave of the kind the texture weaves are built from, and in each case the profile is drawn freely — a huckaback towel’s pattern can be as elaborate as anybody likes — and the shaft count is set by how many distinct kinds of block the design uses, not by how many blocks appear in it.

A weaver working out a summer-and-winter design counts blocks, multiplies by two, adds two for the ties, and reads off the shafts. That is precisely the law above with the repeat of the weaves substituted in, and it is stated in every manual as a recipe rather than as a consequence.

2/2 twill figured on plain. A 5 by 5 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 2/2 twill and the ground is plain, both single cloths on their own; each block is 4 ends and 4 picks. The composite is 1 cloth, with a longest float of 3 and 0 threads lying loose, all counted from the matrix that drew the picture.
Fig. 4 A spot figure — a single motif on a plain ground, which is the simplest block design anybody weaves. The profile has three distinct block-columns out of five, so two of the five are free, and the count beside the draft is what the pair actually costs once the twill’s and the plain weave’s shared columns are taken out. A figure and a ground that share columns are cheaper than the law’s ceiling, and how much cheaper has to be computed rather than assumed.

A stepped figure is the next case up, and it is where the width cost first becomes visible in the rail.

8-end satin figured on 8-end sateen. A 4 by 4 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 8-end satin and the ground is 8-end sateen, both single cloths on their own; each block is 8 ends and 8 picks. The composite is 1 cloth, with a longest float of 8 and 0 threads lying loose, all counted from the matrix that drew the picture.
Fig. 5 A stepped figure of four block-columns in an eight-end satin and its reverse. The rail gives the shafts and the lifts as counted from the composite matrix. Every block-column here is different, so this is the expensive case, and the count is the one the law predicts. What the drawing cannot show is that a design ten times as deep, drawn from the same four columns, would cost precisely the same harness.

The other end of the range is the construction that pays nothing for its figure at all, and it is worth putting beside the two above.

8-end satin figured on 8-end sateen. A 4 by 4 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 8-end satin and the ground is 8-end sateen, both single cloths on their own; each block is 8 ends and 8 picks. The composite is 1 cloth, with a longest float of 8 and 0 threads lying loose, all counted from the matrix that drew the picture.
Fig. 6 Where the cost goes when there is no shaft budget at all. The same check in an eight-end satin and its own reverse, at a block of eight: figure and ground are one weave used two ways, so every end is on a shaft the other region also uses and the harness cost is the ground’s alone. That is the damask bargain, and it is the reason the construction survived.

What was counted, and how

The two families in the opening figure are grown from one seed and nothing about the design changes as they grow except its extent.

The seed is a three-by-three profile with three distinct block-columns and three distinct block-rows, chosen so that it is a figure rather than a stripe in either direction. Wider tiles those three columns; deeper tiles those three rows; richer builds a profile in which every block-column is different, by reading the column’s index off in binary. All three are then filled with the two weaves at a block of eight, the composite matrix is built, and its distinct columns and rows are counted — the same computation this site uses for any other draft’s shafts, applied to a matrix a hundred and twenty ends wide.

Three assertions carry the result, and each of them is one the finding would fail if it were wrong.

The first is that the tiled family’s shaft count does not move: 24, 24, 24, 24, 24 across ends of 24, 48, 72, 96 and 120. The second is that the tiled family’s lift count does not move as the design deepens, which is the same statement turned through a right angle. The third is that the family with every column different rises, and it rises in steps of exactly the block: 16, 24, 32, 40, 48.

And underneath all three, a bound that is checked on every design in both families: the shaft count is at most the distinct block-columns times the block width. That is the statement the law rests on, so it is checked on each of the fifteen designs rather than argued once.

The count that is not the shafts

One quantity has been left implicit and it is worth naming, because it is the number a dobby is actually sold by.

The pattern chain stores distinct lifts, and a design’s lift count is the number of distinct rows of its composite matrix — which by the same argument as the shafts is at most the distinct block-rows times the block depth. So the two budgets have the same form and are computed the same way, one from the columns and one from the rows.

What differs is how they are spent. A dobby’s chain capacity is usually far larger than its shaft count, so a design that is deep and repetitive across the width is cheap in both and a design that is novel in both directions is expensive in both. The binding budget is almost always the shafts, which is why this essay is about them — but the arithmetic that says so is the arithmetic above, and it says so rather than being assumed.

A shallow design cannot be expensive, and the ceiling is a power of two

The law says the cost is the number of distinct block-columns, and there is a second bound on that number which the depth supplies.

A block-column’s pattern is one entry per block-row: figure or ground, so a binary word as long as the design is deep. A design three block-rows deep has at most eight such words available, whatever its width, so it can need at most eight distinct block-columns and therefore at most eight repeats of shafts — sixty-four at a block of eight, and fewer for any real design because a profile that used all eight would be a very odd-looking cloth.

Read the other way it is more useful, because it turns a harness into a design budget. A twenty-four-shaft dobby weaving eight-end weaves affords exactly three distinct block-columns. Three words can be told apart with two block-rows, since two rows give four available patterns — so two rows of blocks is already enough depth to spend the whole of a twenty-four-shaft harness, and every row after the second is free in both senses: it costs no shafts, and it cannot buy any variety the harness could pay for.

That is the sharpest form of the asymmetry. Depth is not merely free; past a very shallow point it is unusable for adding cost, because the harness runs out of shafts long before the design runs out of distinguishable columns.

8-end satin figured on 8-end sateen. A 5 by 5 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 8-end satin and the ground is 8-end sateen, both single cloths on their own; each block is 8 ends and 8 picks. The composite is 1 cloth, with a longest float of 8 and 0 threads lying loose, all counted from the matrix that drew the picture.
Fig. 7 A spot figure at the coarsest block, five block-columns wide and five deep. Its profile has three distinct columns whatever its depth, so it costs three repeats of the ground weave’s shafts and would cost the same three if it were fifty rows deep. The rail counts them off the composite matrix rather than from the drawing, which is what makes the claim checkable.

What the harness actually constrains is an alphabet

If three block-columns is the budget, it is worth asking how much design that buys, and the answer is a great deal more than a designer would expect from the number three.

The harness fixes which columns exist. It says nothing whatever about the order they are used in, because reordering a profile’s columns reuses every one of them and adds no shafts. So a twenty-four-shaft loom weaving eight-end satin figures is a loom with a three-letter alphabet and no constraint on the word.

A design twelve blocks across is a twelve-letter word over three letters: 531,441 profiles, every one of them on the same twenty-four shafts, every one a different cloth. Widen it to twenty blocks and it is over three million. The harness has not moved.

There is a caution attached to that count and it belongs with it. Most of those half-million words are not designs anybody would weave: a profile is meant to draw something, and the arrangements that draw a recognisable figure are a small and unenumerated subset of the arrangements that exist. The number is a statement about what the harness permits rather than about what the design space offers, and the two are not the same size. What it does establish is the direction of the constraint — the loom is not the thing running out first.

So a shaft count is not a limit on how elaborate a figure may be. It is a limit on how many kinds of block it may contain, and those are entirely different constraints. A designer who feels a small dobby as a restriction is usually feeling the second and describing the first, and the remedy — reuse the blocks in a longer and less obvious order — costs nothing and is available on the loom they already have.

Where the model stops

The bound is attained only when the two weaves share no columns. A figure and ground that share columns cost less, exactly as a stripe of two weaves costs their union rather than their sum, and the union has to be computed rather than assumed. The 24-shaft figure above is a satin on its own reverse, which share nothing; a twill on a twill can share a great deal.

The block width has to be at least a repeat, which is the previous rung’s condition and is here for a second reason: below a repeat the end’s phase and the profile’s pattern stop being independent, and the ceiling is no longer a ceiling of the right shape.

Nothing here is about the threading’s shape. How many heddles each shaft carries, and whether the load is even, is a different question with its own answer, and a design that costs twenty-four shafts may put twenty times as many heddles on two of them as on the others.

And nothing here is about the chain’s length in picks. The pattern chain stores distinct lifts, and a design whose rows repeat costs nothing extra in chain — but a loom still has to be told what to do on every pick, and a hundred and twenty picks of design is a hundred and twenty steps of chain however few of them are distinct. The count in this essay is of what has to be stored, which is the quantity a dobby’s capacity is quoted in.

The generalisation

What the block design does is factor a large matrix through a small one. The profile is a map from block-columns to patterns; the weaves are a map from phases to columns; the composite’s column set is the product of the two images. That is a Boolean matrix factorisation, the shaft count is the size of the middle, and everything in this essay is the observation that the middle grows with the distinct parts of the profile rather than with its size.

Stated that way the asymmetry stops being surprising. A factorisation through a small middle is cheap however large the outer objects are; it is only expensive when the middle has to grow. And the two directions are independent because the two factorisations are: one goes through the columns and one through the rows, and a design that is repetitive in one direction and novel in the other pays for exactly one of them.

The practical form is a design rule that inverts the usual instinct. A designer who wants a bigger figure on the same loom should make it deeper, not wider — depth is free and width is not. A designer who wants more variety should reuse block-columns in a new order rather than draw new ones, because an order is free and a column costs a repeat.

Who found it, and when

The block weaves are old, and the arithmetic in the manuals is correct: blocks times shafts-per-block, plus the ties. What the manuals do not say is why that is the rule rather than something involving the size of the design, and the reason is the factorisation above.

The general statement — shaft count is the number of distinct columns — is not new either, and this site made it some time ago. What is new here is running it on a composite matrix rather than on a weave, and finding that the answer separates into a profile part and a weave part exactly, with the profile’s two directions costing two different budgets.

The asymmetry has a practical shadow that predates all of it. Damask designs are traditionally long in the pick direction and repeat across the width; jacquard cards are cheap and hooks are not. Both of those are usually explained by the machine, and the machine explanation is right as far as it goes. Underneath it is a fact about the matrix that would hold on a loom of any construction.

Where the ladder goes next

The next rung is the boundary itself rather than what is inside it. A woven outline is a staircase whose tread is one block, a block is at least one repeat, and so a figured cloth has a resolution — measured in millimetres, decided by the ground weave, and not improvable by a finer machine. The number of distinct directions such an outline can take turns out to be a Farey count, and the directions are not evenly spaced.

Sideways, the same cost question asked of a stripe is what a stripe’s harness costs, which is the one-dimensional case and where the union arithmetic was first written down. The other budget — the pattern chain, and what a dobby actually stores — is in its own essay, and the two bind in different places here for the same reason they do there. And the machine that removes the shaft budget entirely, at the price of making the other one proportional to the width, is the jacquard.

Where this goes next

8-end satin figured on 8-end sateen. A 3 by 3 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 8-end satin and the ground is 8-end sateen, both single cloths on their own; each block is 8 ends and 8 picks. The composite is 1 cloth, with a longest float of 8 and 0 threads lying loose, all counted from the matrix that drew the picture. A figure is not a stripe Two sound weaves side by side always make sound cloth — that is a theorem, and it was proved here. Put one of them inside a *region* instead of a band and it stops being true, and whether it is true or not turns out to depend on a number that appears nowhere in either draft: where the two weaves start relative to each other. A 2/2 threading, run out to three widths. The same four shafts threaded for repeats of increasing width. The threading line runs up and back down; the harness never grows, and the number of ends it carries is bounded by the loom's width rather than by its shafts. The harness does not grow Nearly every account of a loom says the shaft count limits the repeat. It limits the repeat of a straight draw, and of nothing else — a reversed twill on ninety-six ends weaves on the same four shafts as the twill it was made from, and the ratio grows without bound. The draft for herringbone, as a loom holds it. The herringbone written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it. What a dobby stores A pattern chain does not store picks. It stores lifts — the distinct sets of shafts a draft ever raises — and for most drafts worth weaving that number is very much smaller than the number of picks, which is the second half of the reason a wide repeat is affordable.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Block figureDamaskFigure and groundHarnessJacquardLifting planProfile draftRepeatShaftThreading