A check is two stripes and a tartan is one

A check needs two things: an order of coloured ends in the warp, and an order of coloured picks in the weft. They are independent, they are chosen separately, and a mill quoting a check has to quote both.

A tartan is quoted with one. The draft is a matrix either way and the weave is unchanged; what changes is the colouring. The thread count for a tartan — so many blue, so many black, so many green — is a single sequence, and it is used in the warp and in the weft alike. That is not shorthand. It is the definition, and everything that distinguishes a tartan from any other check follows from it.

One order used twice #

Colour and weave established the computation. At every intersection the visible colour is the warp end’s where the warp is on the face and the weft pick’s where it is not, so the surface is a function of three independent inputs: the draft, the warp’s colour order and the weft’s.

Setting the two orders equal collapses three inputs to two and does something a designer notices immediately. The cloth acquires blocks: wherever the warp is in a run of one colour and the weft is in a run of the same colour, every intersection shows that colour whatever the weave does, and a solid square appears. Those squares sit on the diagonal of the pattern, and the pattern is otherwise made of rectangles where the two runs differ.

That is a tartan. Solid squares down the diagonal, mixture rectangles everywhere else, and a single quoted sequence generating both. It is a stripe in each direction, with the two partitions made identical on purpose — and where that essay’s bands carry different weaves, these carry different colours and one weave throughout.

The notation goes one step further. A tartan’s sett is almost always recorded as a half-sett which is then mirrored, without repeating the two threads at the ends — the pivots. Seventy threads are written as thirty-six. The mirroring is done from the half-sett, and the two pivots are the reason it is not simply a reversal appended: repeating them would put a run of double width down the middle of every block, which is the commonest way of getting a sett wrong on paper.

The claim, and how to test it #

What everybody says about this arrangement is that a tartan is symmetric about its diagonal — that the cloth read across is the cloth read down.

That is testable rather than decorative, and the test is one line of algebra. Write C(i, j) for the colour visible at pick i and end j, and c for the single colour order. Then C(i, j) is c[j] where the warp is up and c[i] where it is not. Asking for C(j, i) = C(i, j) gives four cases, and two of them are free.

Where the matrix disagrees with its transpose — the warp up at one of the two intersections and down at the other — both intersections show the same thread’s colour, and the equality holds whatever the colours are. Where the matrix agrees with its transpose, the two intersections show different threads’ colours, and the equality holds only if those two threads happen to be the same colour.

So the whole claim reduces to:

$$C(i,j) = C(j,i) \iff c[i] = c[j] \;\text{ or }\; W_{i \bmod n,\, j \bmod n} \neq W_{j \bmod n,\, i \bmod n}$$

The first clause is the blocks, which is why they are symmetric and why nobody has ever doubted the claim while looking at one. The second clause is the mixture rectangles, and it is a condition on the weave alone.

No weave can do it, and the reason is the diagonal #

The condition asks for the matrix to disagree with its transpose at every pair of residues. Including, unavoidably, the pairs where the two residues are the same.

Two ends whose positions agree modulo four are in the same column of the weave and in quite different parts of the sett — one in the blue block, one in the green. For those, the condition asks that W[p][p] differ from W[p][p], which no matrix does. There are n such residues on an n-end repeat, and they are lost for every weave there is.

So exact diagonal symmetry is impossible. The ceiling is (n − 1)/n of the mixture intersections — three quarters on a four-end weave, four fifths on a five-end one — and it is reached exactly by the weaves whose matrix is a tournament off the diagonal: one of each opposed pair of intersections filled, never both, never neither.

The ceiling is not reached by anything anybody weaves. Over the site’s exhaustive four-by-four sweep, 384 of the 22,874 drafts are tournaments — one in sixty — and not one of them has a name. The 2/2 twill is not among them, and the arithmetic says it could not be at any writing: a shift-rule weave on an even repeat has the offset n/2 as its own negative, so the condition contradicts itself before any drawing is made. The 2/2 twill has no writing at all that reflects, out of its sixteen.

Odd repeats do better. The 2/1 twill reflects at six of its nine writings; the 3/2 twill at ten of its twenty-five. Both are twills as near balanced as an odd repeat permits, which is what the condition amounts to once it is unpicked. Neither is a tartan cloth, and neither has the twill direction a tartan is expected to show.

How rare the ceiling weaves are, and why the count is not the obvious one #

The 384 tournaments among 22,874 drafts is quoted above as one in sixty. It is worth deriving, because the derivation says what happens at larger repeats and because the obvious count gives a different answer.

A tournament fixes one of each opposed pair off the diagonal and leaves the diagonal free. On an n-end repeat that is n(n − 1)/2 pairs and n free cells, so the tournaments are

2 raised to the power −n(n − 1)/2 of all matrices —

one in 64 at four ends, one in 1,024 at five, one in 268 million at eight. The ceiling weaves do not merely fail to include anything anybody weaves; they thin out faster than any search could keep up with. At four ends that is 1,024 matrices out of 65,536.

But the sweep counts 384 out of 22,874, and 384 is not 1,024 times the sweep’s own survival rate. The sweep discards a matrix with a constant row or column — an end that never comes down, a pick that never goes under — which keeps 34.9 per cent of all matrices and 37.5 per cent of the tournaments. Tournaments survive the filter better than matrices at large, and the reason is in their definition.

In a tournament the rows and the columns are complements of one another off the diagonal. So an end that floats over every pick forces the pick of the same index to pass under every end: the two defects arrive together rather than independently. A matrix at large can fail the filter in either of two ways; a tournament fails in both at once or in neither, so there are fewer distinct ways for it to be discarded.

Which is a small result and a useful habit. A filter’s effect on a subset is not its effect on the whole, and the two agree only when the property being filtered for is independent of the property defining the subset. Here it is exactly not.

What the eye is reading instead #

A tartan does look symmetric, and the arithmetic says half its mixture is not. Both are true, and the resolution is a quantity the site already measures.

Take a mixture rectangle where the warp is blue and the weft is green. Blue appears wherever the warp is on the face, so the rectangle is blue in exactly the proportion the weave is warp-faced. Now reflect it: the ends now carry green and the picks blue, so blue appears wherever the warp is not on the face, and the proportion is one minus the first.

The two rectangles hold the same amount of each colour exactly when the weave is balanced. A 2/2 twill is balanced at 0.500, so a mixture rectangle and its mirror image are both half blue and half green — different arrangements of the same two proportions, at thread scale, which at any normal viewing distance is one colour.

That is what a reader sees. Not a symmetry of the surface, but a symmetry of the sett — the blocks, which are exact — with mixtures either side of the diagonal that are optically identical without being identical at all.

The claim has a consequence that can be checked. Weave the same sett in a 3/1 twill, which is warp-faced at 0.750, and the mixture rectangles come out three parts warp colour on one side of the diagonal and one part on the other. The residue count is the same as the 2/2 twill’s — exactly half — and the cloth would not look symmetric for a moment.

What was counted, and how #

Two enumerations, both run while the figures are drawn.

The surface count is tartanSurface. It builds the sett — seventy threads from a half-sett of thirty-six — and walks the whole visible repeat, which for a 2/2 twill on that sett is 140 by 140, or 19,600 intersections. It counts three things separately: where the two threads are the same colour, where they differ, and of those, where the reflection survives. For the 2/2 twill: 67.5 per cent of the whole surface reflects, and exactly 50.0 per cent of the mixtures do. For plain weave and for a 2/2 basket the mixture figure is zero — both matrices are equal to their own transposes, so reflecting a mixture rectangle exchanges the two colours in every intersection of it.

The count is checked against the algebra rather than trusted: the survivors have to be exactly the mixture intersections lying on residue pairs where the matrix disagrees with its transpose, summed pair by pair. The first version of that check compared against a flat fraction and failed on correct arithmetic, because a seventy-thread sett and a five-end weave are not coprime and the residues are not evenly distributed among the mixtures — which is the same trap the sweep’s own censuses set when a count is taken over a set whose members are not equally represented.

The counting is checkCount. With k colours and a repeat of n threads there are kⁿ colour orders, so k²ⁿ checks — an order in each system — and kⁿ tartans, being the checks that use one order twice. A tartan is one check in kⁿ. On three colours and eight threads that is 6,561 orders, 43,046,721 checks and 6,561 tartans.

Distinct to the eye needs an equivalence stated rather than assumed, and this one is: two checks are the same when one becomes the other by renaming the colours and by starting the cloth at a different thread, the shift moving both systems together because that is cutting the piece somewhere else. Shifting one system alone is deliberately not included — colour and weave showed that moving the warp’s phase by a single end turns stripes into a checkerboard, and a change of that size is a different cloth rather than a different view of one.

Reducing by canonical form leaves 146 tartans and 897,370 checks. The ratio, 1 in 6,146, is very nearly the raw 1 in 6,561, because most orders have no repetition to be reduced by.

What the collisions do to the count, which is nothing #

The site already counts one kind of collision in this territory, and the natural expectation is that it would cut the check count down. It does not, and finding out why is the most useful thing in the enumeration.

Colour and weave as a two-colour problem counts blind intersections — the ones where the two crossing threads are the same colour, so the weave leaves no trace. With a one-and-one order half the intersections are blind, and colourCensus reports the consequence: the 22,874 four-by-four drafts collapse onto 256 surfaces, eighty-nine drafts apiece, with a largest class of 256.

Run the same argument the other way and it gives nothing. Fix the weave and vary the colour orders: if two pairs of orders produced the same surface, then for every end there would have to be some pick where that end is on the face — which there is, because every end of a weave interlaces — and the warp colours would have to agree there. Same for the picks. So no two colour orders collide, ever.

The surface hides the weave almost completely and hides the colour order not at all. That asymmetry is a direct consequence of the site’s own minimal definition of a weave, and it is why the check count needs no correction while the draft census does.

Where the model stops #

Every claim here needs a warp somebody threaded and a weave somebody wove. The condition is about one matrix and one colour order, and none of it survives as a statement about patterns in general — a chequered image that was printed rather than woven has no matrix, no residues and no diagonal condition, and nothing above applies to it.

The colour is discrete and the yarn is not. Each thread is one colour and each intersection shows one thread’s colour. Real tartan is woven in woollen yarn that is dyed in the fibre and blended, so a “blue” in a sett is a blend that varies along the thread, and the mixture rectangles are optically softer than any drawing of them.

The sett here is a shape rather than a record. It has the proportions a district sett has — broad ground, narrow guards, a half block at the pivot — and it is nobody’s registered tartan. The arithmetic is about the construction and the numbers move with the sett; the ceiling and the residue count do not.

And the reduction is a count, not a taxonomy. Reducing checks by renaming and re-starting says how many colourings a weaver could not tell apart. It says nothing about what symmetry type any of them has, which is a different question with a different apparatus, and this essay does not ask it.

Who found it, and when #

Chequered wool in this part of the world is older than any account of it. The Falkirk fragment — a piece of two-coloured checked cloth stuffed into a pot of Roman coins, found near Falkirk and dated to about the third century — is the usual starting point, and it is a check rather than a tartan in the modern sense.

The modern sense arrives late and by an odd route. Highland dress was banned by the Act of Proscription of 1746 and the ban was lifted in 1782, and what came back was not what had gone away: the systematic association of a sett with a name is a nineteenth-century business, and its most influential document, the Vestiarium Scoticum of 1842, was presented as a sixteenth-century manuscript and is a forgery. The thread counts in it are nonetheless real thread counts, and the cloth woven from them is real cloth. Since the Scottish Register of Tartans Act of 2008 the setts have had a statutory register, which records each as a sequence of colours and counts and nothing else.

That the record is a single sequence is the whole subject of this essay, and it is a notational fact that the trade takes for granted and never justifies. The justification is that the warp and the weft carry the same order, which is why one sequence suffices — and the “symmetry” that is usually offered as the reason is a property of the sett rather than of the surface.

The condition on the matrix appears to be new here. It is not deep: it is what happens when the colour-and-weave computation is set equal to its own transpose, which takes a line, and the surprise is only that the answer is a condition on the weave and that no weave meets it.

Where the ladder goes next #

Below this rung, a stripe is a partition of the warp does the same partition in structure rather than colour, and finds that the costs there are unions rather than sums.

Beside it, colour and weave as a two-colour problem counts the loss in the other direction, the diagonal is not a thread is the other case of an eye reading a feature that is not in the cloth, and a cord is a stripe with no colour in it is the partition made so small that the two threads of a group act as one.

The symmetry framework all of this sits inside is weaves as plane patterns, which classifies the uncoloured draft and is a different question from the one asked here.

What the pictures here cannot show. Every mixture rectangle is drawn as a mosaic of flat coloured squares, which is the model and not the fabric. The whole argument about what a reader sees turns on optical mixing at thread scale, and a drawing that has already mixed the colours would be assuming its own conclusion — so these figures deliberately do not, and the reader has to step back from the screen to see the thing being claimed.