There is no satin on six ends

There is a five-end satin and an eight-end satin. There is a seven, a nine, a ten and a twelve. There is no six.

Weaving manuals have said so for a long time, usually in the tone of a rule to be learnt. It is not a rule. It is a consequence of one condition on the move number, and the condition has a proof about a line long.

What a satin is, exactly #

A satin places one interlacing per pick — one point where the weft comes to the face — and steps across the ends by a fixed amount from pick to pick. That amount is the move.

An order-$n$ satin has $n$ ends and $n$ picks in its repeat. Starting at end 0 on pick 0, the interlacing on pick $k$ is at end $km \bmod n$.

Two conditions have to hold for that to be a satin at all.

Every end must be tied once. If the sequence $0, m, 2m, \ldots$ revisits an end before it has covered all of them, some ends are never tied down and are left floating over the entire repeat — not woven in at all, and precisely the failure the integrity check catches.

The interlacings must not line up. If they form a visible diagonal, the weave is a twill, not a satin. Whatever else a satin is, it is the weave whose interlacings are scattered.

Why scattering is the point #

The second condition — no diagonal — is easy to state and worth dwelling on, because it is what the whole construction is for.

A twill and a satin can have identical float lengths. A 7/1 twill has warp floats of seven, and so does an eight-end satin. They do not look remotely alike, and the difference is entirely in where the interlacings sit.

In the twill they line up. The eye is extremely good at finding a line, and once found it dominates: a twill reads as a directional cloth, and everything about how it is cut and made up has to take account of the direction.

In the satin they scatter. No line forms, so nothing draws the eye, and the surface reads as an uninterrupted expanse of thread. That is not a small aesthetic difference — it is why satin can be used for large plain areas where a twill would look striped, and why damask uses a satin ground with a differently-oriented satin figure on it, the two reading as flat fields of different lustre rather than as two patterns.

The move condition is how that scattering is achieved with a rule rather than by hand. A single arithmetic progression with a coprime step visits every end once and lands nowhere twice, which is the most even distribution a fixed rule can produce.

The first condition is a statement about factors #

The sequence $0, m, 2m, \ldots \pmod n$ visits every residue exactly once precisely when $m$ and $n$ share no common factor. That is elementary and it is the whole proof.

If $\gcd(m, n) = d > 1$, the multiples of $m$ modulo $n$ are all multiples of $d$, so the sequence visits only $n/d$ of the $n$ ends and returns to the start. The other ends are never tied.

If $\gcd(m, n) = 1$, then $m$ has a multiplicative inverse modulo $n$, so for any target end $j$ there is a pick $k$ with $km \equiv j$. Every end is tied, exactly once.

So the move must be coprime with the order. That is condition one.

The second condition removes two more #

Moves of 1 and $n - 1$ are always coprime with $n$, and both give a twill rather than a satin.

A move of 1 steps one end per pick, which is exactly the definition of a regular twill — the interlacings form a continuous diagonal. A move of $n-1$ is the same thing running the other way, since $n - 1 \equiv -1$.

So the satin moves on $n$ ends are the values $m$ with $1 < m < n - 1$ and $\gcd(m, n) = 1$. There are $\varphi(n) - 2$ of them, where $\varphi$ is Euler’s totient — the count of numbers below $n$ coprime with it.

Where six fails #

Now run it. The candidate moves on six ends are 2, 3 and 4.

$\gcd(2, 6) = 2$. The sequence is $0, 2, 4, 0, \ldots$ — three ends tied, three never tied at all.

$\gcd(3, 6) = 3$. The sequence is $0, 3, 0, \ldots$ — two ends tied and four floating.

$\gcd(4, 6) = 2$. Same as the move of two.

Every candidate fails, and $\varphi(6) - 2 = 2 - 2 = 0$. There is no regular satin on six ends.

Four fails the same way: the only candidate is 2, and $\gcd(2, 4) = 2$. So there is no four-end satin either, which is less often quoted and equally true.

The mirror pair #

A detail that explains why the table’s counts are always even, and why five and eight each give one satin rather than two.

If $m$ is a satin move on $n$ ends then so is $n - m$, because $\gcd(n - m, n) = \gcd(m, n)$. The two produce mirror-image weaves — the same satin with its scatter running the other way — so satin moves always come in pairs, and $\varphi(n) - 2$ is always even for $n > 2$.

The two members of a pair are genuinely different drafts and genuinely the same cloth turned over. Which one a mill uses is a convention, and it matters only where the cloth is being matched to another piece.

Seven is the smallest order with more than one essentially different satin: its moves are 2, 3, 4 and 5, which pair as $\{2,5\}$ and $\{3,4\}$, so there are two distinct seven-end satins. Eleven has four, thirteen five. Higher primes get rapidly more generous, and the reason nobody uses them is that the floats become unusable long before the choices run out.

Which orders work #

Five is the smallest that admits one. Its candidates are 2 and 3, both coprime with 5, so $\varphi(5) - 2 = 2$ moves — and the two are mirror images of one another, giving one satin in two handednesses.

Eight admits moves 3 and 5, again a mirror pair. That is the classic satin of lining and damask, with warp floats of seven.

Every prime order works generously: seven admits four moves, eleven admits eight, thirteen ten. Orders with many small factors are the poor ones — twelve admits only 5 and 7, and any order that is twice an odd number loses every even candidate at once.

Reading the table #

The pattern in the enumeration is worth naming because it is not arbitrary.

$\varphi(n)$ counts the numbers below $n$ that share no factor with it. For a prime that is $n - 1$, so a prime order admits $n - 3$ satin moves. For a power of two it is $n/2$, so eight admits two and sixteen admits six. For a number with several distinct prime factors it collapses fast — thirty has $\varphi(30) = 8$, so only six moves out of twenty-seven candidates.

So the question “which satins exist” is Euler’s totient wearing weaving clothes, and the six-end case is the smallest place the arithmetic bites hard enough to remove every option.

The satin as a cloth #

Arithmetic aside, it is worth remembering what all this is in service of, because the condition exists to make a particular fabric possible.

A satin is the weave that puts as much of one thread system on the face as it can while still being cloth. Seven ends of warp on the face for every one of weft, in the eight-end case — and the one is not optional, because a thread that is never tied is not woven in at all.

Everything a satin is known for follows: the lustre, from an unbroken length of thread reflecting specularly; the drape, from threads that are bent rarely and have compliance left; the density, from few bends needing little room; and the fragility, from the same unbroken length having nothing holding it down. All of it is the float, and the satin condition is the rule that lets a float be made as long as possible without the cloth ceasing to be cloth.

Read that way the theorem is not a curiosity about six. It is the boundary of a construction that is trying to reach a limit, and six is simply a place where the arithmetic will not let it.

What weavers do instead #

The rule is not a wall; it is a fork. There are two standard responses, and both are honest about what they are.

An irregular satin. Give up the single fixed move and place the interlacings by hand so that no two are adjacent and no diagonal forms. Six-end and four-end irregular satins exist and are used; they are not regular satins, and manuals that call them “satins” without qualification are the reason the impossibility is confusing.

A different order. Five or eight are close enough for almost any purpose, and both are available on ordinary machinery.

The interesting thing about the irregular route is that “no two adjacent, no diagonal” is a much weaker condition than a fixed coprime move, and it can be satisfied at six. What cannot be satisfied is the regularity — the property that the interlacings are a single arithmetic progression, which is what makes the weave uniform in every direction and what makes the term “regular satin” mean something.

A count that is a theorem #

This site keeps insisting that counts in this subject are more often theorems than tallies, and the satin table is the clearest case.

Nobody enumerated the satins by trying them. The number of satins on $n$ ends is fixed by a property of $n$ that has nothing to do with cloth, and it was fixed before anybody wove anything. A weaver in the fifteenth century who tried every move on six shafts and gave up was doing an experiment whose result was already determined by the factorisation of six.

Every count on this site is produced the same way: by running the enumeration while the figure is drawn, rather than by recalling it. The table above is not a table of known results. It is the result, computed, and if the condition were altered the figure would throw and the build would stop.

What the theorem does not say #

Three limits, because “there is no six-end satin” is a sentence that travels further than it should.

It does not forbid a six-end weave. Any number of weaves exist on six ends and picks, including good ones. What does not exist is a regular satin.

It does not forbid an irregular satin on six. As above — those exist, and the distinction is the whole content of the word “regular”.

It says nothing about how the cloth behaves. The theorem is about which arrangements are possible, not about whether they are any good. A regular satin on thirteen ends exists in four handednesses and would have floats of twelve, which is far too long to be usable in most yarns — the float sets the practical ceiling, and it usually bites long before the arithmetic does.

Who worked it out #

Attribution is diffuse, which usually means something was noticed independently by people who each thought it obvious.

The rule appears in nineteenth-century weaving manuals in its practical form — the move must not “divide into” the order — and manuals from that period give the surviving moves as tables to be copied rather than as consequences. The connection to coprimality is not stated in the language of number theory, but the condition being tested is exactly $\gcd(m, n) = 1$, arrived at by trying.

The formal treatment came with the twentieth-century mathematical interest in fabrics, and most sharply in the 1980 paper by Branko Grünbaum and Geoffrey Shephard titled, straightforwardly, Satins and twills: an introduction to the geometry of fabrics. Their concern was broader — the symmetry of periodic weaves and the classification of the ones whose symmetry acts transitively on the threads — but the satin condition falls out of it and is stated there in its arithmetic form.

The gap between the practical rule and the proof is about a century, which is not unusual in this subject. The absence of a proof did not stop anybody weaving; it stopped them knowing why.

Where the ladder goes next #

The property a satin is optimising is the float, and the property it is giving up is firmness.

The wider setting is a draft as a periodic plane pattern, where the scattering condition that distinguishes a satin from a twill turns into a statement about symmetry.

What the pictures here cannot show. The table on this page is an enumeration of a condition, not a survey of weaves anybody has made. No drawing can demonstrate that no six-end satin exists — the claim is about a completed search, and a picture can only ever fail to show one.