Textile Structure
how cloth is put together
Knits and other structures

A tube and an interlock balance for different reasons

Two structures come out identically flat on a signed count and are as unalike as two knitted fabrics get. One is two jerseys that curl in opposite directions and are joined at the edges; the other is a fabric whose every course crosses. Telling them apart needs a different question asked of the same grid.
14 min read 7 figures Does it hang togetherA weave is a matrixStructure before fibre

Worth reading first: Which knitted fabrics lie flat · Does a double jersey hang together · Where a two-bed fabric's yarn is.

A criterion that reduces a fabric to a number has to be asked what its number does not see, and the answer is usually easiest to find where two very different things come out the same.

The flatness criterion on this ladder gives a tubular fabric and a one-by-one rib the same answer: zero, balanced, lies flat. It gives interlock the same answer too. Three fabrics, one number, and anybody who has handled all three knows they are not one fabric.

This rung is about what separates them, and about what a zero means.

How much of a fabric's yarn crosses between the beds. The share of half periods that cross from one bed to the other, read off each structure's own traverse rather than quoted. It is the mechanical difference between these fabrics in this account: a half period that crosses climbs the whole bed gap and one that does not climbs a yarn diameter. Single jersey and a tubular fabric come out at zero — the tubular one because its two faces are made on separate courses and never meet — and a one-by-one rib comes out at one, with every sinker loop crossing. A two-by-two rib is at a half, which is the number a reader would guess and is here counted.
Fig. 1 The two censuses disagreeing. A tubular fabric and a one-by-one rib both balance, and here they sit at opposite ends: the rib crosses between the beds at every sinker loop and the tube never crosses at all. One number cannot be doing the work of both.

The two zeros

A signed sum comes out at zero in two quite different ways, and the difference is not visible in the sum.

A cancellation is a sum of large terms with opposite signs. Two jerseys, each of which curls as hard as a fabric can curl, laid back to back so that their moments oppose. Remove one and the other rolls up.

An absence is a sum of terms that were small to begin with, or of terms that were paired locally rather than globally. A one-by-one rib’s balance is a cancellation within every course, between wales a fraction of a millimetre apart. There is nothing to remove.

A tubular fabric is the first kind. A rib is the second. Both are zero.

Why that is not a defect in the criterion

It would be easy to treat this as a flaw and go looking for a better sum, and that would be the wrong lesson.

The criterion answers exactly the question it was asked: does this fabric, as knitted, lie flat? A tube does. The two layers oppose one another and the piece sits on a table without rolling, which is the observable the question was about.

What the criterion does not answer is what happens when the fabric is altered — cut, unpicked, split, or laid out singly. That is a different question, and no sum over an intact repeat could answer it, because the alteration is not in the repeat.

The mistake to avoid is not using a criterion that cannot see everything. It is forgetting which question it answered.

What separates them: connectivity

The question that does separate a tube from a rib was already on this ladder, and it is older than any of the mechanics.

A knitted structure hangs together if it cannot be cut into two pieces along a line no yarn crosses. That is a connectivity question over the graph of loops, and it does not care about beds, gaps, forces or thicknesses at all — it cares whether there is a path.

Run it on a tubular fabric and it comes apart. The front bed’s wales form one component, the back bed’s another, and within the repeat no course of yarn joins them. That is exactly what a tube is, and the criterion names it without being told.

Run it on a one-by-one rib and there is no such partition. Every course visits both beds, so every wale is threaded onto the same continuous length of yarn as its neighbours on the other bed.

Interlock, which is the awkward one

Interlock separates too, and that is the case where the two arguments have to be read carefully rather than quoted.

Interlock as written here is two one-by-one ribs on interlock gating, each using half the needles of each bed. The partition test finds the two ribs, because within one repeat each of them is a closed traverse of its own.

That is a correct reading of the grid and a misleading reading of the fabric. Interlock’s two ribs are interknitted: each rib’s loops pass through the space the other’s occupy, and the fabric is held together by that interference rather than by a shared yarn path. It is a mechanical interlock rather than a topological one — which is where the fabric’s name comes from and is exactly the thing the connectivity test was built not to assume.

So a structure can hang together in the machine and separate on the graph, and interlock is the standing example. That is not a failure of either test; it is the boundary between them, and it is worth having marked.

Three fabrics, three reasons

Putting the three side by side makes the point sharper than any of them alone.

balance crossing share separates on the graph
1×1 rib 0 1.000 no
interlock 0 1.000 yes
tubular 0 0.000 yes

Three identical entries in the first column. Three different fabrics. The second column separates the tube from the other two, and the third separates the rib from the other two — and no single column separates all three.

Where a course of yarn sits through the thickness. The through-thickness position of one course of yarn, half period by half period, for 3 structures at a bed gap of 3 yarn diameters — 0.501 mm — on a 20 tex yarn whose diameter is 0.167 mm. A one-by-one rib crosses between the beds 4 times a course, travelling 0.501 mm through the thickness. An interlock crosses between the beds twice a course, travelling 0.501 mm through the thickness. A tubular fabric never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. The horizontal is a count of half periods rather than a length, because the repeats are not the same width and the comparison is not about their widths.
Fig. 2 The three fabrics of the table, each drawn as the traverse its own yarn takes. They agree on the balance and on nothing else. The rib crosses the gap four times a course; interlock crosses it twice, because half its needles are working on each pass; the tube never crosses at all and its trace is a single jersey’s. The first column of the table is the one quantity these three pictures have in common.

That is the general shape of what a structural census can do. Each sum answers one question, no sum answers all of them, and a fabric’s identity is the vector rather than any of its components.

What the tube’s thickness is, which is nothing

There is a consequence of the tube’s disconnection that this ladder has to report as an absence, and it is the sharpest instance of the difference between the two zeros.

Every other two-bed structure here has a thickness that follows from its geometry: the two loop planes a bed gap apart, plus a radius outside each of them. That works because a piece of yarn runs between the two planes and holds them at a definite distance.

Nothing runs between a tube’s two layers. They lie on one another because of gravity and whatever the finishing did, and there is no length in the fabric that sets their separation. So the model returns no thickness for a tubular fabric — not a small one, not a doubled one, none — with the reason attached.

An absence with a reason is a real output. It says the question is ill-posed for this structure rather than that the answer is hard.

Where a course of yarn sits through the thickness. The through-thickness position of one course of yarn, half period by half period, for 4 structures at a bed gap of 3 yarn diameters — 0.501 mm — on a 20 tex yarn whose diameter is 0.167 mm. Single jersey never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. A one-by-one rib crosses between the beds 4 times a course, travelling 0.501 mm through the thickness. A two-by-two rib crosses between the beds twice a course, travelling 0.668 mm through the thickness. A tubular fabric never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. The horizontal is a count of half periods rather than a length, because the repeats are not the same width and the comparison is not about their widths.
Fig. 3 Where a course of yarn sits through the fabric, for the four structures this rung has been comparing. The two ribs cross the gap — the one-by-one at every sinker loop, the two-by-two at every second one. Single jersey does not, and neither does a tube, because a tube’s course never leaves the bed it started on. The tube’s traverse is a jersey’s traverse exactly, which is the picture of what it is.

What each census costs to run

A tenth of a second for the whole family, on either census, and that is the reason both are worth having rather than one.

A lookup of published values is a reference: it answers about the fabrics somebody measured and extending it means measuring. A sum over a grid is a function: it answers about any structure that can be written, at the same cost, including the ones nobody has knitted.

So there is no reason to economise on which questions get asked. The crossing census and the balance are two walks over the same letters, and a third walk — the connectivity partition — is a graph search over the same object. Three views of one description, and the fabric is the intersection.

Which knitted fabrics lie flat, counted from the structure matrix. The curl balance of every named two-bed structure this site holds, with a tuck counted in full on the bed that took its yarn: the yarn a repeat puts on the front bed minus the yarn it puts on the back, over the total. A fabric lies flat exactly when it is zero, and the criterion has to put single jersey at one end and a one-by-one rib at the other or it is worth nothing — which it does, at 1.00 and 0.00. What it is for is the rest: a tubular fabric balances because it is two jerseys facing opposite ways, both cardigans balance because a tuck holds yarn on the bed that took it, and half-milano and a three-by-one rib come out front-heavy — which is what they are and what they do. 6 of the 13 structures curl.
Fig. 4 The census that merges three fabrics into one number, ranked. Seven structures sit at zero and the chart cannot say which of them cancel and which never had a moment to cancel.

The general rule this is an instance of

A number computed from a structure is a projection, and projections lose things. The useful discipline is to know which.

This collection has the same shape elsewhere. Two drafts with the same float length can be a twill and a satin. Two cloths with the same cover factor can be a close plain weave and an open sateen. Two knits with the same tightness factor really are the same knit in shape — but not in force, because the group has no stiffness in it.

In each case the temptation is to treat the number as the fabric. The remedy is always to find the second number that separates the cases the first one merged, and to say which question each of them answers.

A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted.
Fig. 5 Where a single fabric’s moment would come from. Every course runs from half a diameter behind the mid-surface to half a diameter in front, so which face a stitch’s head is on is which way its contribution points — and a sum of those signs is all the balance is.

Why the tube is the interesting case

A tube is not an exotic structure. It is what every circular machine makes by default, it is how a sock’s leg is knitted, and it is the standard way of producing a seamless body.

So the fabric whose thickness this model refuses is one of the commonest fabrics there is, and it is worth being clear that the refusal is not an edge case tucked away in a corner. It is a statement about a whole class: a fabric made of layers that are not joined within a course has no thickness a per-course model can give it, and the class includes tubes, spacer fabrics knitted without connecting yarn, and any double fabric held together only at its edges.

The thing that would give such a fabric a thickness is a yarn running between its layers. In a spacer fabric that yarn exists by design and it is what the fabric is for. In a tube it does not, and the layers are as far apart as whatever is between them.

What happens when a tube is cut

The criterion’s blindness has an observable consequence and it is a familiar one to anybody who has cut a piece of circular knitting.

Cut a tube along a wale and it does not lie flat. It becomes two jerseys, each with the same one-sided structure as any single jersey, and each rolls at its edges immediately and hard. The flatness that the intact tube had was borrowed from its own other half.

Cut a rib along a wale and nothing happens. It was flat locally, wale by wale, and losing a selvedge does not change any wale’s neighbours.

That is a two-second experiment that distinguishes the two zeros, and it is the reason the distinction is worth making at all rather than being a piece of bookkeeping.

Where the two arguments meet

Connectivity and balance are asked of the same grid and they use different parts of it.

The balance reads which bed each stitch’s yarn went to, and nothing else — not the order, not the neighbours, not the course. A permutation of the grid that kept the bed counts would leave the balance unchanged.

Connectivity reads the order and nothing else — which wales a course visits and in what sequence, so that a path can be traced. A change that kept every path but moved yarn between beds would leave connectivity unchanged.

So the two are close to orthogonal, which is why a fabric can score at either end of one while sitting at a fixed value of the other. That is the formal version of the table above, and it is the reason both sums are worth running rather than one.

What this rung does not settle

Two things, and they are the ones a reader would want next.

How far apart a tube’s layers actually sit. That is a measurement, and it is a measurement of a finishing process rather than of a structure.

Whether interlock’s mechanical interference is enough. The connectivity test says the two ribs are separate on the graph; the fabric says they are not separable in the hand. What decides it is contact and friction between loops that pass through one another’s space, and this collection has no model of that at all. It is the same gap that stops a jersey’s extension ceiling being realistic — courses in this account can pass through one another, and in a fabric they cannot.

What a spacer fabric would do

There is a structure that sits between the two cases and it is worth naming even though this collection does not hold it.

A knitted spacer is two faces held apart by a third yarn running between them, deliberately. Its two faces are joined — so it does not separate on the graph — and the joining yarn is what sets its thickness, which can be several millimetres.

That is the structure a tube would have to become before its thickness was computable. The difference between them is one yarn, and the presence or absence of that yarn is exactly what the connectivity partition detects. So the test that refuses a tube’s thickness is the same test that would accept a spacer’s, which is the strongest reason to believe it is testing the right thing.

What is genuinely new here

One distinction and one refusal.

A zero from cancellation is not a zero from absence, and a signed sum over a repeat cannot tell them apart. The test that can is a cut, and the structural version of a cut is the connectivity partition, which is the oldest check in the collection.

And a tubular fabric has no thickness in this account, reported as an absence with a reason rather than as a plausible number. That is the first time a two-bed quantity here has been refused rather than computed, and it will not be the last.

The third question nobody has asked of the grid

Two censuses and a partition make three walks over the same letters, and there is an obvious fourth that this collection has not run.

Where the yarn is along the fabric, rather than which bed it is on. A structure’s floats, its tucks and its misses all put yarn in places that carry no loop, and a fabric’s snagging, its lustre and its abrasion resistance are decided by those places rather than by its interlacings.

That walk exists elsewhere on this ladder — a second bed changes what a float is counts exactly it — and it has never been put beside the other three. Doing so would give each named structure a four-component description, and the interesting question would be which fabrics the four together still fail to separate.

What the pictures cannot show

The crossing census puts a tube at zero beside single jersey, which reads as though the two fabrics were alike. They share one number and nothing else: a tube has twice the yarn per unit area, twice the layers, a different appearance on each face and a hollow interior.

Nothing in these censuses draws a fabric. They draw sums over grids, and a grid is a description of what a machine does rather than a picture of what comes off it.

Why this matters for the numbers upstream

Every mechanical quantity in the two-bed account is computed from a crossing share and a bed gap, and both of those come from the grid.

A structure that separates has a crossing share of zero within a course and yet has yarn on both beds, so it presents to the arithmetic as a single-bed fabric with twice the material. Its energy, its forces and its thickness would all come back as one layer’s, silently, if nothing were watching.

Something is watching, and it is the partition. That is the practical value of running a test whose answer is already known for the fabrics people knit: it is not there to describe a tube, it is there to stop a tube being reported as a jersey. How thick a knit is carries the same refusal into every table that quotes a thickness.

What is worth taking away

A structural criterion is a projection and every projection merges something. The discipline is not to find a criterion that merges nothing — there is not one — but to know which cases each one merges and to have a second criterion that separates them.

Here three criteria over one grid give three fabrics three different signatures, and no one of them would have done. That is the honest shape of a structural account, and it is why this collection runs several counts over the same letters rather than looking for the one number that describes a fabric.

Which rungs this stands on

The balance, at which knitted fabrics lie flat, which is the criterion being interrogated.

The crossing census, at where a two-bed fabric’s yarn is, which supplies the second column of the table.

And the connectivity test, at does a double jersey hang together, which is older than either and supplies the third.

The rung adds no arithmetic at all. What it adds is the observation that three fabrics need three columns, and the discipline that follows from it: run every census, and say which question each one answered.

The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.
Fig. 6 The measurement a tube does not have. Every other structure’s thickness is two loop planes a bed gap apart with a radius outside each; a tube has two planes and no yarn running between them, so nothing in its own course says how far apart they lie.
A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.
Fig. 7 The other kind of zero. A rib’s opposing contributions sit a fraction of a millimetre apart, wale by wale, and there is nothing to cut away — where a tube’s two large moments are a whole fabric apart and a pair of scissors separates them.

Where the ladder goes next

The two-bed family has now been counted twice and priced once. What it has not been given is a thickness that can be compared with anything, and that comparison is where a model of a knitted fabric is most exposed — because a thickness is easy to measure and this one is predicted outright.

How thick a knit is is that comparison.

Where this goes next

tubular, as a graph of its wales. Every wale of tubular as a node — 2 on the front bed and 2 on the back — with one chain per course joining everything that course takes yarn on, because a course is one traverse of one yarn. The nodes are filled by which component they fall into. This structure comes out as 2 fabrics: F0+F1 and B0+B1. The same answer is obtained a second way, by walking every partition of the wales and asking whether any course straddles it, and the two are required to agree. Does a double jersey hang together Two beds knitting with nothing passing between them are two fabrics that happen to have been made at once. Of 6,561 two-bed arrays, 1,135 are fabrics and 50 of those are two fabrics — and 28 of the 50 split across the beds rather than along them, so neither half is a layer. Which knitted fabrics lie flat, counted from the structure matrix. The curl balance of every named two-bed structure this site holds, with a tuck counted in full on the bed that took its yarn: the yarn a repeat puts on the front bed minus the yarn it puts on the back, over the total. A fabric lies flat exactly when it is zero, and the criterion has to put single jersey at one end and a one-by-one rib at the other or it is worth nothing — which it does, at 1.00 and 0.00. What it is for is the rest: a tubular fabric balances because it is two jerseys facing opposite ways, both cardigans balance because a tuck holds yarn on the bed that took it, and half-milano and a three-by-one rib come out front-heavy — which is what they are and what they do. 6 of the 13 structures curl. Which knitted fabrics lie flat Curl was explained here by counting face changes between courses, which works for stockinette and garter and reaches nothing else. The same question turns out to be a signed sum over a structure's own grid — and it answers for every fabric a two-bed machine can make, including the ones nobody has a rule for. How much of a fabric's yarn crosses between the beds. The share of half periods that cross from one bed to the other, read off each structure's own traverse rather than quoted. It is the mechanical difference between these fabrics in this account: a half period that crosses climbs the whole bed gap and one that does not climbs a yarn diameter. Single jersey and a tubular fabric come out at zero — the tubular one because its two faces are made on separate courses and never meet — and a one-by-one rib comes out at one, with every sinker loop crossing. A two-by-two rib is at a half, which is the number a reader would guess and is here counted. Where a two-bed fabric's yarn is Thirteen named structures, and for each of them the share of its yarn that crosses between the beds — read off its own traverse rather than quoted. It separates the fabrics into three groups, and one of the groups turns out not to be a fabric at all.

What links here

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

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.

Cloth integrityConnectivityCourseCurlHanging togetherInterlockNeedle bedRibTwo-bed