Crochet · Description · Knitting · Structures · Terminology

From lag to loop

Hand knitters employ several techniques for increasing the number of stitches in a row or round of fabric. Instructions for The order how to knit a Hose, published in 1655, make several references to widening a row. Procedural instructions appear regularly in Victorian publications beginning with The Ladies’ Knitting and Netting Book, from 1837. This describes two methods that remain in practice, both working the added stitch into a loop in the preceding row.

The Workwoman’s Guide, from 1840, notes that “increasing the number of loops is generally done in the middle of a pinful of stitches.” It adds to the “various modes of widening” with one “effected by taking up the cross loop, below the next stitch, belonging to the row before, and afterwards continuing the plain knitting.” That cross loop is a central theme of this post and is examined in detail below.

The process of casting on a row of loops at the outset of a piece of knitted fabric is also relevant to this discussion. The first published instructions for it appear in The Knitting Teacher’s Assistant, from 1817. This is framed as a Q&A dialog between a teacher and student. It was retained verbatim in subsequent editions and can be seen in this one from 1881. The book makes no mention of increasing the stitch count in subsequent rows. However, the described cast-on technique simply places a closed loop on the needle and can be applied at any point in the fabric. It is now often termed a “backward loop” or “e-wrap.”

The first illustrated instructions for an increase appear in the The Knitting Book, published in 1847 by Eleanore Riego de la Branchardière. She describes an increase worked into the segment of the yarn or thread between the loop just knitted and the one that is about to be. When lifted up from the preceding row, that segment is the “cross loop” noted above.

It is termed a “sinker loop” in the formally standardized vocabulary of mechanized industrial knitting, in contrast to the “needle loop” (both named for parts of the machine). There has never been any generally accepted term for the former in the glossary of hand knitting, and it is referred to variously as the strand, yarn, or bar between the loops.

When describing other types of looped fabric, the transition from the leading leg of one loop to the trailing leg of the next is termed a “lag.” It is shown here in the context of knitting, highlighted in black.

loops and lag

The same knitted structure is formed by the forward pass in many Tunisian crochet stitches but has no direct correlate in ordinary crochet. In that craft, as well, a stitch is begun by pulling a vertical loop through the one below it. However, it is then joined to the adjacent loop in the preceding row to complete the stitch. Nonetheless, reference to the dichotomy between loop and lag can ease the comparison of looped structures that are typically associated with one craft but on closer examination are seen to be shared with others.

Riego terms the process of converting a lag into a knittable loop “to make a stitch.” There are two methods for doing this. In the one rather cryptically described in 1840, the lag in the preceding row is twisted into a closed loop and a new loop is knitted through it.

The other, as described by Riego, extends this over two rows. It is seeded with a “yarn over” in the first and completed in the second. This approach permits a number of stitches to be made consecutively. The remark that “it will form an open stitch” also implies that Riego recognized twisted stitches as an element of knitting. Continue reading “From lag to loop”

Crochet · Description · Knitting · Terminology

From knitted loop to crocheted stitch

In the past few posts I’ve considered different approaches to the graphic description of looped fabric structures. Although largely in abstract terms thus far, my intention is to apply relevant aspects of them to the analysis of specific objects that have themselves been the focus of other posts or are in the queue for such treatment.

Analytic terminology has been another perennial favorite here. The subject this time around is a formal international standard that both defines and illustrates structural details of knitted fabric in terms that are applicable to other forms of loopcraft, as well. The extent of that applicability will be tested with a comparison between knitted stockinette fabric and slip stitch crochet.

I had previously suggested that crochet could be seen as a handicraft equivalent to warp knitting, using terms taken from the International Standard ISO 4921:2000, Knitting — Basic concepts — Vocabulary. This “defines terms for basic knitting concepts” applicable both to hand and industrial knitting although many of the definitions are only used in the latter context. A related standard ISO 8388:1998Knitted fabrics — Types — Vocabulary, more explicitly defines terms for industrially produced machine knitted fabrics” but is relevant to hand knitted fabric nonetheless.

The vocabularies in both are useful when comparing other aspects of crochet and knitting since they accommodate both symmetrical and asymmetrical loops, and define the terms “loop” and “stitch” separately. Although the symmetrical/asymmetrical distinction is not a factor in the categorization of all hand-knitted structures, it is manifested differently in crochet and is fundamental to all such fabric.

The ISO vocabulary is based on the following differentiation of a loop, a knitted loop, and a stitch. Although the term loop is used in a manner familiar to hand knitters, the combined term “kink of yarn” is preferred to the permissible alternative “loop of yarn.” This is immaterial to the following discussion except for two direct citations from the standard, where kink is replaced with an italicized loop.

A loop of yarn It is defined as  “a length of yarn that has been bent into a shape appropriate for its transformation into a weft-knitted or warp-knitted loop.” Three specimen forms are illustrated.


A “knitted loop” is then defined as “a loop of yarn that is intermeshed at its base.”


The one at the top is an “open loop,” defined as “a knitted loop in which the same thread enters and leaves the loop at opposite sides without crossing over itself” and noting that “the same applies to an open stitch.” The bottom right shows a “closed loop” — “a knitted loop at the base of which the thread crosses over itself” — and again “the same applies to a closed stitch.” The closed loop is also illustrated under its own heading but in neither instance are the two possible directions for the crossover labeled or even noted (“S” as shown here, or “Z” as in a following illustration; both are explained in the preceding post).


Finally, a “stitch” is “a loop of yarn that is intermeshed at its base and at its top.”


This illustration shows a “reverse stitch,” also called a “back stitch” and is explicitly “not the same as a purl stitch” (which means slightly different things in hand and industrial knitting). There is a separate illustration of a “face stitch,” also called a “plain stitch” or a “stocking stitch.” The difference is that the face stitch is “so intermeshed in the fabric that its legs are situated over the top arc of the stitch formed in the same wale in the previous course.”


The terms wale and course correspond to the more familiar column and row but explicitly refer to sequences of stitches and not loops. It is also significant that the term “stitch” is not further specified as a knitted stitch and its definition includes a broad scope note.

“A stitch may be combined with a float, and different types of knitted loops and stitches may be combined in a unit of stitches or an arrangement of stitches.
≠ a knitted loop”

The named arrangements of stitches include a “binding-off course” defined as, “a new row of loops, each one transferred to the adjoining wale and forming a ladderproof chain of loops at the top end of a knitted article.”


The lateral repositioning of the knitted loop changes the fabric structure from symmetrical to asymmetrical but the the loop retains its basic identity. When the loop in the adjoining wale is knitted through it, the initial loop is intermeshed at its base and top, thereby becoming a stitch. The ISO vocabulary doesn’t have a name for it, but it follows from the definition of the binding-off course that it would be called a chain stitch.

The preceding illustration is a detail from the upper end of a piece of knitted fabric that can be extended with an arbitrary number of lower courses of knitted stitches. There is also a type of crocheted fabric that consists of multiple courses of chain stitches identical to those in the binding-off course. This has the slip stitch structure illustrated in numerous previous posts and is seen in this drawing taken from a description of a bootee in the collections of the National Museums of Scotland, by Audrey Henshall. (It is also shown in earlier posts, erroneously drawn with the yarn fed from the front in what crocheter’s call the “inverse” form of a slip stitch, discussed further below.)


All documentation of this fabric prior to the 1820s describes it as “a species of knitting,” with the word “crochet” only used to designate the hook. It can also be seen as a form of knitted fabric according to the ISO definition. Nonetheless, it is now primarily associated with crochet. The vertical intermeshing of one course of chain stitches with another is the definitive attribute of its simplest form, alternately termed slip stitch crochet or single crochet (UK).

A bind-off course fashioned with knitting needles requires all of the knitted loops to be held on the needle until they are worked successively into chain stitches on the next pass. With a crochet hook, the knitted loops are taken onto the tool individually and immediately intermeshed into chain stitches. This is also the more practicable technique for working courses of chain stitches into crocheted fabric.

Regardless of how the fabric illustrated in the two preceding drawings can be produced, the chains in both have the same structural characteristics. The loops all lean to the right, they are all open, and the tool is inserted into the front leg of the preceding stitch. The legs of the new loop pass behind it, forming reverse stitches.

Another of the drawings of slip stitch crochet that’s already been used several times on this blog, by Annemarie Seiler-Baldinger, shows a different configuration. The knitted loops lean to the right here as well but they are closed (with a Z crossing). They are also worked into the back edge of the preceding stitch but their legs pass in front of it, forming face stitches.


The correlation between the variant forms of this structure and the procedural aspects of their production as slip stitch crochet were discussed in depth in the preceding post, deferring a few relevant details for later consideration. One of them is the difference between face and reverse stitches. This correlates basically to whether the yarn is held in back or in front of the fabric, with the hook inserted into it from the front or back, while the loops and stitches are formed.

This maps directly into the colloquial knit and purl of hand knitting. However, the latter term is not widely recognized in crochet and the labeling of reverse stitches is a matter of recurring debate. Slip stitch crocheters commonly refer to them as “inverse slip stitches,” which needn’t be taken any further for now. However, the concept of inverse does not scale as clearly into more complex crochet stitches.

One further property of a slip stitch can make the analysis of fabric produced with it more difficult than that of fabric made with the stocking stitch. In the latter, the initial loop will be propagated into the stitch either open or closed and retained that way in the fabric. With the slip stitch, a new loop that is worked into the front edge of a stitch in the preceding course, applies a vertical force to that stitch which can reverse its open or closed characteristic.

The two-loops-in-one attribute of crochet makes it a compound structure and therefore nominally comparable to the one-loop-over-two compound knitting discussed here, and illustrated with this schematic drawing by Marianne Eriksson.


However, the mechanical dynamics of the intrinsically compound slip stitch and those of the stocking stitch whether compound or not, are fundamentally different. This is one of the limitations on the describability of crochet and knitting using the same terms — but also provides fuel for additional posts.

Crochet · Description · Nalbinding · Structures

Drawing pains: the slip stitch

The preceding two posts present formal numerical and graphical procedures for analyzing and describing looped fabric structures. By intriguing coincidence, the first of the cited publications was issued at the time when attested documentary and material evidence of slip stitch crochet was first beginning to appear. Similarly, the later texts were published when slip stitch crochet was shifting from being a primary means for fabric production to an ancillary technique.

It therefore seems appropriate to examine a few drawings of early fabric with a slip stitch structure that are puzzling in one way or another to see if any aspect of the contemporaneous methodologies might make it easier to understand them. I won’t be going near the mathematics of those approaches but will be considering the applicability of some of their procedural details to the analysis of looped fabric.

In suitably adapted terms, a stitch can be described by the path the thread takes through the loop(s) to which it is anchored and the number of times it crosses over itself before moving into the next anchor loop(s) in the preexisting fabric. This is characterized by the location and direction of the crossovers, permitting a point-by-point comparison of two structures that appear to be similar but may actually differ in some important regard. A typical such question is whether a right-handed and a left-handed worker executing the same instructions from the respective points of view produce fabric structures that are true mirror images of each other.

I’ve devoted several previous posts to slip stitch crochet and will start this one with a reprise of drawings from one of them. Nothing will be said that’s not already familiar to a slip stitch crocheter. However, two of the following illustrations were published as descriptions of nalbinding and this review may be worthwhile from the perspective of that craft. It is otherwise intended as a preliminary exercise in the analysis of illustrated structures that are either not associated with extant fabric or in some other regard are questionable representations of the objects from which they were drawn.

The first of the illustrations shown before is a textbook drawing of the “plain crochet stitch,” by Annemarie Seiler-Baldinger.


The accompanying text says, “the thread is drawn through an upper stitch of the previous row and through the stitch last formed.” However, in the original German from which this was translated, ‘upper stitch’ is obere Maschenschlinge, which is literally ‘upper loop of the stitch.’ In current craft parlance this is the ‘back leg of the loop,’ normally contracted to ‘the back loop’ and abbreviated as BLO (back loop only). Working through the front leg of the loop is similarly abbreviated FLO.

The second repeated drawing, by Audrey Henshall, illustrates the structure of a child’s bootee in the collections of the National Museums of Scotland, in Edinburgh. It also shows a BLO slip stitch but in contrast to Seiler-Baldinger’s drawing, where the back leg of the loop leads forward into the following stitch, in this drawing it is the front leg of the loop that leads forward.


If the legs here are seen as uncrossed, in the Seiler-Baldinger drawing they are crossed. The direction of such crossings is often indicated using the familiar descriptors for the twisting and plying of yarn.


This gives S-crossings and Z-crossings, with Seiler-Baldinger showing the latter. The alternative is to label them as left-over-right and right-over-left, but those designations depend on the point of view.

The path the yarn takes around a crochet hook and the direction in which the loops are worked determine whether their legs are crossed or uncrossed. The variables are normally designated as right-to-left or left-to-right — RTL and LTR — and as yarn-over-hook or yarn-under-hook — YO and YU. Here right and left do indicate direction unambiguously but YO and YU are less clear.

An additional complication pertains to the so-called ‘inverse’ slip stitches, where the yarn is held in front of the fabric and the hook is inserted into its back, also reversing the structural effects of YO and YU. (This additionally causes the legs of a new loop to pass behind the side of the stitch it is anchored to, as seen in Henshall’s drawing, rather in front of it as in all the other drawings shown here.)  The qualifiers clockwise and counterclockwise are therefore sometimes used to avoid confusion. However, doing so requires an explanation of the point of view.

I’m reluctant to suggest coined alternatives (although the following one is not entirely my own) but will note that the S/Z model can also be applied to the direction in which the yarn is wrapped around the hook (or a knitting needle), with YO being an ‘S-wrap’ and YU a ‘Z-wrap’ — YS and YZ. The utility of doing so is worth greater explanation, which I’ll provide in a separate post on the further mechanics of crossovers in slip stitches, but will keep to the familiar abbreviations in the meanwhile.

Seiler-Baldinger’s illustration of the slip stitch structure is oriented LTR rather than RTL as more commonly appears in tutorial contexts. The two forms are mirror images of each other by implication but it is necessary to be certain that they truly are so. Reversing the direction of Seiler-Baldinger’s drawing is easy enough, as shown here by Ella Hildebrand, in a style that more clearly reveals the three-dimensionality of loopwork.


The remaining question is which crossover points need to be inverted to reproduce the illustrated structure in actual fabric. The front and back legs of the loop have the same position in either working direction, leaving the yarn wrap as the only directly controllable variable. As long as we’re dealing with fabric where all rows are worked in the same direction, if the direction of the yarn wrap is changed when the working direction is, everything else falls into place. This was also prescribed in instructions from 1800, describing practice prior to 1780 (discussed further here).

“Hook knitting can also be worked from the left as with ordinary knitting. The only difference is the positioning of the thread. Instead of leading it under the shaft as usual, it is first passed over the shaft and then led under it.”

Since the present-day default for crochet is YO, the reference to YU as being usual before 1780 is significant. In fact, it took a while before crochet instructions regularly prescribed YO as the standard. The earliest known instructions, published in 1785 explicitly illustrate FLO hook knitting being worked YU and RTL, but note that LTR is also possible (fully described here).

Yet another slip stitch variant is shown in a drawing, by Gudrun Böttcher, of a test swatch explicitly illustrating slip stitch crochet (“Häkeln: Kettenmaschen”). The new loop is again worked through the back leg of the corresponding loop in the preceding row, RTL, but is now YO.


Böttcher shows a futher variant of the slip stitch in a drawing of a child’s sock in the collections of the Museum der Kulturen in Basel. The difference this time is that the new loop is worked FLO, again YO in the illustrated RTL working direction. (The published drawing is rotated 180° here to ease the comparison.)


This now brings nalbinding clearly into the discussion. For some enigmatic reason, Böttcher says that the preceding illustration is of a nalbound structure and alternative “techniques such as…crochet [can] immediately be eliminated from consideration.”

Audrey Henshall also described the Edinburgh bootee as nalbinding. However, that was in 1952, when the research community was abuzz with interest in recently published descriptions of that craft, and none of its members were writing about slip stitch crochet or could even be expected to recognize it. I’ve explained my reasons for believing that the bootee is, in fact, archetypal Scottish shepherd’s knitting (the indigenous form of slip stitch crochet) in a previous post titled A Tale of Two Bootees.

Admitting to some poetic license in that title and taking another step toward the telling of the tale’s remaining half, the second bootee is the child’s sock shown in Böttcher’s drawing above. There is no question about the slip stitch being readily produceable with an eyed needle, However, it does not follow that an entire garment with a basic slip stitch structure but also includes shaped construction details such as the toe and heel of a sock, can as plausibly be nalbound as it can be crocheted.

Böttcher doesn’t go beyond the drawing of the stitch structure and says nothing at all about the construction of the sock. However, the article that includes her explicitly labeled drawing of slip stitch crochet also provides an explanation of the general method she used to draw the structures of older pieces of ostensibly nalbound fabric. That’s a blogworthy topic in its own right to which I’ll soon be turning my attention, and will discuss her drawings of the Basel sock further in it (as well as providing full bibliographic details for all of her articles cited above).

Description · Looping · Structures

From loop to knots

The chain of loops illustrated in 1771 by Alexandre-Théophile Vandermonde (discussed in the preceding post) appears in another seminal text on knot theory. Peter Guthrie Tait presented a number of papers on that topic to the Royal Society of Edinburgh during its 1876–77 session, with a condensed version appearing as an article titled “On Knots” in the Society’s Transactions. vol. 38, 1849.


His comments on the first of the illustrated forms state:

“…the supposed number of loops may be any whatever. The free ends must, of course, be joined externally. If we make the crossings alternately + and – it will be seen at a glance that a change of one sign (i.e., that of the extreme crossing at either end) removes the whole knotting; so that there is but one degree of beknottedness. The crossings in this figure are in three rows. Those in the upper row are all copper (the last, of course, becomes silver when the sign is changed)…”

The free ends need to be joined since Tait presents the chain as a mathematical knot, which is closed by definition. All such constructs are analyzed in terms of the number of points where the closed element crosses over or under itself, the direction of each crossing, the effect of selectively changing those directions, and how the knot relates to its mirror image. He uses the the terms ‘copper’ and ‘silver’ to qualify the crossings further.

Tait’s explanation of the lower chain is:

“To give the greatest beknottedness to a knot with the same projection, it is obvious that all we have to do is make the copper crossings into silver ones, i.e., change the sign of each of the upper row of crossings. This gives fig. 9 [unnumbered here]. With five loops it has four degrees of beknottedness.”

In “On Knots. Part II” from 1884, he more clearly defines the concept of beknottedness and how its degrees are counted:

“I still consider that its proper measure is the smallest number of signs which will remove all knottiness.

The discussion then goes further into ‘locking’ and ‘linking,’ concepts introduced in the 1877 publication along with the undefined ‘knotfullness’ and ‘belinkedness.’ The 1884 volume includes On Knots, Part III, which similarly clarifies knottiness. Locking and linking are directly relevant to the description of looped fabric, and the concepts of both beknottedness and belinkedness can usefully be applied to its structural analysis.

Knot theory is largely focused on reducing elaborate looped constructs to their minimum knottiness by eliminating as many extraneous loops as possible. At least one further besomethingedness is therefore needed to quantify the unravelable loops that are deliberately retained in actual fabric. In the spirit of florid Victorian coinage, I’m going to start by suggesting ‘beloopedness’ and pair that with ‘loopfullness.’ Depending on how far they can be taken, it may also prove useful to co-opt the colloquial term for unraveling a sequence of looped stitches by undoing the knot that secures it and pulling the freed end — frogging. This would add ‘degrees of befroggability’ to the extensions of the glossary.

It is likely that the additional terms would have met with Tait’s approval. He took delight in artful terminology and was surely aware of the way his ‘knottiness’ otherwise sounded, adding ‘perversion’ (“deformed into its own perversion”) and ‘screwing’ (“of all kinds”) to his labels for other attributes of knots. Although ‘loopiness’ had yet to acquire the connotations it now has, its latter sense would surely have added to the delight. (His terminology has all been streamlined in the recent literature of knot theory, and I’ll also be making that shift — but not quite yet.)

Other knots that parallel looped fabric structures appear in Tait’s drawings. He provides direct justification for the present excursion by stating that “Some are closely connected with knitting, &c,” explicitly using the following three as examples.


It is further worth noting that the least complex structure beyond a simple ring — ‘the unknot’ — to qualify as a mathematical knot is the ‘trefoil.’ This is also a quintessential loop in the craft sense, rendered mechanically secure by drawing its free end(s) through the middle as described below.

Here is Tait’s illustration of it.tait-trefoil

He doesn’t show anything equivalent to the stocking stitch in Vandermonde’s mathematical framework. However, given Tait’s own mention of knitting, and since the number of loops in a chain “may be any whatever,” he would doubtless have been comfortable seeing his analytic procedures extended to any looped structure found in fabric. The mathematical rigor that was his intent can also be relaxed when adapting his method to such description.

In craft terms, if a loop is added to a fabric structure by leading the end of the working thread into the preceding loop and pulling it entirely through the new loop — as with an eyed needle — the result is a knot. (That’s why calling nalbinding ‘knotless netting’ is fundamentally incorrect; ‘loose-knot netting’ would be far better.) If instead, the thread is pulled into an old loop to form a new one, but the end of the thread is not drawn entirely through it — as with a knitting needle or crochet hook — there will be no knot in the structure until the thread is pulled fully through a subsequent loop. It is also possible to work end-led around a preceding loop but not through it.

Plain knitting (stocking stitch) is loop-led through the penultimate row in the fabric, which is followed with a row of crochet-type slip stitches that successively reduce the number of workable loops to a single one. The end of the thread is then pulled through it to form a knot. (There are alternative ‘bind offs’ but the end effect remains the same.)

A loop-in-loop structure where only the final loop is secured in this way has “but one degree of beknottedness” regardless of how extensive the preceding looping is. A looped structure that is punctuated with knots at regular intervals has greater degrees of beknottedness than one that is homogeneously looped throughout, but again, that does not describe any further detail in its looped component.

However Tait might have felt about any of this, the concept of loopfullness can be used to quantify differences between various forms of looped fabric, based on properties of the individual loops. As counted by Vandermonde, an open-knit loop has four crossover points with the one it’s worked into, two in front and two in back:


Closing the loop adds one more crossover point, thus making the loopfullness of closed-knit fabric 20% greater than that of open-knit fabric.


Working a second row of chains asymmetrically into the one illustrated by Tait produces a structure that is normally seen as plain crochet (aka slip stitch) but also illustrates (left-handedly) the final row of plain knitting.


The first and last loops in this drawing deviate from the configuration of the others in order to set up the free ends for fusing into a mathematical knot (alternating + and – crossings over the join). Otherwise, each of the four complete chains in the second row starts out as an open-knit loop, with its characteristic four crossover points. Another loop is then worked into it laterally, adding four further crossover points, and imparting double the loopfullness to plain crochet than plain knitting has.

Another way of stating this is that plain knitting involves a single loop being worked into another single loop. Plain crochet works one loop into two other loops, and the question is whether the difference between loop-in-loop and loop-in-loops should be taken to indicate different degrees of beloopedness. That distinction applies not just to crochet-type asymmetrical compound looping, but also to symmetrical compound looping as typified by the knit tubes that began to appear in the 5th century CE, and compound nalbinding.

The preceding illustration of plain crochet also seeds a discussion of belinkedness that I’ll continue in a separate post. Instructions for producing such fabric, through to early 20th century (summarized here), prescribe the addition of further rows by placing a knot at the end a completed row and starting the next row afresh from the other edge of the fabric. Although the rows may be identical, each is a separate mathematical knot that is linked to its predecessor. The mode of that linkage is as important to the characterization of the fabric structure as is the configuration of the individual rows, and may prove reasonable to quantify in degrees of belinkedness.

Folding this all back into Tait’s initial concept of beknottedness, nalbound fabric where each stitch is secured with its own knot, has as many degrees of beknottedness as there are stitches. Flatwork plain crochet with each row ending in a knot, has as many degrees of beknottedness as it has rows (discounting its alternate characterization as a sequence of linked knots). Plain knit fabric, as already noted, simply has one degree of beknottedness.

This might be extended into the similar enumeration of degrees of beloopedness by taking ‘one loop in one loop’ structures to have one degree of beloopedness, ‘one loop with multiple loops’ structures to have two degrees of beloopedness, and any such compound structure that encapsulates yet another looped structure as having three degrees of beloopedness.

One illustration of that third case is seen in the family of crochet stitches that interpose multi-looped vertical separators between the anchor loop in the preceding row, and the loop that closes the current stitch. These are normally termed ‘posts’ and their ‘length’ defines the difference between plain crochet and the double, triple, and other modified forms. This attribute of looped fabric is the one that corresponds most directly to Tait’s knottiness, and is would therefore be most correctly designated as loopiness.

That concept also applies to the suggestion that Tait’s conceptual model is applicable to the practice of loop-based crafts. However, I do believe that the methods he developed for analyzing mathematical knots can be useful in the investigation of what Vandermonde called “difficult questions about fabric.”

*          *          *          *

Here are a few particularly clickworthy links to additional material about Tait.

  • A lecture about his academic career and the context in which he developed knot theory can be streamed here.
  • His brother-in-law, Alexander Crum Brown, also a prominent scientist, knit an intricate three-layer model of one of the other knots illustrated by Tait. That model is currently in the collections of the National Museums of Scotland and can be seen here. The museum also posted a two-part blog with background detail about Crum Brown and his method for knitting the model.
  • James Clerk Maxwell, a close friend of Tait’s, commented on the latter’s florid terminology in a poem titled “(Cats) Cradle Song.”
Description · Looping · Structures


The graphic representation of a looped fabric structure can provide a useful complement to its narrative or photographic description. When documenting fabric that includes a variety of stitches that are irregularly juxtaposed, a pattern diagram can provide clarity that is not possible by other means. Numerous systems of notation are based on square-grid diagrams, adapted to the needs of individual crafts. Others, such as ‘symbol crochet’ enable intricate patterns to be represented without the constraints imposed by a grid, and for them to be worked with little or no ability to read the language of the accompanying written instructions.

The structural analysis of, say, a fragment of archeologically recovered fabric is similarly eased by graphic support, and line drawings extrapolated from photographs are commonly included in site reports and independent object descriptions. However, in contrast to the defining aspect of drawings provided by pattern designers, an analytic drawing of an older object may reflect assumptions about details that are obscured by the fabric itself, or are not present in the fragment at all. The difficulty inherent in this is compounded by the ease with which errors can be injected into a drawing, and the low likelihood of their being recognized in the subsequent editorial process.

The earliest generalized illustrations of looped fabric structures appear in an essay by Alexandre-Théophile Vandermonde titled “Remarques sur les problèmes de situation” (Remarks on the problems of location), included in the Proceedings of the French Royal Academy of Sciences from 1771. This is frequently seen as a seminal step toward to the development of ‘knot theory’ later in the following century. In fact, however, the author’s own intention was for it to provide a framework for the description of loop-based yarncraft in both manufacturing and analytic contexts.

“However one or more threads may encircle each other in space [circonvolutions], we can always find a mathematical formula for calculating their dimensions, but this expression would be of no use to the crafts. The worker who makes a plait [or braid, tresse], a mesh [or net, réseau], or knots [nœuds], does not design it on the basis of dimensional relationships, but rather by visually positioning the order in which they are interlaced [entrelacés]. It would therefore be useful to have a system of calculation more in accordance with the worker’s thought process; a notation that only represents the conceptualization of his work and would be sufficient for its reproduction in the same manner at any time…. My objective here is only to suggest the possibility of such a notation and its applicability to difficult questions about fabric.”

Vandermonde’s notation describes the path a thread takes in a looped structure, by assigning numerical coordinates to the points where it crosses over itself. He illustrates this with two drawings. The label he applies to the first of them — tresse — designates a plaited or braided structure, which requires more than the single element it illustrates. However, the depicted structure is identical to the familiar looped chain that pervades many posts on this blog, and provides a more appropriate label for the underlying model. His treatment of it also presages the shorthand now commonly used in nalbinding, which characterizes a stitch by whether the thread crosses over or under itself when entering each successive preexisting loop.


The second drawing is the no less familiar stocking stitch — mailles de bas — as Vandermonde labels it, and one of the earliest appearances (if not the first) of that term in print, in any language.


Vandermonde recognizes that a numerical system for the description of looped structures can easily become complex enough to require adjunct text and drawings. Since the elimination of such need is his primary objective, he identifies each point where the thread crosses over itself with a simple three-digit figure. The coordinates are then presented in a grid paralleling their location in the drawings.

It is apparent that Vandermonde misappraised the potential utility of his system. It was cited extensively, including the two illustrations, in the section on industrial knitting in Platière’s Encyclopédie Méthodique from 1785, where the first illustration of slip stitch crochet is also found. However, the additional illustrations of knit structures there, which clone Vandermonde’s graphic style, dispense with the numerical coordinates altogether.

It is also apparent that the meaningful presentation of identifiers for the crossover points in fabric containing asymmetrical structures, or single loops worked into multiple other loops, will often require the use of drawings. Notwithstanding its limited utility in manufacture, an intriguing question remains about the extent to which the numerical mapping of the crossover points in a pre-existing piece of fabric can increase the accuracy of its structural analysis.

Recognizing the crossover points as an essential attribute and providing ready means for their quantification is reason in itself to see Vandermonde’s system as a watershed contribution to the study of “difficult questions about fabric.” Although likely less intentional, his exclusion of the selvedges and the starting and finishing structures from the primary description, provides additional useful focus to the analytic process.

The vertically interlooped stocking stitch does not require a separate indication of how a horizontal row is anchored to the preceding row. However, basic structures of crafts such as crochet and nalbinding cannot be described without an equally precise indication of how they are interlooped, not only vertically but also laterally. It remains to be determined how easily Vandermonde’s coordinates can be applied or adapted to this. If nothing else, the attention to detail needed for testing their viability may increase the accuracy of the drawings that are still essential to the analytic representation of any fabric structure.

*          *          *          *

  • The entire Vandermonde essay can be found here.
  • He was quite an interesting figure and his biographical details can be found here and here.
  • Further details about the 1785 text can be found here.