Introduction to the special issue on mathematics and fibre arts

Abstract

We begin by outlining our current understanding of mathematical fibre arts. We then describe the mathematics within this special issue, including the confluence of various ideas. We list the motivating questions that are common to mathematical fibre arts papers and contextualize the papers in this issue within that list. Finally, we describe the trajectory of publications in the field.

1. Mathematics and the fibre arts

Welcome to this Special Issue of Journal of Mathematics and the Arts, focusing on mathematics and fibre arts. What is this all about? That is, what does it mean to engage in mathematical fibre art work, as distinct from the general study of mathematics and the arts? While the Special Issue editors have engaged in mathematical study of fibre arts for many years, we did not try to define ‘mathematics and fibre arts’ at the outset of our journey. It was clear to us that knitting, weaving, and crochet ‘counted’ whereas paperfolding did not, despite the fact that paper is made substantively from fibre.

In thinking about this Special Issue, we tried to delineate the boundaries of the field more clearly. Certainly textile arts are included, as they focus on the making of fabrics by techniques such as weaving, knitting, felting, and crochet. But the fibre arts are broader, including embellishments of fabrics by stitching of all sorts (blackwork embroidery, sashiko, needlepoint, and cross stitch, to name a few), alterations of fabrics by sewing or quilting, and crafts that do not involve fabric at all such as macrame. Where does it end? Consulting dictionaries, we see that fibre itself is defined as long and thin, made from filaments, and indeed threadlike. Some definitions require that fibre be spinnable. Thus reeds, for example, are not fibres – they are long and thin, but are not as flexible as thread or as thin as filaments. This means that basketmaking is not a fibre art (as confirmed by Roots: Reclaiming our origins through traditional skills (2023), one of the few sites to offer classes in both fibre arts and basketry).

We now see that any elements unifying mathematics with fibre art have to be those not relying on anything having to do with thickness or rigidity of the components. Materials such as paper or reeds that are intrinsically rigid, so that when manipulated crease or hold fixed forms, are not permitted unless they are incorporated into yarn (see (Habu textiles, 2023) for examples). Similarly, stiff wire is not permitted but flexible wire can be used (see also (Habu textiles, 2023) for knittable wire). Finished objects may have rigidity, as when stiff fabrics are created by tight crochet gauge, but the rigidity comes from the construction rather than from using rigid materials. Notice, for example, that in Gould (2018) fabric layers are embroidered together to create polyhedral facets. In Hermann (2011) plastic pieces are assembled into cubes, but the mathematics analysed is of the patterns on the individual faces rather than over the cube as a whole; the project could have been made with carefully stuffed fabric instead and the mathematical analysis would have remained the same.

It is more straightforward to address the mathematics aspect of the field. Any level of mathematics, even arithmetic, can be used in conjunction with fibre arts to illuminate aspects of craft. For example, knitting a stockinette swatch to measure gauge incorporates multiplication and division in fundamental ways. However, we restrict this Special Issue to sophisticated mathematics in keeping with the standards of the Journal of Mathematics and the Arts.

2. Inspirations, mathematics, and motivations in this special issue

This Special Issue is devoted to items that have been created via a physical process, which is different from either conceptual creation or virtual creation. For example, knitting a wool sweater is a physical process, mentally knitting a sweater is a conceptual process, and writing Mathematica code that creates a model of a sweater is a virtual process. Many of this Issue's authors used computers to help realize their conceptual ideas, perhaps moving through a virtual stage of creation, but all had physical output as the goal. In fact, the physical creation of an item often reveals key observations and unexpected roadblocks to creating art pieces using a particular technique. Resolving the obstacles or explaining the observations is often a matter of more clearly articulating the connection between the mathematics and the fibre arts, which is no small feat.

In this section, we will introduce you to the material elucidated by the authors of this Special Issue. We will categorize the work in three different ways: by what inspired the work, by the content of its mathematical underpinnings, and by the type of mathematical question the work addresses. Slicing the contributions in these different ways will necessitate revisiting the same articles from different perspectives. Therefore, while this section might serve as a prologue to the Issue, it might also make a reasonable epilogue.

2.1. Inspirations

Many mathematicians are also fibre artists. Those who are interested in entering this field might wonder what motivates an investigation in mathematical fibre arts. Two of the authors (Shepherd and Roth) were set on their courses by contemplation of particular artistic pieces. Three of the papers (by Holden, Jensen, and Seaton & Hayes) were inspired by recognizing that their particular craft styles – Japanese braiding, sequence knitting, and hitomezashi stitching, respectively – could be thought of mathematically. Often a crafter's first inklings of such ties are simply that the craft ‘feels mathematical’. Thus, instinct can give a good indication that fertile mathematical ground lies beneath. A different sort of incentive drove authors Baker, Givens, and Gould. They each strove to physically realize mathematical concepts. Interestingly, all three of these authors made wearable works of art, and these works of art inspired further mathematical analysis. Wilmer modelled a rigid physical process (paperfolding) using knitting, and capitalized on the resulting fabric characteristics to create improved patterns for ordinary knitted items. Finally, Wildstrom embarked on a project to investigate what plane and two-colour symmetries are possible with a particular fibre arts technique. We will discuss this type of investigation more below.

We see with Wildstrom's work that the inspiration, mathematics, and underlying questions are closely related. His inquiry is centred around careful analysis of the possible symmetries in intermeshed crochet and subsequent enumeration of such patterns; therefore, the mathematics involves both symmetries and combinatorics, and the questions are: ‘What is possible?’ and ‘How many different variations are there among those that are possible?’. We will find that these questions arise regularly, but first we consider the mathematics of the papers in this Issue.

2.2. Mathematical underpinnings

Both Wildstrom and Shepherd examine two-colour symmetries. Shepherd's quilt-block paper relates group theory, symmetry types, and two-colour symmetries, whereas Wildstrom's paper uses a combinatorial lens. Givens, Jensen, Holden, and Roth also take combinatorial views on their work. Here, Wildstrom employs an algorithmic approach, Givens utilizes enumerative combinatorics, and the last three authors apply group-theoretic techniques to achieve enumerative results. Seaton & Hayes combine number-theoretic and combinatorial perspectives to achieve structure theorems on encoded patterns. The mathematical ideas in the remaining three papers are distinct from the others in this Issue, though the mathematics considered is well situated within the broader literature of the field. Gould's paper on orbifolds is ultimately geometric; Baker, Baker, & Wampler's paper on the triply invertible scarf is topological; Wilmer's paper uses mathematical modelling techniques in the design process.

Finally, we draw the readers' attention to a confluence of ideas across the papers by Seaton & Hayes, Jensen, and Wildstrom in this Issue. Each noted a type of duality while creating a mathematical model of hitomezashi, sequence knitting, and intermeshed crochet, respectively. Independently, the authors essentially encoded stitches in a (finite) row on one side of a piece of fabric as binary states, and noted the reversal of binary states that occurs when turning over the fabric. This aspect of each model is the same, though the papers go on to employ the models for different purposes.

2.3. Underlying motivators

With a variety of mathematical areas represented and a panoply of fibre arts studied, what are the typical questions that motivate the work of mathematical-fibre-arts authors? In the books Making Mathematics with Needlework and Figuring Fibres (Belcastro & Yackel, 2007, 2018), we give a list of such standard motivators. We choose our current favourite wording here.

1. Solving fibre-arts problems using mathematics

2. Illustrating mathematical concepts through the design and fabrication of a fibre-arts piece

3. Proving which mathematical concepts can and cannot be constructed using a given fibre art

4. Describing the mathematics intrinsically present in a given fibre art

5. Analysing of the mathematics intrinsically present in a given fibre art

Motivator (4) deserves additional descriptive detail. Whereas (4a) is about mathematizing a fibre art, (4b) uses that mathematization to analyse the mathematical landscape available to the medium. Mathematization in this context includes identifying physical constraints of the fibre arts medium and encoding those constraints with an appropriate mathematical description. Typically papers that seek to address (4a) go on to address (4b); therefore, we use the inclusive notation (4) to refer to both (4a) and (4b) in what follows.

Now we set the papers of this Issue in the context of our listed motivators.

Wilmer's developed library of knitting techniques to make possible folds of particular kinds uses mathematics to solve the fibre-arts problem of how to mimic origami in knitting; it is the primary example of motivator (1) in this issue.

There are three papers focused on illustrating mathematical concepts (motivator (2)). Baker, Baker, & Wampler describe to the reader the process of physically constructing a scarf made from three symmetrically nested tori, and ground that process in sophisticated topology. While creating the construction, motivator (1) came into play to assure symmetry. Givens instantiated a trinomial triangle as a shawl, and subsequently used the shawl to make mathematical observations (thus straying into the realm of motivator (4b)). Gould was inspired by the orbifolds of The Symmetries of Things (Conway et al., 2008) to create an intricate functional art piece.

Holden and Wildstrom combine motivators (3) and (4). Each mathematically describes a craft (Naiki braiding and interlocking crochet respectively), then uses this description to determine the possible symmetries of patterns made in that craft and to count possible patterns. Within this issue, Wildstrom's paper should be seen as a critical contribution to the set of papers determining the plane symmetries possible within a given form of fibre art (described in Section 3). We wonder, given the mathematical confluence between models for intermeshed crochet, hitomezashi, and sequence knitting, the extent to which his conclusions will carry over to these other forms.

Jensen's paper on sequence knitting is an exemplar of a response to motivator (4), in which she creates a mathematical model to encode sequence knitting and uses this model to prove theorems classifying sequence knitting patterns. Roth, Seaton & Hayes, and Shepherd also respond to motivator (4). Roth mathematized the hue shift afghan pattern to count the number of possibilities for a certain type of hue shift afghan, but was not satisfied with her attempts to fully encode the mathematics she observed in the pattern. Roth's reflection upon her own mathematical modelling attempts represents an explicit discussion of the process involved in (4a). This difficult process is often taken for granted by consumers of the results of mathematical fibre arts work. Seaton & Hayes explain how to think about hitomezashi stitching through a mathematical lens, give an atlas of binary encoded hitomezashi patterns, and justify associated mathematical facts about this type of stitching. Finally, Shepherd uses an observational study of a family of quilting patterns to simultaneously introduce both named and unnamed quilt square arrangements and analyse them from a group theoretic standpoint.

3. The development and growth of mathematical fibre arts as a field

We believe that mathematical fibre arts is beginning to coalesce as a field of study within mathematical art, which has itself become recognized by the AMS as a legitimate field, at least for the purposes of conference sessions (Expanded classification system for JMM 2022 & beyond, 2023). As evidence we note that within mathematical arts:

• There are some characteristic problems on which people work.

• Some people have bodies of work or programmes that they develop over time.

• Researchers in the community respond to one another's work.

All of these aspects are present in the mathematics of fibre arts as well. As shown in this issue, it is typical for mathematical fibre artists to examine the symmetries possible in particular types of crafts or patterns and typical for mathematicians to enumerate variants of a pattern or possibilities resulting from a technique. As selected examples, Holden has written a series of papers on particular types of intertwining fibres through weaving or braiding (Holden, 2017, 2021, 2022; Holden & Holden, 2016); Wildstrom has investigated a range of unusual aspects of crochet (Wildstrom, 2007, 2018, 2019); Gould has repeatedly used innovative sewing techniques to elucidate geometry (Gould, 2009, 2018; Gould & Gould, 2012, 2013, 2020); Goldstine has immersed herself in the study of symmetries in fibre arts (Baker & Goldstine, 2012a, 2012b, 2014; Goldstine, 2017, 2018; Goldstine & Yackel, 2022); and Yackel has elucidated the intersection between mathematics and temari (Yackel, 2005, 2009, 2012, 2022; Yackel & Belcastro, 2011). Wildstrom's paper in this issue was informed by Calderhead (2018). All papers determining the full range of symmetries of a given type expressible using a given fibre arts technique can be thought of as an outgrowth of, and as a response to, Shepherd's foundational work (Shepherd, 2007). As the field continues to gel, more authors respond to one another.

Moreover, there is a significant and growing body of work on mathematical fibre art. The regularly updated Mathematical Fibre Arts Reference List (Belcastro, 2023c) lists 178 papers spanning the years 1867–2001. Of those, 42 appeared prior to 1990, with twice as many about weaving as about other fibre arts. Only eight such papers were published in the 1990s. Then the field really took off, with significant growth decade-over-decade in the non-weaving realm shown by 39 papers appearing in the 2000s and at least 62 appearing in the 2010s. Seventeen papers on mathematical fibre arts have been published in this Journal of Mathematics and the Arts prior to the present issue.

This is the fourth compendium of mathematical fibre arts papers; the first three were Making Mathematics with Needelework (Belcastro & Yackel, 2007), Crafting by Concepts (Belcastro & Yackel, 2011), and Figuring Fibres (Belcastro & Yackel, 2018). These books were based on Joint Mathematics Meetings AMS Special Sessions accompanied by fibre arts exhibits, all of which are described at the website Belcastro (2023b). This Special Issue is our next step in the evolution of the field as suggested by Wilmer (2020), in which we present work in a ‘somewhat more specialized channel,’ targeted at art-experienced and mathematically sophisticated readers. It overlaps with the most recent JMM AMS Special Session, which along with the associated juried fibre arts exhibit is chronicled at a website (Belcastro, 2023a).

We hope that through reading the articles in this issue, you will become energized to engage with mathematical fibre arts or at the very least mathematical art in your chosen medium. This active and inclusive community has space for individuals to be involved at all levels and in different capacities. In addition to this journal, there are opportunities to contribute to the Bridges conference, to exhibit work at the Joint Mathematics Meetings and at the Bridges conference (see Selikoff (2023)), and to network with other mathematical fibre artists at the Joint Mathematics Meetings Knitting Circle.

We look forward to learning about your work in the future.

Best wishes,

Carolyn Yackel and sarah-marie belcastro, March 2023

Disclosure statement

No potential conflict of interest was reported by the author(s).