Mathematical Design for Knotted Textiles

Mathematics reveals facts inherent in nature, e.g., rotational symmetry of flowers, and fractals the system of arteries in the human body. Applications of mathematics can be found not only in biology, chemistry, and physics, but also in art, music, design, and architecture. Mathematical concepts have been adopted in the creation of various forms of art, ranging from geometrical sculptures to computer graphics. In return art can visually convey the phenomena of complex mathematical concepts to a wide audience, enabling people to understand things surrounding or even inside them.

In the field of textile art and design mathematics seems to have advanced the design of textiles, particularly in the field of technical textiles, e.g., models for entangled fibers (Lee and Ockendon 2005). Mathematics can also benefit the textile industry through the use of exact calculation (Woodhouse and Brand 1920, 1921). A more contemporary example is the “132 5. ISSEY MIYAKE” fashion collection that employed the mathematics of folding (Issey Miyake Inc 2018).

On the other hand, mathematicians have noticed the potential for the expression of mathematical concepts through textiles. A complex mathematical idea can be transformed into a material object to demonstrate proof of concept. Crocheting, knitting, and weaving are textile techniques that mathematicians often use to explore and communicate a variety of mathematical concepts. For example, the properties of the Lorenz Manifold are explicitly conveyed by through a crochet piece made from its computer-generated images (Osinga and Krauskopf 2004, 2014). Taimina (2009) uses crocheted models to illustrate the concept of hyperbolic geometry. Harris (1988, 1997) examines the mathematical content of various textile activities and illustrates how skills in mathematics can be acquired through the learning of textile crafts. She developed methods of teaching mathematics through looking at domestic textile craft objects. She used textiles to visualize mathematical concepts, such as symmetry, pairs, patterns, sets, lattices, tension, nets and solids, visual, tactile, and three-dimensional.

This chapter focuses on the relationship between knotted textiles and mathematics. It examines ways in which textile knot practice can be analyzed and discussed through the use of mathematics. Mathematical concepts involved in the analysis and discussion consist of knot theory (braid theory included), Wang tiling, and Rhombille tiling. The chapter aims to shed light on the relationship between these concepts with textile knot practice, especially how knot practice utilizes mathematical knot diagrams and tiling notations as design tools to generate novel knot structures and patterns that would not be created otherwise.

Craft knot practice discussed in this chapter is based on Nithikul Nimkulrat’s knot practice started in 2004 when she began her PhD research that explored the relationship between material and artistic expression in textile art (Nimkulrat 2009). The chosen material was paper string, which Nimkulrat had not used in her textile practice prior to her PhD study. The material had historical significance during the Second World War in Finland and is still locally produced in the country, the location of the study. As the focus of the research lied in the expressivity of material on the creation production, Nimkulrat decided to use no tool or machine to work with the material. Knotting became a natural choice, as it was a basic technique Nimkulrat learned during her childhood in Thailand, in handicraft classes and in scout camps. For over 10 years, Nimkulrat has specialized in knotted textiles, creating large-scale installations such as The White Forest (2008–2016) (Fig. 1). Prior to the work presented in this chapter, the work was exclusively monochrome.

Open image in new window

In Nimkulrat’s knot practice, the reef knot is key in the construction of the repetitive lacy structure. However, each reef knot has two additional central strands passing through the center (Fig. 2) that allow more reef knots to be connected, forming a circle shape (Fig. 3) and subsequently the lacy structure.

Open image in new window

Open image in new window

The design and process of knotting paper string into any three-dimensional forms in Nimkulrat’s knot practice prior to the work presented in this chapter was purely intuitive, as no sketch/drawing was made before the actual knotting. The knotting basically followed the rhythmic and coordinated movement of the left and right hands, employing tacit knowledge. As Sennett (2008: 94) points out: “… much of the knowledge craftsmen possess is tacit knowledge – people know how to do something but they cannot put what they know into words.” It is also the case of Nimkulrat’s knot practice. When communicating how knotting process was made, the articulation tended to be limited to “left hand over” or “right hand starting.” This has been transformed since Nimkulrat’s encountering with a mathematic knot diagram and wondered if there was any relationship between textile knots and mathematical knots and, if so, whether this would reveal ideas for new design and make the process of knotting more explicit. In order to characterize this knot in terms of mathematical knot theory, it is necessary to consider some of the properties of mathematical knots.

“Knotting has been an important adjunct to the everyday life of all people from the earliest days of which we have knowledge” (Ashley 1944: 1). Knots have been commonly used for nearly two thousand years for practical and decorative purposes in many cultures, such as Babylonian, Egyptian, Greek, Byzantine, Celtic, and Chinese art (Jablanand and Sazdanovic 2007). To a mathematician “a knot is a single closed curve that meanders smoothly through Euclidean three-space without intersecting itself” (van de Griend 1996: 205). It is a “closed loop in three-dimensional space,” which has “no free ends” (Devlin 1998: 248–249). To a textile practitioner, knotting or macramé is a craft technique commonly used in textile art and design. While free ends are essential for knotting textile work, “no string with free ends can be knotted” in mathematical knot theory, or topology (Devlin 1999: 234). This difference between craft and mathematical knots generates a question as to whether both types of knots share any similarities.

Knot theory is a subdivision of mathematical topology that examines properties of one-dimensional idealized objects, including knots, links, braids, and tangles (Adams 1994). These idealized objects consist of infinite thread and can be continuously deformed and reformed without breaking (Sossinsky 2002). The study of knots in mathematics focuses on properties relating to the positions of threads in space, the patterns of knots, and the number of crossings (Adams 1994: 2–4). It is not concerned with physical properties, such as tension, size, and the shape of individual loops (Devlin 1998: 247–249. A central problem in knot theory is to determine whether different representations are representations of equivalent (the same) knots. Solving this problem is concerned with seeking a property of each knot that does not change when the knot is subjected to manipulations. For each property, such as the number of crossings, a knot invariant might be defined (Devlin 1998: 249–254).

Mathematical knot diagrams are used as tools to illuminate knot properties in order to confirm whether equivalence exists. Problem solving becomes highly visual through the use of knot diagrams. A knot diagram is a representation or projection of a mathematical knot using simple line drawings to indicate the knot pattern. Broken lines in the diagram show where the knot crosses itself (Fig. 4).

Open image in new window

In Fig. 4, each diagram is an equivalent representation of the same knot. The loop in the first diagram can gradually be removed to show a representation of the figure-eight knot in the fourth diagram.

It can be seen in Table 1 that the differences between textile knot practice and mathematical knots are significant. However, in Nimkulrat’s practice, many of the differences may largely be ignored as the same knot, the same material, spacing, tension and form are employed consistently. The role of loose ends becomes the key difference for further consideration through two approaches, namely, through considering loose ends to be joined as they defined in mathematical knot diagrams and through braid theory, a further branch of topology, where loose ends are permitted.

The next sections will focus on the use of the diagrammatic method in knot theory to analyze textile knot practice in which physical strings are hand-knotted to create three-dimensional artifacts. They will show how diagrams can provide a visual language to interrogate and record practice, visualize or simulate new knot designs prior to making, and inspire the use of new materials.

The analysis of textile knot practice using knot theory confirms the applicability of the diagrammatic method used in knot theory. The recognition of the repeat pattern and the active/passive nature of strands is taken further for the exploration of two aspects: (1) the use of color and (2) changing active and passive strands. The focus is on pattern as opposed to color, as pattern generation plays an important role in textile design practice. Figure 6 is recolored using two colors: black and gray. Gray replaces red and blue and black replaces green and yellow. A repeat pattern is immediately obvious; gray and black knots are on alternate rows. Where the active strands are gray, the knot appears gray. Where the active strands are black, the knot appears black. A further observation is that rectangles of one color appear with an internal area of the other color – this can potentially be a new two-color knot pattern (Fig. 7).

Open image in new window

This design possibility is confirmed by following Fig. 7 to knot black and white paper string in the sequence shown in the diagram: white–black–black–white. White and black strands alternately played an active role in the tying of knots, and eventually black and white circles became apparent (Fig. 8). The rectangles from Fig. 7 have become circles because of the material qualities of the paper string.

Open image in new window

Open image in new window

This step is an example of how diagramming can influence the design of knotted textiles that have always used one color to adopt two colors in textile knot practice.

The subsequent knotting experiment puts the strands in the following positions from left to right: black–black–white–white (Fig. 9a). To link individual knots, one knot is flipped before the tying took place (Fig. 9b), allowing all four strings of the same color to form a pure color knot, black or white (Fig. 9c). The process continues alternately between a row of mixed color knots and that of pure black and white ones, leading to a striped knot pattern (Fig. 9d).

Unlike the circle knot pattern, in this iteration practice comes first, and the diagram (Fig. 10) is used to confirm the intuitive design.

Open image in new window

The new striped design has been used to make three-dimensional artifacts (Fig. 11).

Open image in new window

Knot theory also illuminates the use of a new material in Nimkulrat’s knot practice, encouraging her to work with materials and structures that she did not previously considered. The mathematical definition of a knot according to knot theory provides an opportunity to interrogate textile practice. The definition of a knot as closed curves with no loose ends (Devlin 1999) provokes an idea of using new materials.

Returning to the knot described in Fig. 5, the loose ends of the same strands are joined (Fig. 12). It can be seen that the knot under discussion analyzed in this way represents not a knot with many crossings but a link with four components tangled together. Each component red (a), green (b), yellow (c), and blue (d) is a ring. Examining each ring individually, it may be seen that they do not contain crossings. Rings such as these are the simplest form of knot and are known as the trivial knot or the unknot.

Open image in new window

The diagrammatical representation of a link containing four trivial knots (Fig. 12), where the ends of the same strands of a knot are joined, is the main characteristic and inspiration for a new knot structure. Neoprene cord (5 cm thick), which has very different properties to paper string, is selected as the material. Its thickness and flexibility make loose ends be joined easily with adhesive, mirroring the “joining” in the diagramming (Fig. 13). A closer examination of an individual knot with all ends joined hints that it is possible to unravel it. The unraveling reveals that the knot actually contains four rings or trivial knots (Fig. 14). The unraveling of the component into four trivial knots shows the possibility of making craft knots from flexible materials that are originally in the ring form. Nimkulrat’s knot practice has always utilized thin and stiff paper string as the material. Once tight, paper string knots do not unravel, so this observation is facilitated through the use of neoprene. A series of trivial knots were then formed and links were made using active and passive components as per the diagrams, resulting in a new structure in Fig. 15. This aspect would expand choices of materials for textile knot practice (e.g., use flexible rings instead of lengths of string/cord) and may lead to a possibility of creating spherical or tubular forms from several rings joined together.

Open image in new window

Open image in new window

Open image in new window

This section considers whether a single knot in Fig. 2 can be characterized through braid theory. Both braid theory and knot theory come under the branch of mathematics known as topology. Braid theory however allows for loose ends. A braid may be imagined as a number of threads “attached ‘above’ (to horizontally aligned nails) and hanging ‘down,’ crossing each other without ever going back up; at the bottom, the same threads are also attached to nails, but not necessarily in the same order” (Sossinsky 2002: 15). Two braids may be considered equivalent if their strands can be rearranged without detaching at the top and the bottom or without cutting. A knot or link containing several components may be obtained from a braid by joining the top ends to the lower ends; the resulting knot is called a closed braid. According to Alexander’s theorem, every knot can be represented as a closed braid (Meluzzi et al. 2010). Sossinsky’s (2002) algebraic notation (letter codes) is introduced for describing the process of braiding a simple plait and the craft knot in Fig. 2 to specify which strand crosses over another in which direction. Considering the three strands as the group of a braid shown in Figs. 16 and 17, a strand must always occupy a position and may only move to an adjacent space.

Open image in new window

Open image in new window

A braid diagram may be drawn to show moves and corresponding notation. Figure 18 has been produced for the simple plait. The diagram contains all the information to produce the plait and the notation may be given as aBaBaB…, a shorthand or code.

Open image in new window

This method is applied to the craft knot in Fig. 2. Figure 19 illustrates the knotting steps required to produce this knot. Figure 20 shows the translation of these steps using braid notation.

Open image in new window

Open image in new window

In the formulation of the braid notation in Fig. 20, to complete Steps 3 and 6, the strand must first move upwards the braid to pass through a loop to tie the knot. Such a move is not permitted in braid theory that, by definition, all moves must be in a downward direction. This indicates that pure braid theory cannot be used to characterize the craft knot in question. In order to continue the analysis of this knot using braid theory, the theory is therefore modified to allow the upward move. The notation given for this move is *, which refers to not only the upward move of the strand that is last twisted over in the previous move, but also the passing of it through a loop created by the strand twisting over it. The notation that describes the knotting process of the Fig. 2 knot is abCba*CBaBc*.

The mathematical theory of tilings has to do with covering two-dimensional Euclidean spaces with tiles of various forms without gaps or overlays (Kaplan 2009: 3). Topologically individual tiles are closed continuous disks. A tiling is periodic if it is made up of regions of any size that repeat one after another. The shape of the tiles is arbitrary. Simple elements are regular triangular, square, and hexagonal tiles. The tiles may have matching conditions dictating which may be placed next to each other. These may be implemented as dents and notches but are usually indicated by various colors and coded with numbers.

The set of different kinds of available tiles is called a protoset. The individual tiles in a protoset are called prototiles (Mann 2004). An infinite number of copies of any tile may be used in the tiling. If a protoset can be used to tile the entire Euclidian plane, it is called valid. Each of the aforementioned tiles forms by themselves a valid set. There are also protosets that cannot be used to tile the plane, like the single pentagonal tile. A valid set that allows no regular tilings is an aperiodic set. Although the tiling theory is concerned with the Euclidian plane, the question may be asked about any surfaces. For example, we might be looking for a tiling that covers the surface of a finite cube or infinitely long cylinder. Also the tilings have topologies. For example, the regular square tiling is different from hexagonal tiling. On the other hand, tilings that look different may have similar properties. For example, a brick wall tiling has the same topology as the regular hexagonal tiling, even when the prototiles have different shapes.

An important special case includes the Wang tiles. Mathematically they are defined as unit squares with colored edges and the following matching conditions: (1) tiles may not be mirrored or rotated; (2) they are to be placed in the regular square grid; and (3) touching edges must have the same color (Lagae and Dutre 2006; Nurmi 2016). Regular, aperiodic, and invalid Wang tilesets are known. Since the definition is mathematical and the colors are defined by colors, which in turn may be implemented by shaping the edges, shapes of their actual representations may vary greatly.

Since knot diagrams are essentially planes of individual knots connected by strands, it is very intuitive to model them with tilings. What to be carried out is to assign a prototile for each knot type. In case of the strands being colored, their colors can be used as matching conditions for the tiles. These methods are used to explore some new textile knot patterns and structures. Based on the two-tone knot patterns in Figs. 8 and 9, the reef knot pattern with four strands is identified as a unit cell. The used colors dictate 16 different variations (Fig. 21). As the order of the strands never changes, the list represents an exhaustive binary coding of such designs. The circle and stripe patterns (Figs. 8 and 9) use only variations 0, 3, 6, 9, C, and F. Until now the 10 other variations (1, 2, 4, 5, 7, 8, A, B, D, and E) have not been employed in Nimkulrat’s textile knot practice. Clearly these could be used to create novel knot patterns and structures.

Open image in new window

The first column of Fig. 21 shows the canonical forms of the Wang tiles. As the actual knot patterns are in 45° angle, the second column shows the rotated tile, and the third the corresponding knot diagram. The fourth shows what an individual tightened physical knot would look like.

The 16 variations of knot tiles identified above can easily be used to design knot patterns. Essentially they guarantee regular shape and continuity of colors. For example, the pattern in Fig. 8 might have been designed using the tiling method (Fig. 22).

Open image in new window

Next we explored new possibilities with the same six tile variations (0, 3, 6, 9, C, and F). Variations of the pattern in Fig. 9 with different stripe widths (Fig. 23) take us beyond the checkerboard pattern.

Open image in new window

How about if the work is done the other way around? By using the principle of color matching, but giving up the requirement of regular square grid, valid non-Wang tilings could be produced and valid knot diagrams would also be automatically achieved. Half-step patterns are explored. In the tile space, the continuity of strands is enforced by matching colors. This approach radically alters the structural symmetry of the pattern (Fig. 24). It is also physically very different to knot, as the active and passive strands do not swap regularly – a characteristic of previously used patterns (Nimkulrat and Matthews 2016). Further experimentation revealed that this design can be knotted using three colors (Fig. 25) – of course to model this in tile space we need a larger protoset.

Open image in new window

Open image in new window

Three-dimensional patterns are another application for the tiling based design. In this example (Fig. 26), the common Wang topology (square lattice) is mostly retained, but some seams are marked with lines. Note how the triangle shape transforms a flat design into a three-dimensional form.

Open image in new window

In the next phase, the notion of square grid is abandoned and patterns are explored based on the Rhombille tiling. The opposite corners of a Rhombille tile are 60° or 120°. This property seems to have potential to generate novel knot designs that have different structures and characteristics.

Simple patterns are designed using a single prototile coding a single knot diagram unit (Fig. 27). As the tiles are no longer square, the knot laid out diagram becomes fairly complex. The colored version of the design highlights regular closed loops. It would be easy to implement them with different color, or even with different material, e.g., metal hoops.

Open image in new window

The first design applies the Rhombille tiling rule that each vertex has either six rhombi (all 60°) meeting at their acute corners, or three rhombi (all 120°) meeting at their obtuse corners, to create a Rhombille notation (Fig. 28). The tiles are placed in a way that all marks (i.e., colored strands) match, passing over the edge of the tiling from each tile to its adjacent tiles. While the middle blue-colored strands can pass over the edge of the whole tiling and create circle shapes, the yellow- and green-colored strands cannot do so. Regardless of the color matching, all strands appear continuous.

Open image in new window

Black and white paper string is used to knot Fig. 28: black for blue (passive) strands and white for yellow and green (active) strands. Figure 29 confirms the knottability of this notation. It is a completely new structure that the textile artist would not have been able to generate otherwise. On observing the knotted piece, it can be seen that the black strings do not always remain passive but occasionally are active in tying knots in order to form the circle shape to continue the knotting process. If the material used for the blue strands were in a ring shape instead of string, the blue strands would remain passive throughout the knotting process.

Open image in new window

Open image in new window

The second design combines the Rhombille tiling with a different type of isohedral tilings, P₄-55 (Grünbaum and Shephard 1987), of which all tiles appear in the same orientation (Fig. 30). Although all knots’ strands look continuous, this combined tiling rule ignores the congruity of knot strands’ colors, meaning that the sides of two tiles can be adjacent even when strands’ colors do not match. As a result, the blue strands did not always act as the passive strands in this notation. Figure 30 is used as a tool to knot the piece in Fig. 31. It turns out that the design has an interesting physical property that is not predicted from the diagram. When knotted it curves naturally, suggesting that it might be used to make tubular forms.

Open image in new window

Open image in new window

The third design integrates the P₄-55 into the Rhombille tiling, by using identical larger rhombi, each enclosing four prototiles in Fig. 27 placed in the same orientation (Fig. 32), to create a tiling notation (Fig. 33). Using of this notation to knot black and white paper string is more challenging than the previous notations; it requires a numbering system at the start of the knotting process. Again, the knotted piece naturally becomes three dimensional, a property that is not predicted from the diagram (Fig. 34).

Open image in new window

Open image in new window

This chapter demonstrates that it is possible to explore textile knot practice through the application of mathematics, especially knot theory and tiling concepts.

First, knot theory may be used to examine the properties of knotted textile structures, revealing significant differences between textile craft knots and knots in mathematical knot theory. Second, textile knot practice can adopt the diagrammatic method commonly used in mathematical knot theory as a design tool. Through the coloring of knot diagrams, the positions and roles of strands in a knotted structure become explicit, triggering an exploration of two-tone knot pattern designs. Third, the mathematical characterization of a single craft knot can inspire a new choice of material for knot practice, leading to a new design of knotted textiles with no loose ends. Fourth, modified braid theory can be used to characterize the same craft knot. Although mathematical braid theory in its pure form is invalid as no upward moves are permitted by definition, the characterization of the craft knot in question using a modified theory is useful because it explicitly shows the upward knotting move which is not otherwise obvious. Last, tiling notations that follow the Wang tiling concept and the Rhombille tiling rule can be used as design tools in textile knot practice to generate new knot patterns and structures.

The understanding of mathematical properties of craft knots can facilitate the communication of creative processes in a more objective and detailed way. Mathematical diagrams and notations reveal the nature of a particular knot types and stimulate new ideas, which may not have occurred otherwise. They may be used to design a variety of knotted textile structures that are visually different from one another, yet adopting only a single type of knot in a topological sense.

For future work, there are simple ways to extend tiling based design beyond this work. The braid theory suggests an infinite number of strands flowing in parallel and crossing each other. These may be modeled by knot diagrams running in parallel. These in turn may be reverse engineered into rectangular tiles with colored edges, which allows creative use of the color matching method. On the other hand, the method might be used in different types of tiling. For example, it is easy to create knot diagrams for the two tiles in the aperiodic P3 tiling.

Further, in these examples only one type of knot – the reef knot – is used. The tiles used may have different shapes, numbers of edge colors, and different numbers of colors on the edges. By definition the tiles are ignorant about their decoration and only care about the edge colors. Thus, several tiles with the same edge colors, but different diagrams, may even be achieved.