mathematical design for knotted textiles · 2018-12-18 · knitting, and weaving are textile...

Mathematical Design for Knotted Textiles

Nithikul Nimkulrat and Tuomas Nurmi

Contents

Introduction: Mathematics and Textiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Textile Knot Practice to Be Analyzed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3What is a Knot? Knot Theory and Its Diagrammatic Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5Comparison Between Textile Knot Practice and Mathematical Knot Theory . . . . . . . . . . . . . . 6Analysis of Textile Knot Practice Using Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7New Knot Pattern Designs Based on Knot Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Use of New Materials Inspired by Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Analysis of Textile Knot Practice Using Braid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Definition of Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Analysis of Textile Knot Practice Using Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19New Pattern and Structure Designs Based on Tiling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 19Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Abstract

This chapter examines the relationship between mathematics and textile knotpractice, i.e., how mathematics may be adopted to characterize knotted textilesand to generate new knot designs. Two key mathematical concepts discussed areknot theory and tiling theory. First, knot theory and its connected mathematicalconcept, braid theory, are used to examine the mathematical properties of knottedtextile structures and explore possibilities of facilitating the conceptualization,design, and production of knotted textiles. Through the application of knot

N. Nimkulrat (�)OCAD University, Toronto, Canadae-mail: [email protected]

T. NurmiTurku, Finlande-mail: [email protected]

© Springer Nature Switzerland AG 2019B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_39-1

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https://doi.org/10.1007/978-3-319-70658-0_39-1

2 N. Nimkulrat and T. Nurmi

diagrams, several novel two-tone knotted patterns and a new material structurecan be created. Second, mathematical tiling methods, in particular the Wangtiling and the Rhombille tiling, are applied to further explore the design possibil-ities of new textile knot structures. Based on tiling notations generated, severaltwo- and three-dimensional structures are created. The relationship betweentextile knot practice and mathematics illuminates an objective and detailedway of designing knotted textiles and communicating their creative processes.Mathematical diagrams and notations not only reveal the nature of craft knotsbut also stimulate new ideas, which may not have occurred otherwise.

KeywordsDesign · Knot diagram · Knot theory · Knotted textiles · Rhombille tiling ·Wang tiles

Introduction: Mathematics and 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 mathematicscan be found not only in biology, chemistry, and physics, but also in art, music,design, and architecture. Mathematical concepts have been adopted in the creationof various forms of art, ranging from geometrical sculptures to computer graphics.In return art can visually convey the phenomena of complex mathematical conceptsto a wide audience, enabling people to understand things surrounding or even insidethem.

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

On the other hand, mathematicians have noticed the potential for the expressionof mathematical concepts through textiles. A complex mathematical idea can betransformed into a material object to demonstrate proof of concept. Crocheting,knitting, and weaving are textile techniques that mathematicians often use to exploreand communicate a variety of mathematical concepts. For example, the properties ofthe Lorenz Manifold are explicitly conveyed by through a crochet piece made fromits 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 andillustrates how skills in mathematics can be acquired through the learning of textilecrafts. She developed methods of teaching mathematics through looking at domestictextile craft objects. She used textiles to visualize mathematical concepts, such assymmetry, pairs, patterns, sets, lattices, tension, nets and solids, visual, tactile, andthree-dimensional.

Mathematical Design for Knotted Textiles 3

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

Textile Knot Practice to Be Analyzed

Craft knot practice discussed in this chapter is based on Nithikul Nimkulrat’sknot practice started in 2004 when she began her PhD research that explored therelationship between material and artistic expression in textile art (Nimkulrat 2009).The chosen material was paper string, which Nimkulrat had not used in her textilepractice prior to her PhD study. The material had historical significance duringthe Second World War in Finland and is still locally produced in the country, thelocation of the study. As the focus of the research lied in the expressivity of materialon the creation production, Nimkulrat decided to use no tool or machine to workwith the material. Knotting became a natural choice, as it was a basic techniqueNimkulrat learned during her childhood in Thailand, in handicraft classes and inscout 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.

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

The design and process of knotting paper string into any three-dimensional formsin Nimkulrat’s knot practice prior to the work presented in this chapter was purelyintuitive, as no sketch/drawing was made before the actual knotting. The knottingbasically followed the rhythmic and coordinated movement of the left and righthands, employing tacit knowledge. As Sennett (2008: 94) points out: “ . . . muchof the knowledge craftsmen possess is tacit knowledge – people know how to dosomething but they cannot put what they know into words.” It is also the case ofNimkulrat’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 knotdiagram and wondered if there was any relationship between textile knots andmathematical knots and, if so, whether this would reveal ideas for new design andmake the process of knotting more explicit. In order to characterize this knot interms of mathematical knot theory, it is necessary to consider some of the propertiesof mathematical knots.


Fig. 1 (Left) The White Forest (2016), dimensions: 150 cm(w) × 250 cm(d) × 200 cm(h); (Right)detail of tubular structure. Material: paper string. (Photograph: Nithikul Nimkulrat)

Fig. 2 A single reef knotwith two additional centralstrands passing through thecenter. (Photograph: NithikulNimkulrat)


Fig. 3 A group of knotsforming a circle. (Photograph:Nithikul Nimkulrat)

What is a Knot? Knot Theory and Its Diagrammatic Method

“Knotting has been an important adjunct to the everyday life of all people fromthe earliest days of which we have knowledge” (Ashley 1944: 1). Knots havebeen commonly used for nearly two thousand years for practical and decorativepurposes in many cultures, such as Babylonian, Egyptian, Greek, Byzantine, Celtic,and Chinese art (Jablanand and Sazdanovic 2007). To a mathematician “a knotis a single closed curve that meanders smoothly through Euclidean three-spacewithout 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 textilepractitioner, knotting or macramé is a craft technique commonly used in textile artand design. While free ends are essential for knotting textile work, “no string withfree ends can be knotted” in mathematical knot theory, or topology (Devlin 1999:234). This difference between craft and mathematical knots generates a question asto whether both types of knots share any similarities.

Knot theory is a subdivision of mathematical topology that examines propertiesof one-dimensional idealized objects, including knots, links, braids, and tangles(Adams 1994). These idealized objects consist of infinite thread and can becontinuously deformed and reformed without breaking (Sossinsky 2002). The studyof knots in mathematics focuses on properties relating to the positions of threadsin 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 shapeof individual loops (Devlin 1998: 247–249. A central problem in knot theory is todetermine whether different representations are representations of equivalent (thesame) knots. Solving this problem is concerned with seeking a property of eachknot that does not change when the knot is subjected to manipulations. For each


Fig. 4 Equivalent representations of the same knot. (Diagrams: Janette Matthews)

property, such as the number of crossings, a knot invariant might be defined (Devlin1998: 249–254).

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

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

Comparison Between Textile Knot Practice and MathematicalKnot Theory

A comparison is made between textile knot practice and mathematical knot theory.First, while a mathematical knot is defined as a continuous curve and as suchdoes not have loose ends, knots created through textile knot practice do haveloose ends, which may sometimes be joined. Second, the form of knots createdthrough textile practice is influenced by the characteristics of the materials used,e.g., thickness of strand, elasticity, stiffness, or pliability, whereas mathematicalknot theory is not concerned with any of these. Third, the appearance of knotsused in textile knot practice is determined by the tension or tightness of a knot.In mathematical knot theory, both tight and loose textile knots have equivalentmathematical knot representations. Last, two knots in mathematical knot theory areconsidered equivalent if, after simplification such as the removal of any unnecessarycrossings, they have the same number of crossings and orientation (Fig. 4). Table1 summarizes the similarities and differences between textile knot practice andmathematical knot theory.

It can be seen in Table 1 that the differences between textile knot practice andmathematical knots are significant. However, in Nimkulrat’s practice, many of thedifferences may largely be ignored as the same knot, the same material, spacing,tension and form are employed consistently. The role of loose ends becomes thekey difference for further consideration through two approaches, namely, through


Table 1 A comparison between textile knot practice and mathematical knot theory

Property Textile practice Knot theory

Ends May have loose ends. Continuous curve with no looseends.

Material Textile material dependent, e.g.,thickness, stiffness.

Not concerned with materiality.Cross section of a stranddeemed to be a point.

Tension Textile practice dependent. Atight knot is very different from aloose knot. The space between isas important as the knot itself.

A tight knot has the samerepresentation as a loose knotso they are deemed equivalent.

Form The addition of extra loops orturning a knot over changes itsappearance.

If a knot may be simplified tothe same representation ofanother knot, they areconsidered equivalent.

Representation A photograph or sketch. Knot diagram.

considering loose ends to be joined as they defined in mathematical knot diagramsand through braid theory, a further branch of topology, where loose ends arepermitted.

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

Analysis of Textile Knot Practice Using Knot Theory

A single craft knot used in Nimkulrat’s work is analyzed and described bydiagrammatic representations commonly utilized in knot theory. The difference in“ends” is explored further using a knot diagram and its colorable property. Thecoloring of the diagram, which is a method used in mathematical knot theory todetermine equivalence (Adams 1994: 23–27), aids the visualization of the path ofeach strand. Through coloring the diagram, it becomes immediately obvious that theposition of strands does not change after knotting (Fig. 5). The red (a) strand startsin Position 1 and ends in Position 1, likewise the green (b) strand remains in Position2, the yellow (c) in 3, and the blue (d) in 4. Figure 5 diagrammatically representsthe craft knot in Fig. 2. The red (a) strand is in Position 1 (from the left) or a beforethe knot is tied and afterwards, the green strand in Position 2 or b, the yellow in 3or c, and the blue in 4 or d. This is not obvious from the work (Fig. 2).

As Nimkulrat’s textile structures are constructed from repeats of this knot, thediagram is extended to include four knots across and four knots down (Fig. 6). Onexamining Fig. 6, it is clear that the positions of strands remain constant even afterfour rows of knots have been tied. In addition to start/end positions, it becomes


Fig. 5 Knot diagramshowing positions of strandsof a single knot. (Diagram:Janette Matthews)

apparent that there are active and passive strands. The outer strands, red (a) andblue (d), in the first row of knots are active in knotting, while inner strands green(b) and yellow (c) are passive. This is reversed on the second row. The third rowuses the same strands as the knots on the first row and the fourth repeats thesecond. From the diagram, it is possible to determine both characteristics – start/endand active/passive. This is undetectable through observation of the physical work(Fig. 1) alone. Groups of four strands create knots along a row, and every alternaterow employs the same strands for tying.

New Knot Pattern Designs Based on Knot Diagrams

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


Fig. 6 Diagram of four knots across and four knots down. (Diagram: Janette Matthews)

This design possibility is confirmed by following Fig. 7 to knot black andwhite 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 rectanglesfrom Fig. 7 have become circles because of the material qualities of the paperstring.

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


Fig. 7 Figure 6 recolored using only two colors. (Diagram: Janette Matthews)

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

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

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


Fig. 8 New knot designemerges when followingFig. 7 to knot black and whitepaper string. (Photograph:Nithikul Nimkulrat)

Fig. 9 Black and white paper strings in a different position of strands generate a striped pattern.(Photograph: Nithikul Nimkulrat)

Use of New Materials Inspired by Knot Theory

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 previouslyconsidered. The mathematical definition of a knot according to knot theory providesan opportunity to interrogate textile practice. The definition of a knot as closedcurves 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 strandsare joined (Fig. 12). It can be seen that the knot under discussion analyzed in thisway represents not a knot with many crossings but a link with four components


Fig. 10 Diagram confirming the striped design in Fig. 9. (Diagram: Janette Matthews)

tangled together. Each component red (a), green (b), yellow (c), and blue (d) isa ring. Examining each ring individually, it may be seen that they do not containcrossings. Rings such as these are the simplest form of knot and are known as thetrivial knot or the unknot.

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 maincharacteristic 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. Itsthickness and flexibility make loose ends be joined easily with adhesive, mirroringthe “joining” in the diagramming (Fig. 13). A closer examination of an individual


Fig. 11 (Top) Black & White Striped Armchair (2014). Dimensions: 62 cm × 50 cm × 63 cm.(Bottom) Black & White Striped Knots (2015). Dimensions: 50 cm × 80 cm × 20 cm. (Photograph:Nithikul Nimkulrat)

knot with all ends joined hints that it is possible to unravel it. The unraveling revealsthat the knot actually contains four rings or trivial knots (Fig. 14). The unravelingof the component into four trivial knots shows the possibility of making craft knotsfrom flexible materials that are originally in the ring form. Nimkulrat’s knot practicehas always utilized thin and stiff paper string as the material. Once tight, paper string


Fig. 12 Nimkulrat’s reefknot with all ends joined. Itbecomes a link of four trivialknots. (Diagram: JanetteMatthews)

Fig. 13 Four strands of neoprene cord tied into a knot whose ends are joined to create a link offour trivial knots. (Photograph: Nithikul Nimkulrat)

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 andpassive components as per the diagrams, resulting in a new structure in Fig. 15. Thisaspect would expand choices of materials for textile knot practice (e.g., use flexiblerings instead of lengths of string/cord) and may lead to a possibility of creatingspherical or tubular forms from several rings joined together.

Analysis of Textile Knot Practice Using Braid Theory

This section considers whether a single knot in Fig. 2 can be characterizedthrough braid theory. Both braid theory and knot theory come under the branchof mathematics known as topology. Braid theory however allows for loose ends. Abraid may be imagined as a number of threads “attached ‘above’ (to horizontally


Fig. 14 The unraveling of the individual knot with no loose ends creates four rings or trivial knotsin the mathematics term. (Photograph: Nithikul Nimkulrat)

Fig. 15 A new knotted structure made of neoprene cord. (Photograph: Nithikul Nimkulrat)

Fig. 16 Representations ofmoves using Strands 1 and 2:Move “a” (left) and Move“A” (right). (Diagram: JanetteMatthews)

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 inthe same order” (Sossinsky 2002: 15). Two braids may be considered equivalentif their strands can be rearranged without detaching at the top and the bottom orwithout cutting. A knot or link containing several components may be obtainedfrom a braid by joining the top ends to the lower ends; the resulting knot is calleda closed braid. According to Alexander’s theorem, every knot can be represented asa closed braid (Meluzzi et al. 2010). Sossinsky’s (2002) algebraic notation (lettercodes) is introduced for describing the process of braiding a simple plait and thecraft 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, astrand must always occupy a position and may only move to an adjacent space.


Fig. 17 Representations ofmoves using Strands 2 and 3:Move “b” (left) and Move“B” (right). (Diagram: JanetteMatthews)

Table 2 Possible moves and notation for the group of a three-strand braid

Move Notation 1 Notation 2 Figure

Strand 1 over Strand 2 b1 a Figure 16 leftStrand 2 over Strand 1 b1

−1 A Figure 16 rightStrand 2 over Strand 3 b2 b Figure 17 leftStrand 3 over Strand 2 b2

−1 B Figure 17 right

The possible moves of the three strands are given a notation to describe theirbraiding processes (Table 2). Sossinsky (2002) refers to two different notations. Thefirst notation uses (1) subscripts to label the strands, e.g., b1 for Strands 1 and 2, b2for Strands 2 and 3, and (2) a superscript to identify the inverse move of “right overleft” move, e.g., b1

−1, indicating that Strands 1 and 2 are involved in the move andthat Strand 2 is twisted over Strand 1 from right to left (Fig. 16 right). The secondnotation uses letters a, A, b, and B to represent b1, b1

−1, b2, and b2−1, respectively.

As it is more legible, the second notation was adopted here.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 theinformation to produce the plait and the notation may be given as aBaBaB . . . , ashorthand or code.

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

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 mustbe in a downward direction. This indicates that pure braid theory cannot be usedto characterize the craft knot in question. In order to continue the analysis of thisknot 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 ofthe strand that is last twisted over in the previous move, but also the passing of itthrough a loop created by the strand twisting over it. The notation that describes theknotting process of the Fig. 2 knot is abCba*CBaBc*.


Fig. 18 A braid diagram forthe simple plait and algebraicnotation for each move.(Diagram: Janette Matthews)

Fig. 19 Steps involved in tying a knot in Fig. 2. (Diagram: Janette Matthews)


Fig. 20 The modified braid diagram for Nimkulrat’s knot. (Diagram: Janette Matthews)

Definition of Tilings

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

The set of different kinds of available tiles is called a protoset. The individual tilesin a protoset are called prototiles (Mann 2004). An infinite number of copies of anytile may be used in the tiling. If a protoset can be used to tile the entire Euclidianplane, 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 singlepentagonal tile. A valid set that allows no regular tilings is an aperiodic set. Althoughthe tiling theory is concerned with the Euclidian plane, the question may be askedabout any surfaces. For example, we might be looking for a tiling that covers thesurface 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 theother hand, tilings that look different may have similar properties. For example, abrick wall tiling has the same topology as the regular hexagonal tiling, even whenthe prototiles have different shapes.

An important special case includes the Wang tiles. Mathematically they aredefined 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 regularsquare grid; and (3) touching edges must have the same color (Lagae and Dutre2006; Nurmi 2016). Regular, aperiodic, and invalid Wang tilesets are known. Sincethe definition is mathematical and the colors are defined by colors, which in turnmay be implemented by shaping the edges, shapes of their actual representationsmay vary greatly.

Analysis of Textile Knot Practice Using Tilings

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 aprototile for each knot type. In case of the strands being colored, their colors can beused as matching conditions for the tiles. These methods are used to explore somenew 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 usedcolors dictate 16 different variations (Fig. 21). As the order of the strands neverchanges, the list represents an exhaustive binary coding of such designs. The circleand stripe patterns (Figs. 8 and 9) use only variations 0, 3, 6, 9, C, and F. Until nowthe 10 other variations (1, 2, 4, 5, 7, 8, A, B, D, and E) have not been employed inNimkulrat’s textile knot practice. Clearly these could be used to create novel knotpatterns and structures.

The first column of Fig. 21 shows the canonical forms of the Wang tiles. Asthe 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 individualtightened physical knot would look like.

New Pattern and Structure Designs Based on Tiling Concepts

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


Wangnotation

Wangnotation

Wangnotation

RotatedWang

RotatedWang

RotatedWang

KnotDiagram

KnotDiagram

KnotDiagram

Tightenedknot

Tightenedknot

Tightenedknot

Fig. 21 The 16 knot units identified. (Diagram: Tuomas Nurmi)

Fig. 22 From left: pattern as Wang tiles (6 and 9 in combination), generated knot diagram,predicted, and actual outcome. (Diagram: Tuomas Nurmi. Photograph: Nithikul Nimkulrat)

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) takeus beyond the checkerboard pattern.

How about if the work is done the other way around? By using the principleof color matching, but giving up the requirement of regular square grid, validnon-Wang tilings could be produced and valid knot diagrams would also beautomatically achieved. Half-step patterns are explored. In the tile space, thecontinuity of strands is enforced by matching colors. This approach radically altersthe structural symmetry of the pattern (Fig. 24). It is also physically very differentto knot, as the active and passive strands do not swap regularly – a characteristic of


Fig. 23 Tiling-based designs and sample knots of two stripe patterns. (Diagram: Tuomas Nurmi.Photograph: Nithikul Nimkulrat)

Fig. 24 Tiling-based design that discards the Wang square grid topology. (Diagram: TuomasNurmi. Photographs: Nithikul Nimkurat)

previously used patterns (Nimkulrat and Matthews 2016). Further experimentationrevealed that this design can be knotted using three colors (Fig. 25) – of course tomodel this in tile space we need a larger protoset.

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

In the next phase, the notion of square grid is abandoned and patterns areexplored based on the Rhombille tiling. The opposite corners of a Rhombille tile are60◦ or 120◦. This property seems to have potential to generate novel knot designsthat have different structures and characteristics.

Simple patterns are designed using a single prototile coding a single knot diagramunit (Fig. 27). As the tiles are no longer square, the knot laid out diagram becomesfairly complex. The colored version of the design highlights regular closed loops.


Fig. 25 The design inFig. 24 using three colorsinstead of two. (Photograph:Nithikul Nimkulrat)

Fig. 26 Three-dimensional knot structure based on color coded tiling design. (Diagram: TuomasNurmi. Photographs: Nithikul Nimkurat)

Fig. 27 A single prototilecoding a single knot diagram.The passive strands are in onecolor and the active strandsare in two different colors.(Diagram: Tuomas Nurmi)

It would be easy to implement them with different color, or even with differentmaterial, e.g., metal hoops.

The first design applies the Rhombille tiling rule that each vertex has either sixrhombi (all 60◦) meeting at their acute corners, or three rhombi (all 120◦) meeting attheir obtuse corners, to create a Rhombille notation (Fig. 28). The tiles are placed ina way that all marks (i.e., colored strands) match, passing over the edge of the tilingfrom each tile to its adjacent tiles. While the middle blue-colored strands can passover the edge of the whole tiling and create circle shapes, the yellow- and green-


Fig. 28 Rhombille tiling of a reef knot. (Diagram: Tuomas Nurmi)

colored strands cannot do so. Regardless of the color matching, all strands appearcontinuous.

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 theknottability of this notation. It is a completely new structure that the textile artistwould 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 occasionallyare active in tying knots in order to form the circle shape to continue the knottingprocess. If the material used for the blue strands were in a ring shape instead ofstring, the blue strands would remain passive throughout the knotting process.

The second design combines the Rhombille tiling with a different type ofisohedral tilings, P4-55 (Grünbaum and Shephard 1987), of which all tiles appear


Fig. 29 Knotted workfollowing Fig. 28.(Photograph: NithikulNimkulrat)

Fig. 30 Rhombille tiling of a reef knot – second variation. It combines the Rhombille tiling witha different type of isohedral tilings – P4-55. (Diagram: Tuomas Nurmi)


Fig. 31 Knotted workfollowing Fig. 30.(Photograph: NithikulNimkulrat)

Fig. 32 Four prototilesplaced according to the P4-55tiling rule to form a larger tile.(Diagram: Tuomas Nurmi)

in the same orientation (Fig. 30). Although all knots’ strands look continuous, thiscombined tiling rule ignores the congruity of knot strands’ colors, meaning that thesides of two tiles can be adjacent even when strands’ colors do not match. As aresult, 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 designhas an interesting physical property that is not predicted from the diagram. Whenknotted it curves naturally, suggesting that it might be used to make tubular forms.

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


Fig. 33 Rhombille tiling of a reef knot – third variation. It integrates a different type of isohedraltilings P4-55 into the Rhombille tiling. (Diagram: Tuomas Nurmi)

Fig. 34 Knotted work following Fig. 33. (Photograph: Nithikul Nimkulrat)


Conclusion

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

First, knot theory may be used to examine the properties of knotted textilestructures, revealing significant differences between textile craft knots and knots inmathematical knot theory. Second, textile knot practice can adopt the diagrammaticmethod commonly used in mathematical knot theory as a design tool. Through thecoloring of knot diagrams, the positions and roles of strands in a knotted structurebecome explicit, triggering an exploration of two-tone knot pattern designs. Third,the mathematical characterization of a single craft knot can inspire a new choice ofmaterial for knot practice, leading to a new design of knotted textiles with no looseends. 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 movesare permitted by definition, the characterization of the craft knot in question usinga modified theory is useful because it explicitly shows the upward knotting movewhich is not otherwise obvious. Last, tiling notations that follow the Wang tilingconcept and the Rhombille tiling rule can be used as design tools in textile knotpractice to generate new knot patterns and structures.

The understanding of mathematical properties of craft knots can facilitatethe communication of creative processes in a more objective and detailed way.Mathematical diagrams and notations reveal the nature of a particular knot typesand stimulate new ideas, which may not have occurred otherwise. They may beused to design a variety of knotted textile structures that are visually different fromone 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 thiswork. The braid theory suggests an infinite number of strands flowing in parallel andcrossing 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, themethod might be used in different types of tiling. For example, it is easy to createknot diagrams for the two tiles in the aperiodic P3 tiling.

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

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