Mathematical Design for Knotted Textiles

  • Nithikul NimkulratEmail author
  • Tuomas Nurmi
Living reference work entry


This chapter examines the relationship between mathematics and textile knot practice, i.e., how mathematics may be adopted to characterize knotted textiles and to generate new knot designs. Two key mathematical concepts discussed are knot theory and tiling theory. First, knot theory and its connected mathematical concept, braid theory, are used to examine the mathematical properties of knotted textile structures and explore possibilities of facilitating the conceptualization, design, and production of knotted textiles. Through the application of knot diagrams, several novel two-tone knotted patterns and a new material structure can be created. Second, mathematical tiling methods, in particular the Wang tiling and the Rhombille tiling, are applied to further explore the design possibilities of new textile knot structures. Based on tiling notations generated, several two- and three-dimensional structures are created. The relationship between textile knot practice and mathematics illuminates an objective and detailed way of designing knotted textiles and communicating their creative processes. Mathematical diagrams and notations not only reveal the nature of craft knots but also stimulate new ideas, which may not have occurred otherwise.


Design Knot diagram Knot theory Knotted textiles Rhombille tiling Wang tiles 


  1. Adams C (1994) The knot book: an elementary introduction to the mathematical theory of knots. W H Freeman, New YorkzbMATHGoogle Scholar
  2. Ashley CW (1944) The Ashley book of knots. Faber and Faber, LondonGoogle Scholar
  3. Devlin K (1998) The language of mathematics: making the invisible visible. W H Freeman, New YorkzbMATHGoogle Scholar
  4. Devlin K (1999) Mathematics: the new golden age. Columbia University Press, New YorkzbMATHGoogle Scholar
  5. van de Griend P (1996) A history of topological knot theory. In: Turner JC, van de Griend P (eds) History and science of knots. World Scientific, Singapore, pp 205–260CrossRefGoogle Scholar
  6. Grünbaum B, Shephard GC (1987) Tilings and patterns. W H Freeman, New YorkzbMATHGoogle Scholar
  7. Harris M (1988) Common threads: mathematics and textiles. Math Sch 17(4):24–28Google Scholar
  8. Harris M (1997) Common threads: women, mathematics and work. Trentham Books, Stoke-on-TrentGoogle Scholar
  9. Issey Miyake Inc (2018) 132 5. Issey Miyake. Accessed 20 July 2008
  10. Jablanand S, Sazdanovic R (2007) LinKnot: knot theory by computer, vol 21. World Scientific, SingaporeCrossRefGoogle Scholar
  11. Kaplan CS (2009) Introductory tiling theory for computer graphics. Morgan & Claypool, San RafaelCrossRefGoogle Scholar
  12. Lagae A, Dutre P (2006) An alternative for Wang tiles: colored edges versus colored corners. ACM Trans Graph 25(4):1442–1459CrossRefGoogle Scholar
  13. Lee MEM, Ockendon H (2005) A continuum model for entangled fibres. Eur J Appl Math 16:145–160MathSciNetCrossRefGoogle Scholar
  14. Mann C (2004) Heesch’s tiling problem. Am Math Mon 111(6):509–517MathSciNetCrossRefGoogle Scholar
  15. Meluzzi D, Smith DE, Arya G (2010) Biophysics of knotting. Annu Rev Biophys 39:349–366CrossRefGoogle Scholar
  16. Nimkulrat N (2009) Paperness: expressive material in textile art from an artist’s viewpoint. University of Art and Design Helsinki, HelsinkiGoogle Scholar
  17. Nimkulrat N, Matthews J (2016) Novel textile knot designs through mathematical knot diagrams. In: Torrence E, Torrence B, Séquin C, McKenna D, Fenyvesi K, Sarhangi R (eds) Proceedings of Bridges 2016: mathematics, music, art, architecture, education, culture. Tessellations, Phoenix, pp 477–480Google Scholar
  18. Nurmi T (2016) From checkerboard to cloverfield: using Wang tiles in seamless non-periodic patterns. In: Torrence E, Torrence B, Séquin C, McKenna D, Fenyvesi K, Sarhangi R (eds) Proceedings of Bridges 2016: mathematics, music, art, architecture, education, culture. Tessellations, Phoenix, pp 159–166Google Scholar
  19. Osinga HM, Krauskopf B (2004) Crocheting the Lorenz manifold. Math Intell 26(4):25–37MathSciNetCrossRefGoogle Scholar
  20. Osinga HM, Krauskopf B (2014) How to crochet a space-filling pancake: the math, the art and what next. In: Greenfield G, Hart GW, Sarhangi R (eds) Bridges 2014: mathematics, music, art, architecture, culture. Tessellations, Phoenix, pp 19–26Google Scholar
  21. Sennett R (2008) The craftsman. Yale University Press, New HavenGoogle Scholar
  22. Sossinsky A (2002) Knots: mathematics with a twist. Harvard University Press, Cambridge, MAzbMATHGoogle Scholar
  23. Taimina D (2009) Crocheting adventures with hyperbolic planes. AK Peters, WellesleyCrossRefGoogle Scholar
  24. Woodhouse T, Brand A (1920) Textile mathematics: part I. Blackie & Son, LondonGoogle Scholar
  25. Woodhouse T, Brand A (1921) Textile mathematics: part 2. Blackie & Son, LondonGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.OCAD UniversityTorontoCanada
  2. 2.TurkuFinland

Section editors and affiliations

  • Bharath Sriraman
    • 1
  • Kyeong-Hwa Lee
    • 2
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA
  2. 2.Department of Mathematics Education, College of EducationSeoul National UniversitySeoulSouth Korea

Personalised recommendations