# Knots in Art

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## Abstract

**:**

## 1. Knots in Ancient Art

## 2. Mirror Curves

#### 2.1. Tamil Threshold Designs

#### 2.2. Tchokwe Sand Drawings

#### 2.3. Construction of Mirror-Curves

**Mixed****crossing**- Any mirror placed in a crossing point of two distinct curves connects them into one curve;
**Self-crossing**- Depending on its position, a mirror placed into a self-crossing point of an (oriented) curve either preserves the number of curves, or breaks the curve in two (Figure 13).

#### 2.4. Combinations of Mirror-Curves

**Construction rules:**

- The first rule describes how to combine two mirror curves that share one edge of an open cell on their borders (Figure 16a). Such a composition corresponds to the direct product of $KL$s, and it was probably one of the most exploited constructions in knotwork art. For given mirror curves ${M}_{1}$ and ${M}_{2}$, we will call this kind of direct product ×-direct product and denote it by ${M}_{1}\times {M}_{2}$. If we combine two mirror curves in this way, first with ${c}_{1}$, and the other with ${c}_{2}$ components, the result is a new mirror curve with ${c}_{1}+{c}_{2}-1$ components. Hence, the ×-direct product of two 1-component mirror curves is a new 1-component mirror curve. This idea was used, for example, in the Tchokwe designs and in many Celtic friezes.As a particular application of the first rule, we can add a single square to the border of any monolinear mirror curve. This transformation corresponds to adding an external loop to a $KL$ diagram. It does not change the number of components and can be repeated, since it has a decorative function in knotwork art. For example, the Tamil (unknot) design from Figure 9a is created by a series of external loop additions, beginning from the $RG[1,1]$; the knot design from Figure 9b by adding loops to the $RG[4,3]$; and the knot design from Figure 9c by adding loops to the $RG[5,3]$. The same construction is used for Tchokwe designs (Figure 10a).
- The second rule is the one defining the direct sum ${K}_{1}\#{K}_{2}$ in knot theory (Figure 16b) (Notice that the first and second rule, i.e., ×- and $\left|\right|$-product of mirror curves both correspond to the direct product of knots ${K}_{1}\#{K}_{2}$, but the first with a nugatory crossing. Knots or links obtained by using ×- and $\left|\right|$-product of mirror curves are ambient isotopic to ${K}_{1}\#{K}_{2}$, but not mutually isomorphic as knot or link diagrams.) In the language of mirror curves ${M}_{1}$ and ${M}_{2}$, it means that we cut one external edge of each mirror-curve ${M}_{1}$ and ${M}_{2}$, and reconnect them again to obtain a new mirror-curve that will be denoted by ${M}_{1}\parallel {M}_{2}$.
- The third rule is restricted to plate designs: Two monolinear plate designs whose overlap consists of exactly two cells will give a new monolinear plate design. The schematic interpretation of the third rule is given in Figure 16c.
- Adding plate design ${P}_{1}$ to plate design ${P}_{2}$ is an edge-to-edge identification of their border cells belonging to rectilinear borders (Figure 16d). In the same way, we can add a plate design ${P}_{1}$ to some mirror curve M placed in some polyomino. This operation can be generalized to adding plate designs to polyominoes.The fourth rule tells us that adding any design in a grid $RG[a,b]$, such that $b|a$ to any monolinear mirror curve M (or monolinear plate design ${P}_{2}$) along the edge b, produces a monolinear design (Figure 16d). In particular, any square $RG$ added to a monolinear design gives a new monolinear design.

- if our mirror curve M does not cover P completely, choose a mixed crossing point on M, symmetric to A and put a mirror symmetric to the mirror in A (Figure 17a${}_{1}$). If a symmetric point with this property does not exist, rotate the mirror in A for ${90}^{\circ}$ around its midpoint and then place the mirror symmetric to it (Figure 17a${}_{2}$);

## 3. Symmetry and Classification of Knotwork Designs

- isometric symmetry group of the pattern;
- tangle inscribed in the fundamental region.

## 4. Knot Mosaics and Mirror Curves

## 5. Knots in Modern Sculpture and Architecture

#### 5.1. Polyhedral Knots and Links

## 6. Conclusions

## Acknowledgments

## References

- Przytycki, J.H. Knots: From combinatorics of knot diagrams to combinatorial topology based on knots. Available online: http://arxiv.org/pdf/math/0703096.pdf (accessed on 31 May 2012).
- Bain, G. Celtic Art—The Methods of Construction; Dower: New York, NY, USA, 1973. [Google Scholar]
- Allen, J.R. Celtic Art in Pagan and Cristian Times; Methuen and Co.: London, UK, 1904. [Google Scholar]
- Gerdes, P. Sona Geometry from Angola: Mathematics of an African Tradition; Polimetrica International Science Publishers: Monza, Italy, 2006. [Google Scholar]
- Gerdes, P. Geometry from Africa: Mathematical and Educational Explorations; The Mathematical Association of America: Washington, DC, USA, 1999. [Google Scholar]
- Lissajous Knot. Available online: http://en.wikipedia.org/wiki/Lissajous_knot (accessed on 31 May 2012).
- Conway, J. An Enumeration of Knots and Links and Some of Their Related Properties. In Proceedings of the Conference Computational Problems, AbstractAlgebra, Oxford 1967; Leech, J., Ed.; Pergamon Press: New York, NY, USA, 1970. [Google Scholar]
- Gerdes, P. Reconstruction and extension of lost symmetries. Comput. Math. Appl.
**1989**, 17, 791–813, (Also In Symmetry: Unifying Human Understanding II; Hargittai, I., Ed.; Pergamon Press: Oxford, UK, 1986). [Google Scholar] [CrossRef] - Gerdes, P. On ethnomathematical research and symmetry. Symmetry Cult. Sci.
**1990**, 1, 154–170. [Google Scholar] - Gerdes, P. Une tradition géométrique en Afrique—Les dessins sur le sable, Volume 3: Analyse Comparative; L’Harmattan: Paris, France, 1995. [Google Scholar]
- Gerdes, P. Ethnomathematik dargestellt am Beispiel der Sona Geometrie; Spektrum Verlag: Heidelberg, Germany, 1997. [Google Scholar]
- Gerdes, P. Adventures in the World of Matrices; Nova Science Publishers (Series Contemporary Mathematical Studies): New York, NY, USA, 2008. [Google Scholar]
- Gerdes, P. Lunda Geometry: Mirror Curves, Designs, Knots, Polyominoes, Patterns, Symmetries; Lulu: Morrisville, NC, USA, 2007. [Google Scholar]
- Jablan, S.V. Mirror generated curves. Symmetry Cult. Sci.
**1995**, 6, 275–278. [Google Scholar] - Grünbaum, B.; Shephard, G.C. Tilings and Patterns; W.H.Freeman: New York, NY, USA, 1986. [Google Scholar]
- Cromwell, P.R. Celtic knotwork: Mathematical art. Math. Intell.
**1993**, 15, 36–47. [Google Scholar] [CrossRef] - Martin, G.E. Transformation Geometry; Springer-Verlag: New York, NY, USA, 1980. [Google Scholar]
- Jablan, S.V. Symmetry, Ornament and Modularity, Series on Knots and Everything 30; World Scientific: Singapore, 2002. [Google Scholar]
- Shubnikov, A.V.; Koptsik, V.A. Symmetry in Science and Art; Plenum Press: New York, NY, USA, 1974. [Google Scholar]
- Coxeter, H.S.M.; Moser, W.O.J. Generators and Relations for Discrete Groups; Springer-Verlag: New York, NY, USA, 1980. [Google Scholar]
- Washburn, D.; Crowe, D.W. Symmetries of Culture; University of Washington Press: Seattle, DC, USA, 1988. [Google Scholar]
- Emery, I. The Primary Structures of Fabrics. In Watson-Guptill Publications/Whitney Library of Design; The Textile Museum: Washington, DC, USA, 1995. [Google Scholar]
- Crowe, D.W. Introduction to Slavik Jablan’s Modular Games. Available online: http://www.emis.de/journals/NNJ/Crowe.html (accessed on 31 May 2012).
- Fathauer, R. KnoTiles. Available online: http://mathartfun.com/shopsite_sc/store/html/Tessellations/MakingFaces.html (accessed on 31 May 2012).
- Lomonaco, S.J.; Kauffman, L.H. Quantum knots and mosaics. Quantum Inf. Process.
**2008**, 7, 85–115. [Google Scholar] [CrossRef] - Kuriya, T. On a lomonaco-kauffman conjecture. 2008. Available online: http://arxiv.org/pdf/1106.3784.pdf (accessed on 31 May 2012).
- Sossinsky, A. Knots—Mathematics with a Twist; Harvard University Press: Cambridge, MA, USA, 2002. [Google Scholar]
- Jablan, S.; Radović, L.J.; Sazdanović, R.; Zeković, A. Mirror-curves and knot mosaisc. Comput. Math. Appl.
**2011**, arXiv:1106.3784v2 [math.GT]. [Google Scholar] - Tkalec, U.; Ravnik, M.; Čopar, S.; Žumer, S.; Muševič, I. Reconfigurable knots and links in chiral nematic colloids. Science
**2011**, 333, 62–65. [Google Scholar] [CrossRef] [PubMed] - Grossman, B. Bathsheba Sculpture. Available online: http://www.bathsheba.com/ (accessed on 31 May 2012).
- Van Wijk, J.J. Seifert View. Available online: http://www.win.tue.nl/~vanwijk/seifertview/ (accessed on 31 May 2012).
- Bulatov, A. Bulatov Abstract Creations. Available online: http://bulatov.org/ (accessed on 31 May 2012).
- Vallisser, T. Other Geometries in Architecture: Bubbles, Knots and Minimal Surfaces. Available online: http://www.springer.com/cda/content/document/cda_downloaddocument/978-88-470-1121-2_Wallisser.pdf (accessed on 31 May 2012).
- Mathematica Polyhedron Data. Available online: http://reference.wolfram.com/mathematica/ref/PolyhedronData.html/ (accessed on 31 May 2012).
- Zhang, Y.; Seeman, N.C. Construction of a DNA-truncated octahedron. J. Am. Chem. Soc.
**1994**, 116, 1661–1669. [Google Scholar] [CrossRef] - Qiu, W.-Y.; Zhai, X.-D. Molecular design of Goldberg polyhedral links. J. Mol. Struc. (Theochem)
**2005**, 756, 163–166. [Google Scholar] [CrossRef] - Jablan, S.; Sazdanović. LinKnot—Knot Theory by Computer; World Scientific: New Jersey, NJ, USA; London, UK; Singapore, 2007; Available online: http://math.ict.edu.rs/ or http://www.mi.sanu.ac.rs/vismath/linknot/index.html; (accessed on 31 May 2012). [Google Scholar]
- Kozlov, D. Topological Method of Construction of Point Surfaces as Physical Models. Available online: http://www.marhi.ru/AMIT/2008/spec08/papers/Kozlov/Kozlov02_paper_EAEA2007.pdf (accessed on 31 May 2012).
- Burt, M. Periodical Sponge Structures and Uniform Sponge Polyhedra in Nature and in the Realm of the Theoretically Imaginable. Vismath
**2007**, 9, 4. Available online: http://www.mi.sanu.ac.rs/vismath/burt/index.html (accessed on 31 May 2012). - Scharein, R. KnotPlot. Available online: http://knotplot.com/ (accessed on 31 May 2012).

**Figure 1.**(

**a**) Cylinder seal, Ur, Mesopotamia, ca. 2600–2500 B.C.; (

**b**) stamp seals, Anatolia, ca. 1700 B.C.

**Figure 14.**The successive introduction of internal mirrors in the $RG[2,2]$ that preserves a single curve.

**Figure 15.**Construction of a single mirror curve from the tiling (

**a**) by connecting edge mid-points (

**b**); tracing components (

**c**) and introducing a mirror (

**d**).

**Figure 21.**(

**a**) ‖-direct product in the Tchokwe design; (

**b**,

**c**) the application of Rule 3 in Tchokwe designs.

**Figure 22.**Derivation of Celtic monolinear cross knot design from plate design obtained using the Rule 4.

**Figure 26.**Construction of Celtic monolinear knot design (

**b**) by breaking the symmetry of the two-component symmetric design (

**a**).

**Figure 30.**Figure-eight knot and Borromean rings as knot mosaics [25].

**Figure 32.**Borromean rings ${6}_{2}^{3}$ synthesized in chiral nematic colloids [29] and missing links ${6}_{2}^{2}$ and ${9}_{49}^{2}$.

**Figure 34.**(

**a**) Mid-edge construction; (

**b**) cross-curve and double-line covering construction; (

**c**) edge doubling construction. Broken lines denote edges of the original graph deleted during the construction.

**Figure 35.**Escher’s solid from the Mathematica data base of polyhedra [34] and basic polyhedra obtained from it and its transforms (truncated, stellated and geodesated Escher solid) by mid-edge construction.

**Figure 36.**Basic polyhedra obtained from Escher’s solid and its transforms by cross-curve and double-line covering.

**Figure 38.**A Seifert surface corresponding to knot ${9}_{35}$ and different polyhedral knots and links.

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Jablan, S.; Radović, L.; Sazdanović, R.; Zeković, A.
Knots in Art. *Symmetry* **2012**, *4*, 302-328.
https://doi.org/10.3390/sym4020302

**AMA Style**

Jablan S, Radović L, Sazdanović R, Zeković A.
Knots in Art. *Symmetry*. 2012; 4(2):302-328.
https://doi.org/10.3390/sym4020302

**Chicago/Turabian Style**

Jablan, Slavik, Ljiljana Radović, Radmila Sazdanović, and Ana Zeković.
2012. "Knots in Art" *Symmetry* 4, no. 2: 302-328.
https://doi.org/10.3390/sym4020302