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Projections of Knots and Links

  • Alexander ÅströmEmail author
  • Christoffer ÅströmEmail author
Living reference work entry

Abstract

This chapter presents an introduction to the archaeological and historical aspects of the use of projections of knots and links for decorative purposes. These projections of knots and links have been created by man on various artifacts, for at least some 4500 years. Apparently, the most prominent types of artifacts in most of the studied civilizations and cultures have been decorated with knots and links. Some examples are the Greek and Roman mosaics of the Roman world and various artifacts and art forms in the British Isles in the Anglo-Saxon era. The working hypothesis is that these knots and links are potential projections of real-life three-dimensional objects or at least that the intention of their creators was to present a feeling or sense in the beholder that the images were depictions of actual physical knots or links. The topological study of knots and links consider objects in \(\mathbb {E}^{3}\). However, these projections can in one way be seen as two-dimensional designs and could thus be studied with a geometrical approach in \(\mathbb {E}^{2}\) instead or at least to some extent. From the perspective of decorative art, this may in fact be sufficient and even perhaps more convenient.

Keywords

Archaeology History Geometry Decorative art Interlaced patterns Knotworks Knots Links 

References

  1. Adams CC (2000) The knot book – an elementary introduction to the mathematical theory of knots, 1st edn. W.H Freeman and Company, New YorkGoogle Scholar
  2. Adams CC (2008) A brief introduction to knot theory from the physical point of view. In: Buck D, Flapan E (eds) Proceedings of symposia in applied mathematics – applications of knot theory, vol 66. American Mathematical Society, Providence, pp 1–20Google Scholar
  3. Alexander JW (1928) Topological invariants of knots and links. Trans Am Math Soc 30(2): 275–306MathSciNetCrossRefGoogle Scholar
  4. Allen JR (1883) On the discovery of a sculptured stone at St Madoes, with some notes on interlaced ornament. Proc Soc Antiquaries Scotland 17:211–271Google Scholar
  5. Allen JR (1912) Celtic art in pagan and Christian times, 2nd edn. Dover Publication, New York. Reprinted 2001Google Scholar
  6. Allen JR, Anderson J (1903) The early Christian monuments of Scotland. The Pinkfoot Press, Balgavies, by Forfar, Angus. Reprinted 1993Google Scholar
  7. Anderson J (1881) Scotland in early Christian times. Kessinger Publishing, La Vergne. Reprinted 2009Google Scholar
  8. Arbman H (1940) Birka I die Gräber – Tafeln. Almqvist & Wiksells Boktryckeri Aktiebolag, Uppsala. Kungl. Vitterhets Historie och Antikvitets AkademienGoogle Scholar
  9. Arbman H (1943) Birka I die Gräber - Text. Almqvist & Wiksells Boktryckeri Aktiebolag, Uppsala. Kungl. Vitterhets Historie och Antikvitets AkademienGoogle Scholar
  10. Artin E (1925) Theorie der Zöpfe. Abh Math Semin Univ Hamb 4:47–72MathSciNetCrossRefGoogle Scholar
  11. Arwidsson G (1942) Vendelstile – Email und Glas Im 7.-8 Jahrhundert. Almqvist & Wiksells Boktryckeri Aktiebolag, UppsalaGoogle Scholar
  12. Ashley C (1944) The Ashley book of knots, 1st edn. Doubleday, New YorkGoogle Scholar
  13. Åström A, Åström C (2011) Circular knotworks consisting of pattern No.295: a mathematical approach. J Math Arts 5(4):185–197MathSciNetCrossRefGoogle Scholar
  14. Åström A, Åström C (2015) Circular knotworks II: combining pattern No.295 with Turk’s heads. J Math Arts 9(3–4):91–102MathSciNetCrossRefGoogle Scholar
  15. Åström A, Åström C (2017) A practical approach to circular knotworks consisting of pattern No.295. International Guild of Knottyers/Gipping Press, Needham MarketGoogle Scholar
  16. Atil E (1975) Art of the Arab world. Smithsonian Institution, Washington, DCGoogle Scholar
  17. Bain G (1973) Celtic art the methods of construction. Dover, MineolaGoogle Scholar
  18. Bain I (1986) Celtic knotwork. BAS Printers Ltd, Over WallopGoogle Scholar
  19. Balmelle C, Blanchard-Lemée M, Darmon J, Gozlan S, Raynaud M (2002) Le Décor Géométrique De La Mosaïque Romaine: II. Répertoire graphique et descriptif décors centrés. Picard, ParisGoogle Scholar
  20. Becker L, Kondoleon C (2005) The arts of antioch – art historical and scientific approaches to Roman mosaics and a catalogue of the Worecester art museum antioch collection. Worcester Art Museum, WorcesterGoogle Scholar
  21. Birman JS (1974) Braids, links, and mapping class groups. Annals of mathematics studies, 1st edn. Princeton University Press, PrincetonGoogle Scholar
  22. Black J, Green A (1992) Gods, demons and symbols of ancient Mesopotamia. The British Museum Press, LondonGoogle Scholar
  23. Brochmann D, (1941) Tyrkeknoper. Norsk Sjøfartsmuseum, 1st edn. Oslo, NorwayGoogle Scholar
  24. Buchanan B (1981) Early near eastern seals in the Yale Babylonian collection. Yale University Press, New HavenGoogle Scholar
  25. Bush T (1985) Form and decoration of arrows from the highlands of Papua New Guinea. Rec Aust Mus 37(5):255293CrossRefGoogle Scholar
  26. Canute K (2008) Hypothesis, rule or law? The fourth installment on a philosophy of knots. Knotting Matters 27(98):28–29Google Scholar
  27. Carmichael EK (1922) The elements of Celtic art, 1st edn. An Comunn Gaidhealach, GlasgowGoogle Scholar
  28. Carnegie H (ed) (1908) Catalogue of the collection of antique gems formed by James Ninth Earl of southeast K.T., vol 2. Bernard Quaritch, LondonGoogle Scholar
  29. Carter E, Bahrani Z, André-Salvini B, Caubet A, Tallon F, Aruz J, Deschesne O (1992) The old elamite period. In: Harper PO, Aruz J, Tallon F (eds) The royal city of Susa – ancient near eastern treasures in the Louvre, pp 81–120. The Metropolitan Museum of Art, New YorkGoogle Scholar
  30. Clagett M (1999) Ancient Egyptian science a source book, vol 3. Ancient Egyptian mathematics of memoirs of the American philosophical society, vol 232. American Philosophical Society, PhiladelphiaGoogle Scholar
  31. Coleman J (1997) Turks head knots and the rule of the greatest common factor. Knotting Matters 16(57):22–25Google Scholar
  32. Coleman J (2008) Turks head knots and the role of the greatest common factor. Knotting Matters 27(100):31–33Google Scholar
  33. Conrad NJ (2009) A female figurine from the basal Aurignacian of Hohle Fels Cave in southwestern Germany. Nature 459(7244):248–252CrossRefGoogle Scholar
  34. Conway JH, Burgiel H, Goodman-Strauss C (2008) The symmetries of things. Ak Peters series. CRC Press, Taylor & Francis Group, New YorkGoogle Scholar
  35. Coxeter HSM (1989) Introduction to geometry, 2nd edn. Wiley, New YorkGoogle Scholar
  36. Cromwell PR (1993) Celtic knotwork: mathematical art. Math Intelligencer 15(1):36–47MathSciNetCrossRefGoogle Scholar
  37. Cromwell PR (2004) Knots and links, 1st edn. Cambridge University Press, CambridgeGoogle Scholar
  38. Cromwell PR (2008) The distribution of knot types in celtic interlaced ornament. J Math Arts 2(2):61–68MathSciNetCrossRefGoogle Scholar
  39. Cromwell PR (2010) Islamic geometric designs from the Topkapı scroll II: a modular design system. J Math Arts 4(3):119–136MathSciNetCrossRefGoogle Scholar
  40. Cubbon A (1996) The art of the Manx crosses. Manx National Heritage, The Manx Museum and National Trust, Douglas, Isle of ManGoogle Scholar
  41. Cumming JG (1857) The Runic and other monumental remains of the isle of man. Bell and Daldy, LondonGoogle Scholar
  42. de Morgan J (1924) Prehistoric man – an general outline of prehistory. The Edinburgh Press, EdinburghGoogle Scholar
  43. Delaporte L (1923) Catalogue des Cylindres – Cachets et Pierres Gravées de Style Oriental. Librairie Hachette, ParisGoogle Scholar
  44. Denham S (2013) The meanings of late Neolithic stamp seals in North Mesopotamia. PhD dissertation, University of ManchesterGoogle Scholar
  45. Dunbabin MK (2006) Mosaics of the Greek and Roman world. Cambridge University Press, CambridgeGoogle Scholar
  46. Dunham D (2000) Hyperbolic celtic knot patterns. In: Sarhangi R (ed) Bridges: mathematical connections in art, music, and science. Southwestern College, Winfield, pp 13–22Google Scholar
  47. El-Said I, Parman A (1976) Geometric concepts in Islamic art. World of Islam Festival Publishing Company Ltd, LondonGoogle Scholar
  48. Finlay I (1973) Celtic art – an introduction. Noyes Press, New JerseyGoogle Scholar
  49. Fisher G, Mellor B (2004) On the topology of celtic knot designs. In: Sarhangi R, Séquin C (eds) Bridges: mathematical connections in art, music, and science. Southwestern College, Winfield, pp 37–44Google Scholar
  50. Frankfort H (1939) Cylinder seals – a documentary essay on the art and religion of the ancient near east. Macmillan and Co., LondonGoogle Scholar
  51. Frankfort H (1955) Stratified cylinder seals from the Diyala region. The University of Chicago Oriental Institute Publications, vol LXXII. The University of Chicago Press, ChicagoGoogle Scholar
  52. Fyfe A (2008) Gender, mobility and population history: exploring material culture distributions in the Upper Sepik and Central New Guinea. PhD dissertation, The University of AdelaideGoogle Scholar
  53. Gayet A (1893) L’Art Arabe. Ancienne Maison Quantin, ParisGoogle Scholar
  54. Gerdes P (1989) Reconstruction and extension of lost symmetries: examples from the tamil of South India. Comp Math App 17(4–6):791–813Google Scholar
  55. Gerdes P (1999) Geometry from Africa: mathematical and educational explorations. The Mathematical Association of America, WashingtonGoogle Scholar
  56. Gerdes P (2007a) Drawings from Angola: living mathematics. Privately Published, MaputoGoogle Scholar
  57. Gerdes P (2007b) LUNDA geometry: mirror curves, designs, knots, polyominoes, patterns, symmetries, 2nd edn. Lulu Enterprises, MorrisviHe. First published in 1996Google Scholar
  58. Grant B (2002) Encyclopedia of rawhide and leather braiding, 1st edn. Cornell Maritime Press, CentrevilleGoogle Scholar
  59. Grünbaum B, Shephard GC (1985) Symmetry groups of knots. Math Mag 58(3):161–165MathSciNetCrossRefGoogle Scholar
  60. Grünbaum B, Shephard GC (1989) Tilings and patterns – an introduction, 1st edn. W. H Freeman and Company, New YorkGoogle Scholar
  61. Grünbaum B, Shephard GC (1992) Interlace patterns in islamic and moorish art. Leonardo 25(3/4):331–339. Visual mathematics: special double issueCrossRefGoogle Scholar
  62. Hall T (1996) Introduction to Turk’s-head knots, 1st edn. Privately PublishedGoogle Scholar
  63. Heath STL (1956a) The thirteen books of Euclid’s elements, 2nd edn. Books I and II, vol 1. Dover Publications Inc., New YorkGoogle Scholar
  64. Heath STL (1956b) The thirteen books of Euclid’s elements, 2nd edn. Books III and IX, vol 2. Dover Publications Inc., New YorkGoogle Scholar
  65. Heath STL (1956c) The thirteen books of Euclid’s elements, 2nd edn. Books X and XIII, vol 3. Dover Publications Inc., New YorkGoogle Scholar
  66. Herzfeld EE (1941) Iran in the ancient east – archaeological studies presented in the Lowell lectures at Boston. Oxford University Press, LondonGoogle Scholar
  67. Hiebert FT (1994) Origins of the bronze age oasis civilization in Central Asia. American School of Prehistoric Research Bulletin 42. Peabody Museum of Archaeology and Ethnology Harvard University, CambridgeGoogle Scholar
  68. Home CE, Hann MA (1998) The geometrical basis of patterns and tilings: a review of conceptual developments. J Text Inst 89(1):27–46CrossRefGoogle Scholar
  69. Horne CE (2000) Geometric symmetry in patterns and tilings, 1st edn. Woodhead Publishing Ltd/CRC Press LLC, CornwallCrossRefGoogle Scholar
  70. Jablan S (2001) Mirror curves. In: Sarhangi R, Séquin C (eds) Bridges: mathematical connections in art, music, and science. Southwestern College, Winfield, pp 233–246Google Scholar
  71. Jablan S (2012) Mirror-curves and knot mosaics. Comput Math Appl 64(4):527–543MathSciNetCrossRefGoogle Scholar
  72. Jablan SV (1995) Curves generated by mirror reflections. Filomat 9(2):143–148Google Scholar
  73. Jablan SV (2002) Symmetry, ornament and modularity. Series on knots and everything, vol 30. World Scientific, SingaporeGoogle Scholar
  74. Jablan SV, Radović L, Sazdanović R, Zeković A (2012) Knots in art. Symmetry 4(4):302–328MathSciNetCrossRefGoogle Scholar
  75. Kaplan M, Cohen E (2003) Computer generated celtic design. In: Proceedings of the 14th Eurographics workshop on rendering, EGRW’03, pp 9–19Google Scholar
  76. Kauffman LH (1987) On knots. Annals of mathematics studies, 1st edn, vol 115. Princeton University Press, New JerseyGoogle Scholar
  77. Kauffman LH (1993) Knots and physics. Series on knots and everything, 2nd edn, vol 1. World Scientific, SingaporeGoogle Scholar
  78. Kawauchi A (1996) A survey of knot theory. Birkhäser Basel, BaselzbMATHGoogle Scholar
  79. Kendrick TD, Kitzinger E, Allen D (1939) The sutton hoo finds. Br Mus Q 13(4):ii+111–136Google Scholar
  80. Kermode P (1907) Manx crosses. The Pinkfoot Press, Balgavies, Angus. Reprinted 1994Google Scholar
  81. Kline M (1972) Mathematical thought from ancient to modern times. Oxford University Press, New YorkGoogle Scholar
  82. Knoll E, Taylor T, Landry W, Carreiro P, Puxley K, Harrison K (2017) The aesthetics of colour in mathematical diagramming. In: Swart D, Séquin, CH, Fenyvesi K (eds) Proceedings of bridges 2017: mathematics, art, music, architecture, education, culture, pp 563–570Google Scholar
  83. Köhler EC (2011) The rise of the Egyptian state. In: Teeter E (ed) Before the pyramids – the origins of Egyptian civilization. Oriental Institute Museum Publications 33. The Oriental Institute of the University of Chicago, Chicago, pp 123–125Google Scholar
  84. Krötenheerdt O (1964) Über eine speziellen typ alternierender knoten. Mathematische Annalen 153(4):270–284Google Scholar
  85. Krötenheerdt O (1971) Zur lösung des isotopieproblems der rosettenknoten. In: Herrmann M, Kertész A, Krötenheerdt O (eds) Beiträge zur Algebra und Geometrie 1. Springer, Berlin/Heidelberg, pp 19–31CrossRefGoogle Scholar
  86. Kvavadze E, Bar-Yosef O, Belfer-Cohen A, Boaretto E, Jakeli N, Matskevich Z, Meshveliani T (2009) 30,000-year-old wild flax fibers. Science 325(5946):1359CrossRefGoogle Scholar
  87. Lasić VD (1995) Pleterni Ukras – Od Najstarijih Vremena Do Danas Njegov Likovni Oblik I Značenje. Ziral, Chicago Translated: the twist or guilloche as ornament from ancient times to the present: its exterior form and inner meaning.Google Scholar
  88. Lever D (1819) The young sea officer’s sheet anchor, or a key to the leading of rigging, and to practical seamanship, 2nd edn. Dover Publications, Inc., Mineola. Reprinted in 1998Google Scholar
  89. Liu Y, Toussaint GT (2010) Unraveling roman mosaic meander patterns: a simple algorithm for their generation. J Math Arts 4(1):1–11CrossRefGoogle Scholar
  90. Livingston C (1993) Knot theory. Carus mathematical monographs, vol 24, 1st edn. The Mathematical Association of America, WashingtonGoogle Scholar
  91. Lomonaco SJ, Kauffman LH (2008) Quantum knots and mosaics. Quantum Inf Process 7(2–3): 85–115MathSciNetCrossRefGoogle Scholar
  92. Lund K (1968) Måtter og rosetter, 1st edn. Borgen, RingkøbingGoogle Scholar
  93. Lurker M (2005) An illustrated dictionary of the gods and symbols of ancient Egypt. Thames and Hudson, LondonGoogle Scholar
  94. Mackay EJH (1937a) Further excavations at Mohenjo-Daro, vol. I: text. Government of India Press, New DelhiGoogle Scholar
  95. Mackay EJH (1937b) Further excavations at Mohenjo-Daro, vol. II: plate I – CXLVI. Government of India Press, New DelhiGoogle Scholar
  96. Mackay EJH (1943) Chanhu-Daro excavations 1935-36. American oriental series, vol 20. American Oriental Society, New HavenGoogle Scholar
  97. Madden AM (2014) Corpus of Byzantine church mosaic pavements from Israel and the Palestinian territories. Peeters, LeuvenGoogle Scholar
  98. Megaw R, Megaw V (2001) Celtic art – from its beginnings to the book of kells. Thames and Hudson, New YorkGoogle Scholar
  99. Mellaart J (1964) Excavations at Çatal hüyük, 1963, third preliminary report. Anatolian Studies 14(1):39–119CrossRefGoogle Scholar
  100. Mellaart J (1975) The neolithic of the near East. Thames and Hudson Ltd., LondonGoogle Scholar
  101. Moortgat A (1945) Die Entstehung der Sumerischen Hochkultur. Der Alte Orient, Band 43. J. C. Hinrichs Verlag, LeipzigGoogle Scholar
  102. Müller-Karpe H (1968a) Handbuch der Vorgeschichte – Jungsteinzeit, vol 2. Text. C.H. Beck’sche Verlagsbuchhandlung, MünchenGoogle Scholar
  103. Müller-Karpe H (1968b) Handbuch der Vorgeschichte – Jungsteinzeit, vol 2. Tafeln. C.H. Beck’sche Verlagsbuchhandlung, MünchenGoogle Scholar
  104. Murasugi K (1965) Remarks on rosette knots. Math Ann 158(5):290–292Google Scholar
  105. Murasugi K (1971) On periodic knots. Commentarii mathematici Helvetici 46:162–177MathSciNetCrossRefGoogle Scholar
  106. Neugebauer O (1969) The exact sciences in antiquity, 2nd edn. Dover Publications Inc., MineolaGoogle Scholar
  107. Neugebauer O, Sachs A (1945) Mathematical cuneiform texts. Pub. jointly by the American Oriental Society and the American Schools of Oriental Research, New HavenGoogle Scholar
  108. Nilsen KW (1978) Om tyrkerknop slått i handa. In: Molaug S, Kolltveit B, Dahl GB (eds) Norsk Sjøfartsmuseum, 1th edn. Aktietrykkeriet i Stavanger, Oslo, pp 105–154Google Scholar
  109. Öhrvall H (1908) Om knutar, 1st edn. Bonniers, StockholmGoogle Scholar
  110. Olsén P (1945) Die Saxe Von Valsgärde I. Almqvist & Wiksells Boktryckeri AB, UppsalaGoogle Scholar
  111. Ovadiah A (1980) Geometric and floral patterns in ancient Mosaics: a study of their origin in the Mosaics from the classical period to the age of Augustus. L’Erma di Bretscheider, RomeGoogle Scholar
  112. Parzysz B (2009) Using key diagrams to design and construct roman geometric mosaics? Nexus Netw J 11(2):273–288CrossRefGoogle Scholar
  113. Petrie WMF (1920) Egyptian decorative art – a course of lectures delivered by the royal institution, 2nd edn. Methuen & Co, LTD., LondonGoogle Scholar
  114. Piggott S (1950) Prehistoric India – to 1000 B.C. Penguin Books LTD, HarmondsworthGoogle Scholar
  115. Pike AWG, Hoffmann DL, García-Diez M, Pettitt PB, Alcolea J, De Balbín R, González-Sainz C, de las Heras C, Lasheras JA, Montes R, Zilhão J (2012) U-series dating of paleolithic art in 11 caves in Spain. Science 336(6087):1409–1413CrossRefGoogle Scholar
  116. Pittman H (1987) Ancient art in miniature – near eastern seals from the collection of Martin and Sarah Cherkasky. The Metropolitan Museum of Art, New YorkGoogle Scholar
  117. Porada E, Hansen DP, Dunham S, Babcock SH (1992) The chronology of mesopotamia, ca. 7000 - 1600 B.C. In: Ehrich RW (ed) Chronologies in old world archaeology, 3rd edn, vol 1. University of Chicago Press, Chicago, pp 77–121Google Scholar
  118. Przytycki JH (2009) The trieste look at knot theory. In: Kauffman LH, Lambropoulou S, Jablan S, Przytycki J (eds) Introductory lectures on knot theory: selected lectures presented at the advanced school and conference on knot theory and its applications to physics and biology. Series on knots and everything, vol 46. World Scientific Publishing Co, Singapore, pp 407–441CrossRefGoogle Scholar
  119. Reidemeister K (1948) Knotentheorie. Ergebnisse Der Mathematik und Ihrer Grenzgebiete, vol 1. Chelsea Publishing Company, New YorkGoogle Scholar
  120. Rice DT (1986) Islamic art. Thames and Hudson, LondonGoogle Scholar
  121. Richter GMA (1959) A handbook of Greek art. Phaidon Press Limited, London. Reprinted 1994Google Scholar
  122. Röding JH (1798a) Allgemeines Wortenbuch Der Marine, vol 3. Licentiat Nemnich und Adam Freidrich Böhme, HamburgGoogle Scholar
  123. Röding JH (1798b) Allgemeines Wortenbuch Der Marine, vol 4. Licentiat Nemnich und Adam Freidrich Böhme, HamburgGoogle Scholar
  124. Rosen J (2008) Symmetry rules. How science and nature are founded on symmetry, 1st edn. Springer, BerlinGoogle Scholar
  125. Sarianidi V (1986) Die Kunst des alten Afghanistan. VEB E.A. Seemann Verlag, LeipzigGoogle Scholar
  126. Sarianidi VI (1981a) Margiana in the bronze age. In: Kohl PL (ed) The bronze age civilization of Central Asia – recent soviet discoveries. M. E. Sharpe, Inc., New York, pp 165–193Google Scholar
  127. Sarianidi VI (1981b) Seal-amulets of the murghab style. In: Kohl PL (ed) The bronze age civilization of Central Asia – recent Soviet discoveries. M. E. Sharpe, Inc., New York, pp 221–255Google Scholar
  128. Schaake GA, Hall T (1995) Braiding instructions and their presentation. Braider 1(1):2–12Google Scholar
  129. Schaake GA, Hall T, Turner JC (1992) Braiding – standard herringbone knots. A series of books on braiding, book 3/1, 1st edn. Department of Mathematics and Statistics University of Waikato, HamiltonGoogle Scholar
  130. Schaake GA, Turner JC (1988) A new theory of braiding. Research report 1/1, No. 165, 1st edn. Privately Published, HamiltonGoogle Scholar
  131. Schaake GA, Turner JC (1991) An introduction to flat braids. Pamphlet No. 5, 1st edn. Privately Published, HamiltonGoogle Scholar
  132. Schaake GA, Turner JC, Sedgwick DA (1988) Braiding – regular knots. A series of books on braiding, book 1/1, 1st edn. Department of Mathematics and Statistics University of Waikato, HamiltonGoogle Scholar
  133. Schaake GA, Turner JC, Sedgwick DA (1990) Braiding – regular fiador knots. A series of books on braiding, book 2/1, 1st edn. Department of Mathematics and Statistics University of Waikato, HamiltonGoogle Scholar
  134. Schaake GA, Turner JC, Sedgwick DA (1991) Braiding – standard herringbone pineapple knots. A series of books on braiding, book 4/1, 1st edn. Department of Mathematics and Statistics University of Waikato, HamiltonGoogle Scholar
  135. Soffer O (1997) The mutability of upper paleolithic art in central and Eastern Europe: patterning and significance. In: Conkey M, Soffer O, Stratmann D, Jablonski N (eds) Beyond art: pleistocene image and symbol, Wattis symposium series in anthropology, Memoirs of the California academy of sciences, vol 23. University of California Press, San Francisco, pp 239–262Google Scholar
  136. Stormes C, Reeves D (2010) Luis Ortega’s rawhide artistry – braiding in the California tradition. University of Oklahoma Press, OklahomaGoogle Scholar
  137. Teissier B (1984) Ancient near eastern cylinder seals – from the Marcopoli collection. University of California Press, Los AngelesGoogle Scholar
  138. Thurston WP, Levy S (1997) Three-dimensional geometry and topology, vol 1. Princeton mathematical series, 35. Princeton University Press, New JerseyCrossRefGoogle Scholar
  139. Turner JC, Schaake GA (1991) A proof of the law of the common divisor in braids. Knotting Matters 10(35):6–10Google Scholar
  140. Turner JC, Schaake GA, Sedgwick DA (1991) Introducing grid-diagrams in braiding, 1st edn. Privately Published, HamiltonGoogle Scholar
  141. Van de Griend P (1994) Ketupat knot designs. Privately Published, ÅrhusGoogle Scholar
  142. Van De Griend P (1996) A history of topological knot theory. In: History and science of knots. Series on knots and everything, vol 11. World Scientific Publishing Co. Pte. Ltd., Singapore, pp 205–260Google Scholar
  143. Von der Osten HH (1934) Ancient oriental seals in the collection of Mr. Edward T. Newell. The University of Chicago Oriental Institute Publications, vol XXII. The University of Chicago Press, ChicagoGoogle Scholar
  144. Wallis H (1898) Egyptian ceramic art – the Macgregor collection. Taylor and Francis, LondonGoogle Scholar
  145. Ward WH (1899) The hittite gods in hittite art. Am J Archaeol 3(1):1–39CrossRefGoogle Scholar
  146. Ward WH (1909) Cylinders and other ancient oriental seals – in the library of J. Pierpont Morgan. Privately published, New YorkGoogle Scholar
  147. Ward WH (1910) The seal cylinders of western Asia. Carnegie Institution of Washington, Washington, DCGoogle Scholar
  148. Washburn DK, Crowe DW (1991) Symmetries of culture: theory and practice of plane pattern analysis, 1st edn. University of Washington Press, Seattle/LondonzbMATHGoogle Scholar
  149. Westwood JO (1988) The art of illuminated manuscripts – illustrated sacred writings. Arch Cape Press, New YorkGoogle Scholar
  150. Weyl H (1952) Symmetry, 1st edn. Princeton University Press, PrincetonCrossRefGoogle Scholar
  151. Williams D (1999) Greek vases, 2nd edn. British Museum Press, LondonGoogle Scholar

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Authors and Affiliations

  1. 1.GothenburgSweden
  2. 2.UcklumSweden

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