A Research from the Past to Present On the Concept of Symmetry and Periodic Surface Design

Yıl 2023, Cilt: 8 Sayı: 3, 625 - 653, 30.08.2023

Korcan Gülfidan , İpek Fitöz

https://doi.org/10.26835/my.1271644

Öz

Symmetry is a unifying concept for various fields of interest, an interdisciplinary link based on both qualitative and quantitative values. The act of designing symmetry has been involved for thousands of years. The designer was often fed by natural sciences such as astronomy, physics, biology, mathematical discoveries, philosophical texts and theological elements while using symmetry. Today, the concept of symmetry, which is at the intersection of many fields of study, has parallel but unique definitions in various fields of science. By following definitions, technical expressions and chronological findings, the study reaches the current meaning and importance of periodic surface design. In the study, the ideas of scientists and thinkers who put forward qualitative and quantitative information on symmetry were evaluated in chronological order, and following this, floor patterns, which are important in terms of architecture and art history, and works that can set an example on symmetrical surface are included in the study.
It can be said that symmetry is a cost-effective way due to the possibilities it provides in the design setup and production stages. This efficiency factor is also visible in the process of recording symmetric transformations as ready-made method packages in the algorithm-aided design process. Two-dimensional symmetry group theory, which is valid today and has been proved mathematically recently, has been a binding element in the study. This theory is well suited for use with computer-aided design and generative design tools, because of its precise theoretical boundaries, both in the process of generating new patterns, and opportunities to identify existing periodic patterns. By means of numerical design, group algorithms that will produce more than one result can be configured, historical symmetrical tiling patterns can be classified and different method sequences can be developed to produce new results.

Anahtar Kelimeler

Symmetry, Surface Design, Tesselations

Simetri Kavramı ve Periyodik Yüzey Tasarımları Üzerine Geçmişten Günümüze Bir Araştırma

Öz

Simetri çeşitli uğraşı alanları için birleştirici bir kavram, nitel ve nicel değerlere bağlı disiplinlerarası bir bağlantı yoludur. Simetri tasarlama eylemine binlerce yıldır dahildir. Geçmişte tasarımcılar simetri kullanımında astronomi, fizik, biyoloji gibi doğa bilimlerinden, matematiksel keşiflerden, felsefi metinlerden ve teolojik unsurlardan sıklıkla beslenmiştir. Bugün de pek çok çalışma alanının kesişiminde bulunan simetri kavramı, çeşitli bilim alanlarında birbirlerine paralel fakat anlam olarak özgün tanımlamalara sahiptir. Çalışmanın amacı tanımlar, teknik ifadeler ve bulguları kronolojik sırada takip ederek simetrinin periyodik yüzey tasarımı için günümüzdeki teknolojik araçlarla kullanım sınırlarını değerlendirmektir. Çalışmada simetriye ve periyodik yüzeylere dair nitel ve nicel bilgiler sunmuş bilim insanları ve düşünürlerin fikirleri değerlendirilmiş, bu düşüncelere paralel olarak mimarlık ve sanat tarihi açısından önem atfedilen döşeme örüntülerine ve simetrik yüzey konusunda örnek teşkil edebilecek eserlere yer verilmiştir. Simetrinin ve periyodikliğin tasarım kurgusunda ve üretim aşamalarında sağladığı olanaklar nedeniyle tasarruflu bir yol olduğu söylenebilir. Bu verim etkeni, algoritma destekli tasarım sürecinde simetrik dönüşümlerin hazır yöntem paketleri halinde kaydedilmesi sürecinde de görünür olmaktadır. Günümüzde geçerli olan ve matematiksel doğruluğu yakın bir dönemde ispatlanmış iki boyutlu simetri grup ) çalışmada bağlayıcı bir öğe olmuştur. Kesin kuramsal sınırlar belirtmesi, hem yeni örüntülerin üretilmesi sürecinde, hem de mevcut periyodik örüntüleri kimliklendirme fırsatları nedeniyle mekânsal anlamda faydalı görülmelidir. Bu kuram bilgisayar destekli tasarım ve üretken tasarım araçlarıyla kullanıma oldukça uygundur. Sayısal tasarım yoluyla birden fazla sonuç üretecek grup algoritmaları yapılandırılabilir, tarihsel simetrik döşeme örüntüleri sınıflandırılabilir ve yeni sonuç verecek farklı yöntem dizinleri geliştirilebilir.

Anahtar Kelimeler

Simetri, Yüzey Tasarımı, Döşeme Örüntüleri

Kaynakça

• Abbott, E. A. (2015). Düzülke. Alfa Yayıncılık, Çevirmen: Hasan Fehmi Nemli, Orijinal Adı: FlatlaAbbott, E. A. (2015). Düzülke. Alfa Yayıncılık, Çevirmen: Hasan Fehmi Nemli, Orijinal Adı: Flatland: A Romance of Many Dimensions.
• Acocella, A. (2006). Stone Architecture: Ancient and Modern Construction Skills. Skira Editore.
• Agkathidis, A. (2015). Generative Design. Laurance King Publishing.
• Agnello, F. (2010). The Painted Ceiling of the Nave of the Capella Palatina in Palermo: An Essay on Its Geometric and Constructive Features. In G. Necipoğlu & K. A. Leal (Eds.), Muqarnas An Annual on the Visual Cultures of the Islamic World (Publ. 2009, Vol. 27, p. 407;447). Brill, Leiden, The Netherlands.
• Al-Khalili, J. ve McFadden, J. (2016). Kuantum Sınırında Yaşam: Yaklaşan Kuantum Biyolojisi Çağı. Domingo Yayınevi.
• Algorithm. (2021). Google Books Ngram Viewer. https://books.google.com/ngrams/graph?content=algorithm&year_start=1800&year_end=2019&corpus=26&smoothing=3&direct_url=t1%3B%2Calgorithm%3B%2Cc0#t1%3B%2Calgorithm%3B%2Cc0, Erişim Tarihi: 20.12.2020
• Arık, R. (2018). Selçuklu Sarayları ve Köşkleri. Ankara Üniversitesi Yayınları, 2017, Ankara.
• Arık, R. ve Arık, O. (2008). Tiles, Treasures of Anatolian Soil: Tiles of the Seljuk and Beylik Periods. Kale Grubu.
• Artstor. (2021). Artstor Digital Library, JSTOR®, ITHAKA®, Artstor®. https://0310n1zr1-y-https-library-artstor-org.msgsu.proxy.deepknowledge.io/#/home
• Baglivo, J. A. ve Graver, J. E. (1983). Incidence and Symmetry in Design and Architecture; Cambridge Urban & Architectural Studies (C. U. Press (Ed.)).
• Bakışım. (2021). Türk Dil Kurumu Sözlükleri. https://sozluk.gov.tr/ üzerinden Erişim Tarihi: 12.02.2021
• Ball, P. (2011). In retrospect: On the Six-Cornered Snowflake. Nature, 480. https://doi.org/https://doi.org/10.1038/480455a
• Ball, P. (2016). Why Nature Prefers Hexagons: The Geometric Rules Behind Fly Eyes, Honeycombs and Soap Bubbles. Nautius. https://nautil.us/issue/35/boundaries/why-nature-prefers-hexagons
• Barabe, D. ve Jean, R. V. (1998). Symmetry in Plants (Mathematical Biology and Medicine). World Scientific Publishing Company (April 11, 1998).
• Becker, J. A. (2021). Greek Architectural Orders. Khan Academy. https://www.khanacademy.org/humanities/ancient-art-civilizations/greek-art
• Bentley, W. A. (1999). Snowflake Bentley. https://snowflakebentley.com/images
• Blake, S. P. (2016). Astronomy and Astrology in the Islamic World (The New Ed). Edinburgh University Press.
• Bonner, J. ve Kaplan, C. S. (2017). Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction. Springer.
• Bool, F. H., Ernst, B., Kist, J. R., Locher, J. L. ve Wierda, F. (1995). Escher : The Complete Graphic Work (J. L. Locher (Ed.)).
• Broug, E. (2008). Islamic Geometric Patterns. Thames & Hudson.
• Bush, O. (2018). Reframing the Alhambra: Architecture, Poetry, Textiles and Court Ceremonial. Edinburgh University Press. https://doi.org/10.3202/caa.reviews.2019.19
• Codex Atlanticus. (2017). The Visual Agency. https://www.codex-atlanticus.it/
• Conway, J. H., Burgiel, H. ve Goodman-Strauss, C. (2008). The Symmetries of Things. A K Peters, Ltd. Wellesley, Massachusetts.
• Coxeter, H. S. M. (1998). Non-Euclidean Geometry (Mathematical Association of America Textbooks) (Ed.: Sixth). Mathematical Association of America. https://ekutuphane.msgsu.edu.tr/eds/detail/retrieve?An=759515&dbId=nlebk
• Cromwell, P. R. (2008). The Search for Quasi-Periodicity in Islamic 5-fold Ornament. The Mathematical Intelligencer 2008 Springer Science+Business Media, LLC., 31(36), p.36;56.
• Crowe, D. W. (2001). Symmetries of Culture. Bridges, Mathematical Connections in Art, Music, and Science. https://archive.bridgesmathart.org/2001/bridges2001-1.pdf
• Demaine, E. D., Demaine, M. L. ve Koschitz, D. (2011). Reconstructing David Huffman’s Legacy in Curved-Crease Folding. In P. Wang-Iverson, R. J. Lang, & M. Yim (Eds.), Origami 5 (1st Editio, p. 14). A K Peters/CRC Press. https://doi.org/10.1201/b10971-6
• Du Sautoy, M. (2008a). Symmetry, Everywhere: A Mathematician Surveys The New Frontier Of Science. Science & Spirit(Vol. 19, Issue 2), Publisher: Heldref Publications, 19(2).
• Du Sautoy, M. (2008b). Symmetry: A Journey into Patterns of Nature. Harper Collins e-books.
• Du Sautoy, M. (2009). Simetri, Gerçekliğin Bilmecesi. TED, Youtube. https://www.youtube.com/watch?v=415VX3QX4cU, Erişim Tarihi: 05.02.2021
• Emmer, M. (2005). The Visual Mind II (Leonardo Book Series) (M. Emmer (Ed.); 2005th ed.). MIT Press.
• Ernst, B. (2007). The Magic Mirror of M.C. Escher. TASCHEN.
• Erten Bilgiç, D., Bal, B. (2019). İslam Mimarisinde Geometri ve Elhamra Sarayı Örneği, UBAK ,4. Uluslararası Sosyal Ve Eğitim Bilimleri Araştırmaları Kongresi, Yalova, Türkiye, 14 - 17 Şubat 2019, Cilt.1
• Escher, M. C. (2001). M. C. Escher : The Graphic Work Introduced and Explained by the Artist ; English Translation John E. Brigham (E. translation J. E. Brigham (Ed.)). Köln : Taschen, 2001.
• Fraser, G. J., Shono, T., Thiery, A., Cooper, R. L., Britz, R. ve Okabe, M. (2019). Evolution and Developmental Diversity of Skin Spines in Pufferfishes. IScience, 19, 1248–1259. https://doi.org/https://doi.org/10.1016/j.isci.2019.06.003
• Frazer, J. (1995). An Evolutionary Architecture. John Frazer and the Architectural Association, London.
• Glaeser, G. (2013). Nature and Numbers: A Mathematical Photo Shooting (Edition An). Ambra Verlag. Vienna, Austria, 2013.
• Goehring, L. ve Mahadevan, L. (2008). Cracking the Giant’s Causeway: Solving a 300 Year Old Geology Problem Using Kitchen Materials. https://www8.physics.utoronto.ca/~nonlin/PNASpress/PNASpress.html
• Gregotti, V. (1996). Inside Architecture. MIT Press.
• Groß, B., Bohnacker, H., Laub, J. ve Lazzeroni, C. (2018). Generative Design, Visualize, Program and Create with Javascript in p5.js. Princeton Architectural Press, New York.
• Grünbaum, B. (2006). What Symmetry Groups Are Present in the Alhambra? Notices of the American Mathematical Society, 53(6), 670–673.
• Grünbaum, B., Grünbaum, Z. ve Shephard, G. C. (1986). Symmetry in Moorish and Other Ornaments. Comp. & Maths. with Appls., 12(3–4), 641–653. https://doi.org/https://doi.org/10.1016/0898-1221(86)90416-5
• Grünbaum, B. ve Shephard, G. C. (1987). Tiling And Patterns. W. H. Freeman and Company, New York.
• Gülfidan, K. (2015). Bildirişim Aracı Olarak İç Mekân Tasarımı: Gösterge Kavramı Kapsamında ''2015 Evrensel Sergisi'' Örneği, Yüksek Lisans Tezi, Mimar Sinan Güzel Sanatlar Üniversitesi
• Hahn, A. J. (2012). Mathematical Excursions to the World’s Great Buildings. Princeton University Press, Princeton, New Jersey.
• Hankin, E. H. (1925). Memoirs of The Archaelogical Survey of India No.15: The Drawing of Geometric Patterns in Saracenic Art (Original E). Archaelogical Survey of India, Janpath, New Delhi. https://archive.org/details/HankinSaracenicArt
• Harari, Y. N. (2016). Homo Deus: Yarının Kısa Bir Tarihi. Kolektif. Kitap, Bilişim ve Tasarım ltd. şti.
• Hargittai, I. ve Hargittai, M. (1994). Symmetry: A Unifying Concept. Shelter Publications, Inc.
• Hargittai, M. ve Hargittai, I. (2009). Visual Symmetry. World Scientific Publishing Co. Pte. Ltd.
• Harris, J. (2012). Fractal Architecture: Organic Design Philosophy in Theory and Practice (1st Editio). University of New Mexico Press.
• Hon, G. ve Goldstein, B. R. (2008). From Summetria To Symmetry: The Making Of A Revolutionary Scientific Concept (C. I. Jed Z. Buchwald, Dreyfuss Professor of History & U. of Technology, Pasadena, CA (Eds.); Archimedes). 2008 Springer Science+Business Media B.V.
• Horne, C. E. (2000). Geometric Symmetry in Pattern and Tilings. Woodhead Publishing Ltd and CRC Press LLC.
• Jablan, S. V. (2002). Symmetry, Ornament and Modularity. River Edge, NJ : World Scientific, c2002.
• Janson, A. ve Tigges, F. (2014). Fundamental Concepts of Architecture, The Vocabulary of Spatial Situations. Birkhäuser. https://doi.org/https://doi.org/10.1515/9783034608923
• Johnston, H. (2007). Islamic ‘Quasicrystals’ Predate Penrose Tiles. Physics World, IOP Publishing. https://physicsworld.com
• Kaplan, C. S. (2005). Islamic Star Patterns from Polygons in Contact. Conference: Proceedings of the Graphics Interface 2005 Conference, May 9-11, 2005, Victoria, British Columbia, Canada. 10.1145/1089508.1089538
• Kaplan, C. S. (2009). Introductory Tiling Theory for Computer Graphics. In Synthesis Lectures on Computer Graphics and Animation (p. 102). A Publication in the Morgan & Claypool Publishers series. https://doi.org/10.2200/S00207ED1V01Y200907CGR011
• Kaplan, C. S. ve Salesin, D. H. (2004). Islamic Star Patterns in Absolute Geometry. ACM Transactions on Graphics, 23(2), 97–119.
• Kepler, J. (1619). Harmonicus Mundi, Book II.
• Kolarevic, B. (2003). Architecture in the Digital Age Design and Manufacturing (B. Kolarevic (Ed.). Spon Press Taylor & Francis Group.
• Lange, C. ve Mecit, S. (2012). The Seljuqs: Politics, Society and Culture. Edinburgh University Press.
• Le Corbusier. (2014). Modulor I-II. Yapı Endüstri Merkezi Yayınları, Mimarlık Dizisi, Çevirmen: Aziz Ufuk Kılıç.
• Lederman, L. M. ve Hill, C. T. (2004). Symmetry And The Beautiful Universe. Prometheus Books.
• Legendre, G. L. (2011). Mathematics of Space. Architectural Design.
• Liu, Y. ve Collins, R. (1999). Frieze and Wallpaper Symmetry Groups Classification under Affine and Perspective Distortion. Group Theory-based Regularity Perception in Human and Computer Vision. https://www.researchgate.net/publication/2596375_Frieze_and_Wallpaper_Symmetry_Groups_Classification_under_Affine_and_Perspective_Distortion
• Liu, Y., Collins, R. ve Tsin, Y. (2002). Gait Sequence Analysis Using Frieze Patterns. ECCV 2002, 657–671.
• Liu, Y., Hel-Or, H., Kaplan, C. S. ve Van Gool, L. (2010). Computational Symmetry in Computer Vision and Computer Graphics. Foundations and Trends® in Computer Graphics and Vision, 5(Nos. 1–2 (2009)), 1–195. https://doi.org/10.1561/0600000008
• Liu, Y., Tsin, Y. ve Lin, W. C. (2005). The Promise and Perils of Near-Regular Texture. International Journal of Computer Vision 2005 Springer Science + Business Media Inc., 62(1/2), 145–159.
• Lu, P. J. ve Steinhardt, P. J. (2007). Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture. Science, 315, 1106. https://doi.org/10.1126/science.1135491
• Mainzer, K. (1996). Symmetries of Nature : A Handbook for Philosophy of Nature and Science. De Gruyter. https://doi.org/10.1515/9783110886931
• Mainzer, K. (2005). Symmetry and Complexity The Spirit and Beauty of Nonlinear Science. World Scientific Series on Nonlinear Science Series A: Volume 51, 51, 448. https://doi.org/https://doi.org/10.1142/5770
• Makovicky, E. (2016). Symmetry Through the Eyes of the Old Masters. De Gruyter.
• Makovicky, E. (2018). Vault Mosaics of the Kukeldash Madrasah, Bukhara, Uzbekistan. Nexus Network Journal Architecture and Mathematics, 20, 309–320.
• Makovicky, E. (2019). Byzantine Floor Patterns of Interlocking Twisted Cable Loops. Nexus Network Journal Architecture and Mathematics, 22(2), 449–469. https://doi.org/10.1007/s00004-019-00473-0
• Mann, T. (1998). Büyülü Dağ. Can Yayınları, Özgün Adı: Der Zauberberg, Çevirmen: İris Kantemir. Massey, J. (2012). Buckminster Fuller’s Reflexive Modernism. The Journal of the Design Studies Forum, Design and(4:3), 325–344. https://doi.org/10.2752/175470812X13361292229159
• Mazzoleni, I. ve Price, S. (2013). Architecture Follows Nature, Biomimetic Principles for Innovative Design (Biomimetic). CRC Press Taylor & Francis Group.
• McClary, R. P. (2017). Rum Seljuq Architecture, 1170-1220: The Patronage of Sultans. Edinburgh University Press.
• McNeel, R. (2014). McNeel Grasshopper 3D (6.0). Robert McNeel and Associates.
• Necipoğlu, G. (1995). The Topkapı Scroll - Geometry and Ornament in Islamic Architecture; Topkapı Palace Museum Library MS H. 1956 (J. Bloomfield, T. F. Reese, & S. Settis (Eds.)).
• Necipoğlu, G., Hogendijk, J. P., Kheirandish, E., Özdural, A., ve Thackston, W. M. (2017). The Arts of Ornamental Geometry : A Persian Compendium on Similar and Complementary Interlocking Figures = Fī Tadākhul al-Ashkāl al-Mutashābiha aw al-Mutawāfiqa (Bibliothèque Nationale de France, Ms. Persan 169, fols. 180r-199r) : A Volume Commemorating A (G. Necipoğlu (Ed.)). Leiden : Brill, 2017.
• Örüntü. (2021). Türk Dil Kurumu Sözlükleri. https://sozluk.gov.tr/ üzerinden “‘örüntü’”, “ ‘örmek’”; Erişim Tarihi: 04.10.2020
• Özgan, S. Y. ve Özkar, M. (2017). A Thirteenth-Century Dodecahedron in Central Anatolia: Geometric Patterns and Polyhedral Geometry. Nexus Network Journal Architecture and Mathematics, 19, 455–471. https://doi.org/10.1007/s00004-017-0341-0
• Paoletti, I., ve Gülfidan, K. (2020). Symmetry + Tesselation. Material Balance Portfolio. https://www.materialbalance.polimi.it/portfolio_page/symmetry/ Erişim Tarihi: 22.12.2021
• Pawlyn, M. (2016). Biomimicry in Architecture (F. Gibbons & K. Mackillop (Eds.); Second Edi). Riba Publishing.
• Penrose, R. (2017). Preface. In J. Bonner & C. S. Kaplan (Eds.), Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction (p. 595). Springer.
• Picon, A. (2013). Ornament, The Politics of Architecture and Subjectivity (AD Primers (Ed.)). John Wiley & Sons Ltd.
• Pólya, G. (1924). Über die Analogie der Kristallsymmetrie in der Ebene. Akademische Verlagsgesellschaft. https://books.google.com.tr/books/about/Über_die_Analogie_der_Kristallsymmetrie.html?id=hYMgHQAACAAJ&redir_esc=y
• Rael, R., San Fratello, V., Curth, S., Arja, L. ve Piracci, T. (2021). Emerging Objects. Emerging Objects. http://emergingobjects.com/
• Roth, L. M. (2006). Mimarlığın Öyküsü, Öğeleri, Tarihi ve Anlamı. Kabalcı Yayınevi, Çeviren: Ergün Akça, Leland M. Roth, Understanding Architecture It’s Elements, History and Meaning, 1993.
• Schattschneider, D. (1978). The Plane Symmetry Groups: Their Recognition and Notation. The American Mathematical Monthly, 85(6).
• Schattschneider, D. (1987). The Pólya-Escher Connection. Mathematics Magazine, 60(5), 293–298. https://doi.org/10.1080/0025570X.1987.11977327
• Schattschneider, D., & Emmer, M. (2003). M.C. Escher’s Legacy, A Centennial Celebration: Collection of articles coming from the M.C. Escher Centennial Conference, Rome 1998 (1st editio). Springer.
• Sieden, L. S. (2000). Buckminster Fuller’s Universe: His Life and Work. Omni International Publications Ltd.
• Smith, L. (2014). Kaos. Kültür Kitaplığı: 139, Bilim: 7, Çeviri: Hakan Gür, This Translation of “‘Chaos’” originally published in English in 2007 is published by arrangement with Oxford University Pres.
• Souza, E. (2020). Rethinking Artificial Reef Structures through 3D Clay Printing. ArcDaily. https://www.archdaily.com/947495/rethinking-artificial-reef-structures-through-3d-clay-printing
• Spaulding, S. ve Edlund, M. (2008). Diatoms of North America. https://diatoms.org/genera
• Stamper, J. W. (2005). The Architecture of Roman Temples : The Republic to the Middle Empire. Cambridge University Press.
• Stewart, I. ve Golubitsky, M. (1993). Fearful Symmetry, Is God A Geometer. Blackwell Publishers, by arrangement with Penguin Books.
• Symmetry. (2021a). Online Etymology Dictionary. https://www.etymonline.com/ içinde “‘symmetry’”; “‘sym-’”; “ ‘meter’”; Erişim Tarihi: 04.09.2019
• Symmetry. (2021b). Merriam-Webster Dictionary.
• Tectum. (2021). Online Etymology Dictionary. https://www.etymonline.com/ içinde “‘architect’”; “ ‘tectum’” Erişim Tarihi: 13.02.2020
• Tedeschi, A. (2014). AAD_Algorithms-Aided Design: Parametric Strategies Using Grasshopper. Le Penseur Publisher.
• Teges. (2021). Wiktionary The Free Dictionary. https://en.wiktionary.org/wiki/teges Date Accessed: 13.02.2021
• Tekeli, İ. (2010). Mekânsal ve Toplumsal Olanın Bilgibilimi Yazıları (İlhan Teke). Tarih Vakfı Yurt Yayınları.
• Tekeli, İ. (2017). Postmodernizm Tartışmaları Üzerine Düşünceler. POSSEIBLE Felsefe Dergisi / Journal of Philosophy, 10.
• Teks. (2021). Online Etymology Dictionary. https://www.etymonline.com/ içinde “textile”; “texture”; “tectum” Erişim Tarihi: 13.02.2021
• Thompson, D. W. (1942). On Growth And Form. Cambridge University Press.
• Tile. (2021). Online Etymology Dictionary. https://www.etymonline.com/ içinde “tegula”; “tile”; “teges” Erişim Tarihi: 03.12.2021
• Van Der Helm, P. (2015). Symmetry Perception. In J. Wagemans (Ed.), Oxford Handbook of Perceptual Organization (pp. 108–128). Oxford University Press, Oxford, UK.
• Vincent, J. F. V., Bogatyreva, O. A., Bogatyrev, N. R., Bowyer, A. ve Pahl, A.-K. (2006). Biomimetics: Its Practice and Theory. Interface : Journal of The Royal Society, 3, 471–482.
• Walker, A. (2010). Middle Byzantine Aesthetics of Power and the Incomparability of Islamic Art: The Architectural Ekphraseis of Nikolaos Mesarites. In G. Necipoğlu & K. A. Leal (Eds.), Muqarnas An Annual on the Visual Cultures of the Islamic World (Publ. 2009, Vol. 27, p. s.79;101). Brill, Leiden, The Netherlands.
• Washburn, D. K. (2011). Pattern Symmetries of the Chaco Phenomenon. American Antiquity, 76(2), 252–284.
• Washburn, D. K. ve Crowe, D. W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press.
• Washburn, D. K., Crowe, D. W. ve Ahlstrom, R. V. N. (2010). A Symmetry Analysis of Design Structure: 1,000 Years of Continuity and Change in Puebloan Ceramic Design. American Antiquity, 75(4), 743–772. https://doi.org/10.7183/0002-7316.75.4.743
• Weyl, B. Y. H. (1989). Symmetry. Princeton University Press, Princeton, New Jersey. nd: A Romance of Many Dimensions.