A study of the concept and background of golden ratios in written sources

Document Type : Research Paper

Author

Research Institute of History of Science, University of Tehran, Tehran

Abstract

One of the most popular theories in aesthetics is the golden ratio theory in either theoretical or practical geometry. The golden ratio, the golden proportion which is called the extreme and mean ratio by Euclid, goes by other several various names over history. The golden ratio is well known in the artistic composition and architecture of buildings, statues, and sculptures, as still used today. There are a lot of myths about the golden ratio; Some attribute it to sacred numbers among the Pythagoreans, and some figure out an excellent placement in geometry and architecture for it, mentioning that particular ancient temples and structures were built on the basis of these numbers. The present study aims to examine the written background of the golden ratio through the written heritage of the Islamic period including accessible manuscripts and treatises as primary sources, besides the articles and reports on ancient Greek works, mainly in Greek, Latin, or other languages as secondary sources. In this article, we first explain the background of this issue in the case of ancient Greece, and then we review some textbooks of the Islamic period on the subject of arithmetic or geometry that addressed this theory directly or indirectly. Since this article relies on the Islamic period in medieval and early modern times, only the treatises and some of the efforts spared including the existing written works and the visual arts from the Islamic period to the present century are introduced. Moreover, the special shapes famed as sacred or golden due to their specific geometric properties are described

Keywords

Main Subjects


[1] G. Markowsky, Misconceptions about the Golden Ratio, The College Mathematics Journal (Mathemat-ical Association of America), 23 (1992) 2–19.
[2] T. Dantzig, The Bequest of The Greeks, Charles Scribner’s Sons, New York, 1955 191 pp.
[3] M. Livio, The Golden Ratio, The Story of PHI, 2002.
[4] G. Sarton, When did the term ”golden section” or its equivalent in other languages originate? [Queries and Answers], Isis, 42 (1951) p. 47.
[5] B. Fett, An In-depth Investigation of the Divine Ratio, The Montana Mathematics Enthusiast (TMME), 3 (2006) 157-175.
[6] R. Herz-Fischler, A Very Pleasant Theorem, The College Mathematics Journal (Mathematical Associa-tion of America), 24 (1993) 318–324.
[7] L. M. Dabbour, Geometric proportions:The underlying structure of design process for Islamic geometric patterns, Frontiers of Architectural Research, Elsevier, 1 (2012) 380–391.
[8] A. Bennett, Follow the golden ratio from Africa to the Bauhaus for a cross-cultural aesthetic for images, 2012. http://www.researchgate.net/publication/269709899.
[9] W. K. Chorbachi, In the Tower of Babel: Beyond Symmetry in Islamic Design, Computers Math., Harvard University 17 (1989) 751–789.
[10] L. Freitas, Notes on Some Pentagonal “Mysteries” in Egyptian and Christian Iconography, Fivefold Symmetry, by Istavan Hargittai, Scientific World, Singapore (1992) 307–332.
[11] J. Pottage, The Mathematical Gazette: Review, The Mathematical Association, (1989) 265–267.
[12] Plato, Timaeus, Desmond Lee, Penguin Books, 1977.
[13] J. Kappraff, The Relationship Between Mathematics and Mysticism of the Golden Mean Through History, Fivefold Symmetry, by Istvan Hargittai, World Scientific Publishing Co.Pte.Ltd Singapore, (1992) 33–66.
[14] H. F. Verheyen, The Icosahedral Design of the Great Pyramid, Fivefold Symmetry, by Istvan Hargittai, World Scientific Publishing Co.Pte.Ltd, Singapore, (1992) 333–359.
[15] D. E. Smith, History Of Mathematics, 2, Book contributor Osmania University, Dover Publications Inc., 1951.
[16] B. L. Waerden, Geometry and Algebra in Ancient Civilization, Berlin Heidelberg, Springer-Verlag, 1983.
[17] J. L. Heiberg, Euclid’s Elements of Geometry, Translated by Richard Fitzpatrick, 1883–1885.
[18] A. Seidenberg, The Origin of Mathematics, Archive for History of Exact Sciences, Springer, 18 (1974) 301–342.
[19] R. Herz-Fischler, Theorem xiv of the First “Supplement” to The Elements, Archives internationales d’histoire des sciences 38 (1988) 3–66.
[20] W. K. Chorbachi and A. L. Loeb, An Islamic Pentagonal Seal (From Scientific Manuscripts of the Geometry of Design), Fivefold Symmetry, by Istavan Hargittai, Scientific World, Singapore, (1992) 283–305.
[21] E. Kheirandish, An Early Tradition in Practical Geometry: The Telling Lines of Unique Arabic and Persian Sources, The Arts of Ornamental Geometry; A Persian Compendium on Similar and Comple-mentary Interlocking Figures, Ed. by Gulru Necipuglu, Brill, Leiden, Boston, 2017 79–144.
[22] N. al-Din Tusi, al-Magala al-Rabi’s, Sitat Magalat min Kitab Tahrir ‘Uqlidis, Al-Matb’a al-Hindya, Calcutta (1924) 116–117.
[23] Paris Bibl. Nat. ancient fond person MS169, starting folio141.
[24] ج. ل. برگرن، بررسی گزیده ای از پژوهش‌­های منتشر شده در تاریخ ریاضیات دوره اسلامی، میراث علمی اسلام و ایران، ترجمۀ حمید بهلول، سال سوم 1 (1393).
[25] ی. کرامتی، اصول اقلیدس، دایرة المعارف بزرگ اسلامی، مرکز دایرة المعارف بزرگ اسلامی، تهران.
[26] ابوریحان بیرونی، التفهیم لأوائل صناعة التنجیم، شرح و تصحیح جلال‌الدین همایی، چاپخانۀ بهمن، تهران، 1351.
[27] ابن سینا، الشفاء، به کوشش ابراهیم مدکور، قم، 1405.
[28] ع. انوار، سیر هندسه اقلیدسی از اقلیدس تا شیخ الرئیس ابوعلی سینا، تاریخ علم، 2 (1383) 119- 135.
 
[29] م. مختاری، دانش‌نامۀ علایی، پایان نامه، دانشکدۀ علوم ریاضی شریف، تهران، 1374.
[30] نصیرالدین طوسی، تحریر اصول اقلیدس، دست‌نویس ش. 4078 کتابخانۀ مجلس شورای اسلامی.
[31] نصیرالدین طوسی، تحریر اصول اقلیدس، دست‌نویس کتابخانۀ ملی قطر. https://www.wdl.org/en/item/10666
[32] قطب‌الدین شیرازی، درة التاج لغرة الدباج، به کوشش حسن مشکان طبسی، تهران، 1324.
[33] قطب‌الدین شیرازی، درة التاج لغرة الدباج، دست‌نویس 698 کتابخانۀ مجلس شورای اسلامی.
[34] م. کاوه­‌یزدی، رساله­‌ای از ثابت بن قرّه در هندسه، میراث علمی اسلام و ایران، سال اول 1 (1391).
[35] غ. جونپوری، جامع بهادرخانی، چاپ سنگی کلکته، 1835.
[36] عبدالغفار نجم‌الدوله، اصول هندسه، تهران، 1318.
[37] ع. جذبی، هندسۀ ایرانی-کاربرد هندسه در عمل، سروش، تهران، 1376.
[38] ا. قربانی و م. شیخان، بوزجانی‌نامه، انتشارات و آموزش انقلاب اسلامی، تهران، 1371.
[39] ابواسحاق کوبنانی، ترجمۀ فی ما یحتاج الیه الصانع من اعمال الهندسه، میکروفیلم ش. 775، کتابخانۀ مرکزی دانشگاه تهران.
[40] م. اسلام‌پناه، گره‌سازی به روایت استاد غلامرضا، فرهنگ ایران زمین، 29 (1375).
[41] غیاث‌الدین جمشید کاشانی، مفتاح الحساب، به کوشش نادر نابلسی، دمشق، 1977.
[42] غیاث‌الدین جمشید کاشانی، دست‌نویس ش. 6148، کتابخانۀ مجلس شورای اسلامی.
Volume 7, Issue 3 - Serial Number 3
September 2022
Pages 25-55
  • Receive Date: 30 April 2022
  • Revise Date: 25 December 2022
  • Accept Date: 22 January 2023
  • Publish Date: 22 November 2022